n–1
The table above shows only the t -tests for population means. Another common t -test is for correlation coefficients . You use this t -test to decide if the correlation coefficient is significantly different from zero.
When you define the hypothesis, you also define whether you have a one-tailed or a two-tailed test. You should make this decision before collecting your data or doing any calculations. You make this decision for all three of the t -tests for means.
To explain, let’s use the one-sample t -test. Suppose we have a random sample of protein bars, and the label for the bars advertises 20 grams of protein per bar. The null hypothesis is that the unknown population mean is 20. Suppose we simply want to know if the data shows we have a different population mean. In this situation, our hypotheses are:
$ \mathrm H_o: \mu = 20 $
$ \mathrm H_a: \mu \neq 20 $
Here, we have a two-tailed test. We will use the data to see if the sample average differs sufficiently from 20 – either higher or lower – to conclude that the unknown population mean is different from 20.
Suppose instead that we want to know whether the advertising on the label is correct. Does the data support the idea that the unknown population mean is at least 20? Or not? In this situation, our hypotheses are:
$ \mathrm H_o: \mu >= 20 $
$ \mathrm H_a: \mu < 20 $
Here, we have a one-tailed test. We will use the data to see if the sample average is sufficiently less than 20 to reject the hypothesis that the unknown population mean is 20 or higher.
See the "tails for hypotheses tests" section on the t -distribution page for images that illustrate the concepts for one-tailed and two-tailed tests.
For all of the t -tests involving means, you perform the same steps in analysis:
.pdf version of this page
In this review, we’ll look at significance testing, using mostly the t -test as a guide. As you read educational research, you’ll encounter t -test and ANOVA statistics frequently. Part I reviews the basics of significance testing as related to the null hypothesis and p values. Part II shows you how to conduct a t -test, using an online calculator. Part III deal s with interpreting t -test results. Part IV is about reporting t -test results in both text and table formats and concludes with a guide to interpreting confidence intervals.
What is Statistical Significance?
The terms “significance level” or “level of significance” refer to the likelihood that the random sample you choose (for example, test scores) is not representative of the population. The lower the significance level, the more confident you can be in replicating your results. Significance levels most commonly used in educational research are the .05 and .01 levels. If it helps, think of .05 as another way of saying 95/100 times that you sample from the population, you will get this result. Similarly, .01 suggests that 99/100 times that you sample from the population, you will get the same result. These numbers and signs (more on that later) come from Significance Testing, which begins with the Null Hypothesis.
Part I: The Null Hypothesis
We start by revisiting familiar territory, the scientific method . We’ll start with a basic research question: How does variable A affect variable B? The traditional way to test this question involves:
Step 1. Develop a research question.
Step 2. Find previous research to support, refute, or suggest ways of testing the question.
Step 3. Construct a hypothesis by revising your research question:
H1: A = B | There is no relationship between A and B | Null |
H2: A ≠ B | There is a relationship between A and B. Here, there is a relationship, but we don’t know if it is positive or negative. | Alternate |
H3: A < B | There is a negative relationship between A and B. Here, the < suggests that the less A is involved, the better B. | Alternate |
H4: A > B | There is a positive relationship between A and B. Here, the > suggests that the more B is involved, the better A. | Alternate |
Step 4. Test the null hypothesis. To test the null hypothesis, A = B, we use a significance test. The italicized lowercase p you often see, followed by > or < sign and a decimal ( p ≤ .05) indicate significance. In most cases, the researcher tests the null hypothesis, A = B , because is it easier to show there is some sort of effect of A on B, than to have to determine a positive or negative effect prior to conducting the research. This way, you leave yourself room without having the burden of proof on your study from the beginning.
Step 5. Analyze data and draw a conclusion. Testing the null hypothesis leaves two possibilities:
A = B | Fail to reject the null. We find no relationship between A and B. | Null |
A =, <, or > B | Reject the null. We find a relationship between A and B. | Alternate |
Step 6. Communicate results. See Wording results, below.
Part II: Conducting a t -test (for Independent Means)
So how do we test a null hypothesis? One way is with a t -test. A t -test asks the question,
“Is the difference between the means of two samples different (significant) enough to say that some other characteristic (teaching method, teacher, gender, etc.) could have caused it?”
To conduct a t-test using an online calculator, complete the following steps:
Step 1. Compose the Research Question.
Step 2. Compose a Null and an Alternative Hypothesis.
Step 3. Obtain two random samples of at least 30, preferably 50, from each group.
Step 4. Conduct a t -test:
Step 5. Interpret the results (see below).
Step 6. Report results in text or table format (see below).
Part III. Interpreting a t-test (Understanding the Numbers)
tells you a test was used. | |
(98) | tells you the (the sample of tests performed). |
3.09 | is the “ statistic” – the result of the calculation. |
≤ .05 | is the probability of getting the observed score from the sample groups. |
.05 | likely to be a result of chance (same as saying A = B) |
difference is not significant | |
null is correct | |
“fail to reject the null” | |
There is no relationship between A and B. | |
≤ .05 | not likely to be a result of chance (same as saying A ≠ B) |
difference is significant | |
null is incorrect | |
“reject the null” | |
There is a relationship between A and B. |
Note: We acknowledge that the average scores are different. With a t -test we are deciding if that difference is significant (is it due to sampling error or something else?).
Understanding the Confidence Interval (CI)
The Confidence Interval (CI) of a mean is a region within which a score (like mean test score) may be said to fall with a certain amount of “confidence.” The CI uses sample size and standard deviation to generate a lower and upper number that you can be 95% sure will include any sample you take from a set of data.
Consider Georgia’s AYP measure, the CRCT . For a science CRCT score, we take several samples and compare the different means. After a few calculations, we could determine something like. . .the average difference (mean) between samples is -7.5, with a 95% CI of -22.08 to 6.72. In other words, among all students’ science CRCT scores, 95 out of 100 times we take group samples for comparison (for example by year, or gender, etc.), one of the groups, on average will be 7.5 points lower than the other group. We can be fairly certain that the difference in scores will be between -22.08 and 6.72 points.
Part IV. Wording Results
Wording Results in Text
In text, the basic format is to report: population ( N ), mean ( M ) and standard deviation ( SD ) for both samples, t value, degrees freedom ( df ), significance ( p ), and confidence interval (CI .95 )* .
Example 1: p ≤ .05, or Significant Results
Among 7th graders in Lowndes County Schools taking the CRCT reading exam ( N = 336), there was a statistically significant difference between the two teaching teams, team 1 ( M = 818.92, SD = 16.11) and team 2 ( M = 828.28, SD = 14.09), t (98) = 3.09, p ≤ .05, CI .95 -15.37, -3.35. Therefore, we reject the null hypothesis that there is no difference in reading scores between teaching teams 1 and 2.
Example 2: p ≥ .05, or Not Significant Results
Among 7th graders in Lowndes County Schools taking the CRCT science exam ( N = 336), there was no statistically significant difference between female students (M = 834.00, SD = 32.81) and male students (841.08, SD = 28.76), t (98) = 1.15 p ≥ .05, CI .95 -19.32, 5.16. Therefore, we fail to reject the null hypothesis that there is no difference in science scores between females and males.
Wording Results in APA Table Format
Table 1. Comparison of CRCT 7 th Grade Science Scores by Gender
|
|
|
|
|
|
| |
Female |
50
|
834.00 |
32.81 |
– |
– |
– |
– |
Male |
50
|
841.08 |
28.76 |
– |
– |
– |
– |
Total |
100
|
837.54 |
30.90 |
1.14 |
98 |
.2540 |
-19.32 – 5.16 |
Note: On the Web site, this appears blocked and should not be. See the .pdf for the correct format.
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Research Rundowns was made possible by support from the Dewar College of Education at Valdosta State University .
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Statistics By Jim
Making statistics intuitive
By Jim Frost 12 Comments
T-tests are statistical hypothesis tests that you use to analyze one or two sample means. Depending on the t-test that you use, you can compare a sample mean to a hypothesized value, the means of two independent samples, or the difference between paired samples. In this post, I show you how t-tests use t-values and t-distributions to calculate probabilities and test hypotheses.
As usual, I’ll provide clear explanations of t-values and t-distributions using concepts and graphs rather than formulas! If you need a primer on the basics, read my hypothesis testing overview .
The term “t-test” refers to the fact that these hypothesis tests use t-values to evaluate your sample data. T-values are a type of test statistic. Hypothesis tests use the test statistic that is calculated from your sample to compare your sample to the null hypothesis. If the test statistic is extreme enough, this indicates that your data are so incompatible with the null hypothesis that you can reject the null. Learn more about Test Statistics .
Don’t worry. I find these technical definitions of statistical terms are easier to explain with graphs, and we’ll get to that!
When you analyze your data with any t-test, the procedure reduces your entire sample to a single value, the t-value. These calculations factor in your sample size and the variation in your data. Then, the t-test compares your sample means(s) to the null hypothesis condition in the following manner:
Read the companion post where I explain how t-tests calculate t-values .
The tricky thing about t-values is that they are a unitless statistic, which makes them difficult to interpret on their own. Imagine that we performed a t-test, and it produced a t-value of 2. What does this t-value mean exactly? We know that the sample mean doesn’t equal the null hypothesis value because this t-value doesn’t equal zero. However, we don’t know how exceptional our value is if the null hypothesis is correct.
To be able to interpret individual t-values, we have to place them in a larger context. T-distributions provide this broader context so we can determine the unusualness of an individual t-value.
A single t-test produces a single t-value. Now, imagine the following process. First, let’s assume that the null hypothesis is true for the population. Now, suppose we repeat our study many times by drawing many random samples of the same size from this population. Next, we perform t-tests on all of the samples and plot the distribution of the t-values. This distribution is known as a sampling distribution, which is a type of probability distribution.
Related posts : Sampling Distributions and Understanding Probability Distributions
If we follow this procedure, we produce a graph that displays the distribution of t-values that we obtain from a population where the null hypothesis is true. We use sampling distributions to calculate probabilities for how unusual our sample statistic is if the null hypothesis is true.
Luckily, we don’t need to go through the hassle of collecting numerous random samples to create this graph! Statisticians understand the properties of t-distributions so we can estimate the sampling distribution using the t-distribution and our sample size.
The degrees of freedom (DF) for the statistical design define the t-distribution for a particular study. The DF are closely related to the sample size. For t-tests, there is a different t-distribution for each sample size.
Related posts : Degrees of Freedom in Statistics and T Distribution: Definition and Uses .
T-distributions assume that the null hypothesis is correct for the population from which you draw your random samples. To evaluate how compatible your sample data are with the null hypothesis, place your study’s t-value in the t-distribution and determine how unusual it is.
The sampling distribution below displays a t-distribution with 20 degrees of freedom, which equates to a sample size of 21 for a 1-sample t-test. The t-distribution centers on zero because it assumes that the null hypothesis is true. When the null is true, your study is most likely to obtain a t-value near zero and less liable to produce t-values further from zero in either direction.
On the graph, I’ve displayed the t-value of 2 from our hypothetical study to see how our sample data compares to the null hypothesis. Under the assumption that the null is true, the t-distribution indicates that our t-value is not the most likely value. However, there still appears to be a realistic chance of observing t-values from -2 to +2.
We know that our t-value of 2 is rare when the null hypothesis is true. How rare is it exactly? Our final goal is to evaluate whether our sample t-value is so rare that it justifies rejecting the null hypothesis for the entire population based on our sample data. To proceed, we need to quantify the probability of observing our t-value.
Related post : What are Critical Values?
Hypothesis tests work by taking the observed test statistic from a sample and using the sampling distribution to calculate the probability of obtaining that test statistic if the null hypothesis is correct. In the context of how t-tests work, you assess the likelihood of a t-value using the t-distribution. If a t-value is sufficiently improbable when the null hypothesis is true, you can reject the null hypothesis.
I have two crucial points to explain before we calculate the probability linked to our t-value of 2.
Because I’m showing the results of a two-tailed test, we’ll use the t-values of +2 and -2. Two-tailed tests allow you to assess whether the sample mean is greater than or less than the target value in a 1-sample t-test. A one-tailed hypothesis test can only determine statistical significance for one or the other.
Additionally, it is possible to calculate a probability only for a range of t-values. On a probability distribution plot, probabilities are represented by the shaded area under a distribution curve. Without a range of values, there is no area under the curve and, hence, no probability.
Related posts : One-Tailed and Two-Tailed Tests Explained and T-Distribution Table of Critical Values
Considering these points, the graph below finds the probability associated with t-values less than -2 and greater than +2 using the area under the curve. This graph is specific to our t-test design (1-sample t-test with N = 21).
The probability distribution plot indicates that each of the two shaded regions has a probability of 0.02963—for a total of 0.05926. This graph shows that t-values fall within these areas almost 6% of the time when the null hypothesis is true.
There is a chance that you’ve heard of this type of probability before—it’s the P value! While the likelihood of t-values falling within these regions seems small, it’s not quite unlikely enough to justify rejecting the null under the standard significance level of 0.05.
Learn how to interpret the P value correctly and avoid a common mistake!
Related posts : How to Find the P value: Process and Calculations and Types of Errors in Hypothesis Testing
The sample size for a t-test determines the degrees of freedom (DF) for that test, which specifies the t-distribution. The overall effect is that as the sample size decreases, the tails of the t-distribution become thicker. Thicker tails indicate that t-values are more likely to be far from zero even when the null hypothesis is correct. The changing shapes are how t-distributions factor in the greater uncertainty when you have a smaller sample.
You can see this effect in the probability distribution plot below that displays t-distributions for 5 and 30 DF.
Sample means from smaller samples tend to be less precise. In other words, with a smaller sample, it’s less surprising to have an extreme t-value, which affects the probabilities and p-values. A t-value of 2 has a P value of 10.2% and 5.4% for 5 and 30 DF, respectively. Use larger samples!
Click here for step-by-step instructions for how to do t-tests in Excel !
If you like this approach and want to learn about other hypothesis tests, read my posts about:
To see an alternative to traditional hypothesis testing that does not use probability distributions and test statistics, learn about bootstrapping in statistics !
May 25, 2021 at 10:42 pm
what statistical tools, is recommended for measuring the level of satisfaction
May 26, 2021 at 3:55 pm
Hi McKienze,
The correct analysis depends on the nature of the data you have and what you want to learn. You don’t provide enough information to be able to answer the question. However, read my hypothesis testing overview to learn about the options.
August 23, 2020 at 1:33 am
Hi Jim, I want to ask about standardizing data before the t test.. For example I have USD prices of a big Mac across the world and this varies by quite a bit. Doing the t-test here would be misleading since some countries would have a higher mean… Should the approach be standardizing all the usd values? Or perhaps even local values?
August 24, 2020 at 12:37 am
Yes, that makes complete sense. I don’t know what method is best. If you can find a common scale to use for all prices, I’d do that. You’re basically using a data transformation before analysis, which is totally acceptable when you have a good reason.
April 3, 2020 at 4:47 am
Hey Jim. Your blog is one of the only few ones where everything is explained in a simple and well structured manner, in a way that both an absolute beginner and a geek can equally benefit from your writing. Both this article as well as your article on one tailed and two tailed hypothesis tests have been super helpful. Thank you for this post
March 6, 2020 at 11:04 am
Thank you, Jim, for sharing your knowledge with us.
I have a 2 part question. I am testing the difference in walking distance within a busy environment compared with a simple environment. I am also testing walking time within the 2 environments. I am using the same individuals for both scenarios. I was planning to do a paired ttest for distance difference between busy and simple environments and a 2nd paired ttest for time difference between the environments.
My question(s) for you is: 1. Do you feel that a paired ttest is the best choice for these? 2. Do you feel that, because there are 2 tests, I should do a bonferroni correction or do you believe that because the data is completely different (distance as opposed to time), it is okay not to do a multiple comparison test?
August 13, 2019 at 12:43 pm
thank you very eye opening on the use of two or one tailed test
April 19, 2019 at 3:49 pm
Hi Mr. Frost,
Thanks for the breakdown. I have a question … if I wanted to run a test to show that the medical professionals could use more training with data set consisting of questions which in your opinion would be my best route?
January 14, 2019 at 2:22 pm
Hello Jim, I find this statement in this excellent write up contradicting : 1)This graph shows that t-values fall within these areas almost 6% of the time when the null hypothesis is true I mean if this is true the t-value =0 hypothesis is rejected.
January 14, 2019 at 2:51 pm
I can see how that statement sounds contradictory, but I can assure that it is quite accurate. It’s often forgotten but the underlying assumption for the calculations surrounding hypothesis testing, significance levels, and p-values is that the null hypothesis is true.
So, the probabilities shown in the graph that you refer to are based on the assumption that the null hypothesis is true. Further, t-values for this study design have a 6% chance of falling in those critical areas assuming the null is true (a false positive).
Significance levels are defined as the maximum acceptable probability of a false positive. Usually, we set that as 5%. In the example, there’s a large probability of a false positive (6%), so we fail to reject the null hypothesis. In other words, we fail to reject the null because false positives will happen too frequently–where the significance level defines the cutoff point for too frequently.
Keep in mind that when you have statistically significant results, you’re really saying that the results you obtained are improbable enough assuming that the null is true that you can reject the notion that the null is true. But, the math and probabilities are all based on the assumption that the null is true because you need to determine how unlikely your results are under the null hypothesis.
Even the p-value is defined in terms of assuming the null hypothesis is true. You can read about that in my post about interpreting p-values correctly .
I hope this clarifies things!
November 9, 2018 at 2:36 am
Jim …I was involved in in a free SAT/ACT tutoring program that I need to analyze for effectiveness .
I have pre test scores of a number of students and the post test scores after they were tutored (treatment ).
Glenn dowell
November 9, 2018 at 9:05 am
It sounds like you need to perform a paired t-test assuming.
Topics: Hypothesis Testing , Data Analysis
T-tests are handy hypothesis tests in statistics when you want to compare means. You can compare a sample mean to a hypothesized or target value using a one-sample t-test. You can compare the means of two groups with a two-sample t-test. If you have two groups with paired observations (e.g., before and after measurements), use the paired t-test.
How do t-tests work? How do t-values fit in? In this series of posts, I’ll answer these questions by focusing on concepts and graphs rather than equations and numbers. After all, a key reason to use statistical software like Minitab is so you don’t get bogged down in the calculations and can instead focus on understanding your results.
In this post, I will explain t-values, t-distributions, and how t-tests use them to calculate probabilities and assess hypotheses.
T-tests are called t-tests because the test results are all based on t-values. T-values are an example of what statisticians call test statistics. A test statistic is a standardized value that is calculated from sample data during a hypothesis test. The procedure that calculates the test statistic compares your data to what is expected under the null hypothesis .
Each type of t-test uses a specific procedure to boil all of your sample data down to one value, the t-value. The calculations behind t-values compare your sample mean(s) to the null hypothesis and incorporates both the sample size and the variability in the data. A t-value of 0 indicates that the sample results exactly equal the null hypothesis. As the difference between the sample data and the null hypothesis increases, the absolute value of the t-value increases.
Assume that we perform a t-test and it calculates a t-value of 2 for our sample data. What does that even mean? I might as well have told you that our data equal 2 fizbins! We don’t know if that’s common or rare when the null hypothesis is true.
By itself, a t-value of 2 doesn’t really tell us anything. T-values are not in the units of the original data, or anything else we’d be familiar with. We need a larger context in which we can place individual t-values before we can interpret them. This is where t-distributions come in.
When you perform a t-test for a single study, you obtain a single t-value. However, if we drew multiple random samples of the same size from the same population and performed the same t-test, we would obtain many t-values and we could plot a distribution of all of them. This type of distribution is known as a sampling distribution .
Fortunately, the properties of t-distributions are well understood in statistics, so we can plot them without having to collect many samples! A specific t-distribution is defined by its degrees of freedom (DF) , a value closely related to sample size. Therefore, different t-distributions exist for every sample size. You can graph t-distributions u sing Minitab’s probability distribution plots .
T-distributions assume that you draw repeated random samples from a population where the null hypothesis is true. You place the t-value from your study in the t-distribution to determine how consistent your results are with the null hypothesis.
The graph above shows a t-distribution that has 20 degrees of freedom, which corresponds to a sample size of 21 in a one-sample t-test. It is a symmetric, bell-shaped distribution that is similar to the normal distribution, but with thicker tails. This graph plots the probability density function (PDF), which describes the likelihood of each t-value.
The peak of the graph is right at zero, which indicates that obtaining a sample value close to the null hypothesis is the most likely. That makes sense because t-distributions assume that the null hypothesis is true. T-values become less likely as you get further away from zero in either direction. In other words, when the null hypothesis is true, you are less likely to obtain a sample that is very different from the null hypothesis.
Our t-value of 2 indicates a positive difference between our sample data and the null hypothesis. The graph shows that there is a reasonable probability of obtaining a t-value from -2 to +2 when the null hypothesis is true. Our t-value of 2 is an unusual value, but we don’t know exactly how unusual. Our ultimate goal is to determine whether our t-value is unusual enough to warrant rejecting the null hypothesis. To do that, we'll need to calculate the probability.
The foundation behind any hypothesis test is being able to take the test statistic from a specific sample and place it within the context of a known probability distribution. For t-tests, if you take a t-value and place it in the context of the correct t-distribution, you can calculate the probabilities associated with that t-value.
A probability allows us to determine how common or rare our t-value is under the assumption that the null hypothesis is true. If the probability is low enough, we can conclude that the effect observed in our sample is inconsistent with the null hypothesis. The evidence in the sample data is strong enough to reject the null hypothesis for the entire population.
Before we calculate the probability associated with our t-value of 2, there are two important details to address.
First, we’ll actually use the t-values of +2 and -2 because we’ll perform a two-tailed test. A two-tailed test is one that can test for differences in both directions. For example, a two-tailed 2-sample t-test can determine whether the difference between group 1 and group 2 is statistically significant in either the positive or negative direction. A one-tailed test can only assess one of those directions.
Second, we can only calculate a non-zero probability for a range of t-values. As you’ll see in the graph below, a range of t-values corresponds to a proportion of the total area under the distribution curve, which is the probability. The probability for any specific point value is zero because it does not produce an area under the curve.
With these points in mind, we’ll shade the area of the curve that has t-values greater than 2 and t-values less than -2.
The graph displays the probability for observing a difference from the null hypothesis that is at least as extreme as the difference present in our sample data while assuming that the null hypothesis is actually true. Each of the shaded regions has a probability of 0.02963, which sums to a total probability of 0.05926. When the null hypothesis is true, the t-value falls within these regions nearly 6% of the time.
This probability has a name that you might have heard of—it’s called the p-value! While the probability of our t-value falling within these regions is fairly low, it’s not low enough to reject the null hypothesis using the common significance level of 0.05.
Learn how to correctly interpret the p-value.
As mentioned above, t-distributions are defined by the DF, which are closely associated with sample size. As the DF increases, the probability density in the tails decreases and the distribution becomes more tightly clustered around the central value. The graph below depicts t-distributions with 5 and 30 degrees of freedom.
The t-distribution with fewer degrees of freedom has thicker tails. This occurs because the t-distribution is designed to reflect the added uncertainty associated with analyzing small samples. In other words, if you have a small sample, the probability that the sample statistic will be further away from the null hypothesis is greater even when the null hypothesis is true.
Small samples are more likely to be unusual. This affects the probability associated with any given t-value. For 5 and 30 degrees of freedom, a t-value of 2 in a two-tailed test has p-values of 10.2% and 5.4%, respectively. Large samples are better!
I’ve showed how t-values and t-distributions work together to produce probabilities. To see how each type of t-test works and actually calculates the t-values, read the other post in this series, Understanding t-Tests: 1-sample, 2-sample, and Paired t-Tests .
If you'd like to learn how the ANOVA F-test works, read my post, Understanding Analysis of Variance (ANOVA) and the F-test .
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The t-test assesses whether the means of two groups are statistically different from each other. This analysis is appropriate whenever you want to compare the means of two groups, and especially appropriate as the analysis for the posttest-only two-group randomized experimental design .
Figure 1 shows the distributions for the treated (blue) and control (green) groups in a study. Actually, the figure shows the idealized distribution – the actual distribution would usually be depicted with a histogram or bar graph . The figure indicates where the control and treatment group means are located. The question the t-test addresses is whether the means are statistically different.
What does it mean to say that the averages for two groups are statistically different? Consider the three situations shown in Figure 2. The first thing to notice about the three situations is that the difference between the means is the same in all three . But, you should also notice that the three situations don’t look the same – they tell very different stories. The top example shows a case with moderate variability of scores within each group. The second situation shows the high variability case. the third shows the case with low variability. Clearly, we would conclude that the two groups appear most different or distinct in the bottom or low-variability case. Why? Because there is relatively little overlap between the two bell-shaped curves. In the high variability case, the group difference appears least striking because the two bell-shaped distributions overlap so much.
This leads us to a very important conclusion: when we are looking at the differences between scores for two groups, we have to judge the difference between their means relative to the spread or variability of their scores. The t-test does just this.
The formula for the t-test is a ratio. The top part of the ratio is just the difference between the two means or averages. The bottom part is a measure of the variability or dispersion of the scores. This formula is essentially another example of the signal-to-noise metaphor in research: the difference between the means is the signal that, in this case, we think our program or treatment introduced into the data; the bottom part of the formula is a measure of variability that is essentially noise that may make it harder to see the group difference. Figure 3 shows the formula for the t-test and how the numerator and denominator are related to the distributions.
The top part of the formula is easy to compute – just find the difference between the means. The bottom part is called the standard error of the difference . To compute it, we take the variance for each group and divide it by the number of people in that group. We add these two values and then take their square root. The specific formula for the standard error of the difference between the means is:
Remember, that the variance is simply the square of the standard deviation .
The final formula for the t-test is:
The t -value will be positive if the first mean is larger than the second and negative if it is smaller. Once you compute the t -value you have to look it up in a table of significance to test whether the ratio is large enough to say that the difference between the groups is not likely to have been a chance finding. To test the significance, you need to set a risk level (called the alpha level ). In most social research, the “rule of thumb” is to set the alpha level at .05 . This means that five times out of a hundred you would find a statistically significant difference between the means even if there was none (i.e. by “chance”). You also need to determine the degrees of freedom (df) for the test. In the t-test , the degrees of freedom is the sum of the persons in both groups minus 2 . Given the alpha level, the df, and the t -value, you can look the t -value up in a standard table of significance (available as an appendix in the back of most statistics texts) to determine whether the t -value is large enough to be significant. If it is, you can conclude that the difference between the means for the two groups is different (even given the variability). Fortunately, statistical computer programs routinely print the significance test results and save you the trouble of looking them up in a table.
The t-test, one-way Analysis of Variance (ANOVA) and a form of regression analysis are mathematically equivalent (see the statistical analysis of the posttest-only randomized experimental design ) and would yield identical results.
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Neag School of Education
An introduction to statistics usually covers t tests, anovas, and chi-square. for this course we will concentrate on t tests, although background information will be provided on anovas and chi-square., a powerpoint presentation on t tests has been created for your use..
The t test is one type of inferential statistics. It is used to determine whether there is a significant difference between the means of two groups. With all inferential statistics, we assume the dependent variable fits a normal distribution . When we assume a normal distribution exists, we can identify the probability of a particular outcome. We specify the level of probability (alpha level, level of significance, p ) we are willing to accept before we collect data ( p < .05 is a common value that is used). After we collect data we calculate a test statistic with a formula. We compare our test statistic with a critical value found on a table to see if our results fall within the acceptable level of probability. Modern computer programs calculate the test statistic for us and also provide the exact probability of obtaining that test statistic with the number of subjects we have.
When the difference between two population averages is being investigated, a t test is used. In other words, a t test is used when we wish to compare two means (the scores must be measured on an interval or ratio measurement scale ). We would use a t test if we wished to compare the reading achievement of boys and girls. With a t test, we have one independent variable and one dependent variable. The independent variable (gender in this case) can only have two levels (male and female). The dependent variable would be reading achievement. If the independent had more than two levels, then we would use a one-way analysis of variance (ANOVA).
The test statistic that a t test produces is a t -value. Conceptually, t -values are an extension of z -scores. In a way, the t -value represents how many standard units the means of the two groups are apart.
With a t tes t, the researcher wants to state with some degree of confidence that the obtained difference between the means of the sample groups is too great to be a chance event and that some difference also exists in the population from which the sample was drawn. In other words, the difference that we might find between the boys’ and girls’ reading achievement in our sample might have occurred by chance, or it might exist in the population. If our t test produces a t -value that results in a probability of .01, we say that the likelihood of getting the difference we found by chance would be 1 in a 100 times. We could say that it is unlikely that our results occurred by chance and the difference we found in the sample probably exists in the populations from which it was drawn.
Assumptions underlying the t test.
This is concerned with the difference between the average scores of a single sample of individuals who are assessed at two different times (such as before treatment and after treatment). It can also compare average scores of samples of individuals who are paired in some way (such as siblings, mothers, daughters, persons who are matched in terms of a particular characteristics).
Note: The F-Max test can be substituted for the Levene test. The t test Excel spreadsheet that I created for our class uses the F -Max.
A bit of history… William Sealy Gosset (1905) first published a t-test. He worked at the Guiness Brewery in Dublin and published under the name Student. The test was called Studen t Test (later shortened to t test).
t tests can be easily computed with the Excel or SPSS computer application. I have created an Excel Spreadsheet that does a very nice job of calculating t values and other pertinent information.
Chris nickson.
Quantitative data is data which can be expressed numerically to indicate a quantity, amount, or measurement
Quantitative data collection involves measurement of variables
A variable is a characteristic of a unit being observed that may assume more than one of a set of values to which a numerical measure or a category from a classification can be assigned (e.g. age, weight, etc.). In other words, they are “data points” that vary numerically between measurements.
Variables are dependent (outcome or response variable) or independent (predictor variable)
Quantitative variables can be continuous or discrete and have varying levels of measurement
Continuous variables
Levels of measurement (lowest to highest)
Categorical variables are qualitative data, not quantitative, even though they may be labelled as numbers
DATA COLLECTION
Data collection involves either:
and experimental research can have either:
Variation in data from quantitative research can be systematic or unsystematic
Many statistical tests work by identifying systematic and unsystematic variation in data and making a comparison
CORRELATION AND CAUSATION
Correlation is the presence of an association between dependent variables and an independent variable.
Causation is when the independent variable is the cause and the dependent variable is the effect
note that Hume’s requirements are simplistic
Mill proposed an additional criterion for causation
DATA ANALYSIS
Quantative data analysis involves both:
DESCRIPTIVE STATISTICS
Describe the distribution of data
INFERENTIAL STATISTICS
STATISTICAL TESTS
Parametric Tests
Non-parametric tests
REFERENCES AND LINKS
FOAM and web resources
Chris is an Intensivist and ECMO specialist at the Alfred ICU in Melbourne. He is also a Clinical Adjunct Associate Professor at Monash University . He is a co-founder of the Australia and New Zealand Clinician Educator Network (ANZCEN) and is the Lead for the ANZCEN Clinician Educator Incubator programme. He is on the Board of Directors for the Intensive Care Foundation and is a First Part Examiner for the College of Intensive Care Medicine . He is an internationally recognised Clinician Educator with a passion for helping clinicians learn and for improving the clinical performance of individuals and collectives.
After finishing his medical degree at the University of Auckland, he continued post-graduate training in New Zealand as well as Australia’s Northern Territory, Perth and Melbourne. He has completed fellowship training in both intensive care medicine and emergency medicine, as well as post-graduate training in biochemistry, clinical toxicology, clinical epidemiology, and health professional education.
He is actively involved in in using translational simulation to improve patient care and the design of processes and systems at Alfred Health. He coordinates the Alfred ICU’s education and simulation programmes and runs the unit’s education website, INTENSIVE . He created the ‘Critically Ill Airway’ course and teaches on numerous courses around the world. He is one of the founders of the FOAM movement (Free Open-Access Medical education) and is co-creator of litfl.com , the RAGE podcast , the Resuscitology course, and the SMACC conference.
His one great achievement is being the father of three amazing children.
On Twitter, he is @precordialthump .
| INTENSIVE | RAGE | Resuscitology | SMACC
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Edward barroga.
1 Department of General Education, Graduate School of Nursing Science, St. Luke’s International University, Tokyo, Japan.
2 Department of Biological Sciences, Messiah University, Mechanicsburg, PA, USA.
The development of research questions and the subsequent hypotheses are prerequisites to defining the main research purpose and specific objectives of a study. Consequently, these objectives determine the study design and research outcome. The development of research questions is a process based on knowledge of current trends, cutting-edge studies, and technological advances in the research field. Excellent research questions are focused and require a comprehensive literature search and in-depth understanding of the problem being investigated. Initially, research questions may be written as descriptive questions which could be developed into inferential questions. These questions must be specific and concise to provide a clear foundation for developing hypotheses. Hypotheses are more formal predictions about the research outcomes. These specify the possible results that may or may not be expected regarding the relationship between groups. Thus, research questions and hypotheses clarify the main purpose and specific objectives of the study, which in turn dictate the design of the study, its direction, and outcome. Studies developed from good research questions and hypotheses will have trustworthy outcomes with wide-ranging social and health implications.
Scientific research is usually initiated by posing evidenced-based research questions which are then explicitly restated as hypotheses. 1 , 2 The hypotheses provide directions to guide the study, solutions, explanations, and expected results. 3 , 4 Both research questions and hypotheses are essentially formulated based on conventional theories and real-world processes, which allow the inception of novel studies and the ethical testing of ideas. 5 , 6
It is crucial to have knowledge of both quantitative and qualitative research 2 as both types of research involve writing research questions and hypotheses. 7 However, these crucial elements of research are sometimes overlooked; if not overlooked, then framed without the forethought and meticulous attention it needs. Planning and careful consideration are needed when developing quantitative or qualitative research, particularly when conceptualizing research questions and hypotheses. 4
There is a continuing need to support researchers in the creation of innovative research questions and hypotheses, as well as for journal articles that carefully review these elements. 1 When research questions and hypotheses are not carefully thought of, unethical studies and poor outcomes usually ensue. Carefully formulated research questions and hypotheses define well-founded objectives, which in turn determine the appropriate design, course, and outcome of the study. This article then aims to discuss in detail the various aspects of crafting research questions and hypotheses, with the goal of guiding researchers as they develop their own. Examples from the authors and peer-reviewed scientific articles in the healthcare field are provided to illustrate key points.
A research question is what a study aims to answer after data analysis and interpretation. The answer is written in length in the discussion section of the paper. Thus, the research question gives a preview of the different parts and variables of the study meant to address the problem posed in the research question. 1 An excellent research question clarifies the research writing while facilitating understanding of the research topic, objective, scope, and limitations of the study. 5
On the other hand, a research hypothesis is an educated statement of an expected outcome. This statement is based on background research and current knowledge. 8 , 9 The research hypothesis makes a specific prediction about a new phenomenon 10 or a formal statement on the expected relationship between an independent variable and a dependent variable. 3 , 11 It provides a tentative answer to the research question to be tested or explored. 4
Hypotheses employ reasoning to predict a theory-based outcome. 10 These can also be developed from theories by focusing on components of theories that have not yet been observed. 10 The validity of hypotheses is often based on the testability of the prediction made in a reproducible experiment. 8
Conversely, hypotheses can also be rephrased as research questions. Several hypotheses based on existing theories and knowledge may be needed to answer a research question. Developing ethical research questions and hypotheses creates a research design that has logical relationships among variables. These relationships serve as a solid foundation for the conduct of the study. 4 , 11 Haphazardly constructed research questions can result in poorly formulated hypotheses and improper study designs, leading to unreliable results. Thus, the formulations of relevant research questions and verifiable hypotheses are crucial when beginning research. 12
Excellent research questions are specific and focused. These integrate collective data and observations to confirm or refute the subsequent hypotheses. Well-constructed hypotheses are based on previous reports and verify the research context. These are realistic, in-depth, sufficiently complex, and reproducible. More importantly, these hypotheses can be addressed and tested. 13
There are several characteristics of well-developed hypotheses. Good hypotheses are 1) empirically testable 7 , 10 , 11 , 13 ; 2) backed by preliminary evidence 9 ; 3) testable by ethical research 7 , 9 ; 4) based on original ideas 9 ; 5) have evidenced-based logical reasoning 10 ; and 6) can be predicted. 11 Good hypotheses can infer ethical and positive implications, indicating the presence of a relationship or effect relevant to the research theme. 7 , 11 These are initially developed from a general theory and branch into specific hypotheses by deductive reasoning. In the absence of a theory to base the hypotheses, inductive reasoning based on specific observations or findings form more general hypotheses. 10
Research questions and hypotheses are developed according to the type of research, which can be broadly classified into quantitative and qualitative research. We provide a summary of the types of research questions and hypotheses under quantitative and qualitative research categories in Table 1 .
Quantitative research questions | Quantitative research hypotheses |
---|---|
Descriptive research questions | Simple hypothesis |
Comparative research questions | Complex hypothesis |
Relationship research questions | Directional hypothesis |
Non-directional hypothesis | |
Associative hypothesis | |
Causal hypothesis | |
Null hypothesis | |
Alternative hypothesis | |
Working hypothesis | |
Statistical hypothesis | |
Logical hypothesis | |
Hypothesis-testing | |
Qualitative research questions | Qualitative research hypotheses |
Contextual research questions | Hypothesis-generating |
Descriptive research questions | |
Evaluation research questions | |
Explanatory research questions | |
Exploratory research questions | |
Generative research questions | |
Ideological research questions | |
Ethnographic research questions | |
Phenomenological research questions | |
Grounded theory questions | |
Qualitative case study questions |
In quantitative research, research questions inquire about the relationships among variables being investigated and are usually framed at the start of the study. These are precise and typically linked to the subject population, dependent and independent variables, and research design. 1 Research questions may also attempt to describe the behavior of a population in relation to one or more variables, or describe the characteristics of variables to be measured ( descriptive research questions ). 1 , 5 , 14 These questions may also aim to discover differences between groups within the context of an outcome variable ( comparative research questions ), 1 , 5 , 14 or elucidate trends and interactions among variables ( relationship research questions ). 1 , 5 We provide examples of descriptive, comparative, and relationship research questions in quantitative research in Table 2 .
Quantitative research questions | |
---|---|
Descriptive research question | |
- Measures responses of subjects to variables | |
- Presents variables to measure, analyze, or assess | |
What is the proportion of resident doctors in the hospital who have mastered ultrasonography (response of subjects to a variable) as a diagnostic technique in their clinical training? | |
Comparative research question | |
- Clarifies difference between one group with outcome variable and another group without outcome variable | |
Is there a difference in the reduction of lung metastasis in osteosarcoma patients who received the vitamin D adjunctive therapy (group with outcome variable) compared with osteosarcoma patients who did not receive the vitamin D adjunctive therapy (group without outcome variable)? | |
- Compares the effects of variables | |
How does the vitamin D analogue 22-Oxacalcitriol (variable 1) mimic the antiproliferative activity of 1,25-Dihydroxyvitamin D (variable 2) in osteosarcoma cells? | |
Relationship research question | |
- Defines trends, association, relationships, or interactions between dependent variable and independent variable | |
Is there a relationship between the number of medical student suicide (dependent variable) and the level of medical student stress (independent variable) in Japan during the first wave of the COVID-19 pandemic? |
In quantitative research, hypotheses predict the expected relationships among variables. 15 Relationships among variables that can be predicted include 1) between a single dependent variable and a single independent variable ( simple hypothesis ) or 2) between two or more independent and dependent variables ( complex hypothesis ). 4 , 11 Hypotheses may also specify the expected direction to be followed and imply an intellectual commitment to a particular outcome ( directional hypothesis ) 4 . On the other hand, hypotheses may not predict the exact direction and are used in the absence of a theory, or when findings contradict previous studies ( non-directional hypothesis ). 4 In addition, hypotheses can 1) define interdependency between variables ( associative hypothesis ), 4 2) propose an effect on the dependent variable from manipulation of the independent variable ( causal hypothesis ), 4 3) state a negative relationship between two variables ( null hypothesis ), 4 , 11 , 15 4) replace the working hypothesis if rejected ( alternative hypothesis ), 15 explain the relationship of phenomena to possibly generate a theory ( working hypothesis ), 11 5) involve quantifiable variables that can be tested statistically ( statistical hypothesis ), 11 6) or express a relationship whose interlinks can be verified logically ( logical hypothesis ). 11 We provide examples of simple, complex, directional, non-directional, associative, causal, null, alternative, working, statistical, and logical hypotheses in quantitative research, as well as the definition of quantitative hypothesis-testing research in Table 3 .
Quantitative research hypotheses | |
---|---|
Simple hypothesis | |
- Predicts relationship between single dependent variable and single independent variable | |
If the dose of the new medication (single independent variable) is high, blood pressure (single dependent variable) is lowered. | |
Complex hypothesis | |
- Foretells relationship between two or more independent and dependent variables | |
The higher the use of anticancer drugs, radiation therapy, and adjunctive agents (3 independent variables), the higher would be the survival rate (1 dependent variable). | |
Directional hypothesis | |
- Identifies study direction based on theory towards particular outcome to clarify relationship between variables | |
Privately funded research projects will have a larger international scope (study direction) than publicly funded research projects. | |
Non-directional hypothesis | |
- Nature of relationship between two variables or exact study direction is not identified | |
- Does not involve a theory | |
Women and men are different in terms of helpfulness. (Exact study direction is not identified) | |
Associative hypothesis | |
- Describes variable interdependency | |
- Change in one variable causes change in another variable | |
A larger number of people vaccinated against COVID-19 in the region (change in independent variable) will reduce the region’s incidence of COVID-19 infection (change in dependent variable). | |
Causal hypothesis | |
- An effect on dependent variable is predicted from manipulation of independent variable | |
A change into a high-fiber diet (independent variable) will reduce the blood sugar level (dependent variable) of the patient. | |
Null hypothesis | |
- A negative statement indicating no relationship or difference between 2 variables | |
There is no significant difference in the severity of pulmonary metastases between the new drug (variable 1) and the current drug (variable 2). | |
Alternative hypothesis | |
- Following a null hypothesis, an alternative hypothesis predicts a relationship between 2 study variables | |
The new drug (variable 1) is better on average in reducing the level of pain from pulmonary metastasis than the current drug (variable 2). | |
Working hypothesis | |
- A hypothesis that is initially accepted for further research to produce a feasible theory | |
Dairy cows fed with concentrates of different formulations will produce different amounts of milk. | |
Statistical hypothesis | |
- Assumption about the value of population parameter or relationship among several population characteristics | |
- Validity tested by a statistical experiment or analysis | |
The mean recovery rate from COVID-19 infection (value of population parameter) is not significantly different between population 1 and population 2. | |
There is a positive correlation between the level of stress at the workplace and the number of suicides (population characteristics) among working people in Japan. | |
Logical hypothesis | |
- Offers or proposes an explanation with limited or no extensive evidence | |
If healthcare workers provide more educational programs about contraception methods, the number of adolescent pregnancies will be less. | |
Hypothesis-testing (Quantitative hypothesis-testing research) | |
- Quantitative research uses deductive reasoning. | |
- This involves the formation of a hypothesis, collection of data in the investigation of the problem, analysis and use of the data from the investigation, and drawing of conclusions to validate or nullify the hypotheses. |
Unlike research questions in quantitative research, research questions in qualitative research are usually continuously reviewed and reformulated. The central question and associated subquestions are stated more than the hypotheses. 15 The central question broadly explores a complex set of factors surrounding the central phenomenon, aiming to present the varied perspectives of participants. 15
There are varied goals for which qualitative research questions are developed. These questions can function in several ways, such as to 1) identify and describe existing conditions ( contextual research question s); 2) describe a phenomenon ( descriptive research questions ); 3) assess the effectiveness of existing methods, protocols, theories, or procedures ( evaluation research questions ); 4) examine a phenomenon or analyze the reasons or relationships between subjects or phenomena ( explanatory research questions ); or 5) focus on unknown aspects of a particular topic ( exploratory research questions ). 5 In addition, some qualitative research questions provide new ideas for the development of theories and actions ( generative research questions ) or advance specific ideologies of a position ( ideological research questions ). 1 Other qualitative research questions may build on a body of existing literature and become working guidelines ( ethnographic research questions ). Research questions may also be broadly stated without specific reference to the existing literature or a typology of questions ( phenomenological research questions ), may be directed towards generating a theory of some process ( grounded theory questions ), or may address a description of the case and the emerging themes ( qualitative case study questions ). 15 We provide examples of contextual, descriptive, evaluation, explanatory, exploratory, generative, ideological, ethnographic, phenomenological, grounded theory, and qualitative case study research questions in qualitative research in Table 4 , and the definition of qualitative hypothesis-generating research in Table 5 .
Qualitative research questions | |
---|---|
Contextual research question | |
- Ask the nature of what already exists | |
- Individuals or groups function to further clarify and understand the natural context of real-world problems | |
What are the experiences of nurses working night shifts in healthcare during the COVID-19 pandemic? (natural context of real-world problems) | |
Descriptive research question | |
- Aims to describe a phenomenon | |
What are the different forms of disrespect and abuse (phenomenon) experienced by Tanzanian women when giving birth in healthcare facilities? | |
Evaluation research question | |
- Examines the effectiveness of existing practice or accepted frameworks | |
How effective are decision aids (effectiveness of existing practice) in helping decide whether to give birth at home or in a healthcare facility? | |
Explanatory research question | |
- Clarifies a previously studied phenomenon and explains why it occurs | |
Why is there an increase in teenage pregnancy (phenomenon) in Tanzania? | |
Exploratory research question | |
- Explores areas that have not been fully investigated to have a deeper understanding of the research problem | |
What factors affect the mental health of medical students (areas that have not yet been fully investigated) during the COVID-19 pandemic? | |
Generative research question | |
- Develops an in-depth understanding of people’s behavior by asking ‘how would’ or ‘what if’ to identify problems and find solutions | |
How would the extensive research experience of the behavior of new staff impact the success of the novel drug initiative? | |
Ideological research question | |
- Aims to advance specific ideas or ideologies of a position | |
Are Japanese nurses who volunteer in remote African hospitals able to promote humanized care of patients (specific ideas or ideologies) in the areas of safe patient environment, respect of patient privacy, and provision of accurate information related to health and care? | |
Ethnographic research question | |
- Clarifies peoples’ nature, activities, their interactions, and the outcomes of their actions in specific settings | |
What are the demographic characteristics, rehabilitative treatments, community interactions, and disease outcomes (nature, activities, their interactions, and the outcomes) of people in China who are suffering from pneumoconiosis? | |
Phenomenological research question | |
- Knows more about the phenomena that have impacted an individual | |
What are the lived experiences of parents who have been living with and caring for children with a diagnosis of autism? (phenomena that have impacted an individual) | |
Grounded theory question | |
- Focuses on social processes asking about what happens and how people interact, or uncovering social relationships and behaviors of groups | |
What are the problems that pregnant adolescents face in terms of social and cultural norms (social processes), and how can these be addressed? | |
Qualitative case study question | |
- Assesses a phenomenon using different sources of data to answer “why” and “how” questions | |
- Considers how the phenomenon is influenced by its contextual situation. | |
How does quitting work and assuming the role of a full-time mother (phenomenon assessed) change the lives of women in Japan? |
Qualitative research hypotheses | |
---|---|
Hypothesis-generating (Qualitative hypothesis-generating research) | |
- Qualitative research uses inductive reasoning. | |
- This involves data collection from study participants or the literature regarding a phenomenon of interest, using the collected data to develop a formal hypothesis, and using the formal hypothesis as a framework for testing the hypothesis. | |
- Qualitative exploratory studies explore areas deeper, clarifying subjective experience and allowing formulation of a formal hypothesis potentially testable in a future quantitative approach. |
Qualitative studies usually pose at least one central research question and several subquestions starting with How or What . These research questions use exploratory verbs such as explore or describe . These also focus on one central phenomenon of interest, and may mention the participants and research site. 15
Hypotheses in qualitative research are stated in the form of a clear statement concerning the problem to be investigated. Unlike in quantitative research where hypotheses are usually developed to be tested, qualitative research can lead to both hypothesis-testing and hypothesis-generating outcomes. 2 When studies require both quantitative and qualitative research questions, this suggests an integrative process between both research methods wherein a single mixed-methods research question can be developed. 1
Research questions followed by hypotheses should be developed before the start of the study. 1 , 12 , 14 It is crucial to develop feasible research questions on a topic that is interesting to both the researcher and the scientific community. This can be achieved by a meticulous review of previous and current studies to establish a novel topic. Specific areas are subsequently focused on to generate ethical research questions. The relevance of the research questions is evaluated in terms of clarity of the resulting data, specificity of the methodology, objectivity of the outcome, depth of the research, and impact of the study. 1 , 5 These aspects constitute the FINER criteria (i.e., Feasible, Interesting, Novel, Ethical, and Relevant). 1 Clarity and effectiveness are achieved if research questions meet the FINER criteria. In addition to the FINER criteria, Ratan et al. described focus, complexity, novelty, feasibility, and measurability for evaluating the effectiveness of research questions. 14
The PICOT and PEO frameworks are also used when developing research questions. 1 The following elements are addressed in these frameworks, PICOT: P-population/patients/problem, I-intervention or indicator being studied, C-comparison group, O-outcome of interest, and T-timeframe of the study; PEO: P-population being studied, E-exposure to preexisting conditions, and O-outcome of interest. 1 Research questions are also considered good if these meet the “FINERMAPS” framework: Feasible, Interesting, Novel, Ethical, Relevant, Manageable, Appropriate, Potential value/publishable, and Systematic. 14
As we indicated earlier, research questions and hypotheses that are not carefully formulated result in unethical studies or poor outcomes. To illustrate this, we provide some examples of ambiguous research question and hypotheses that result in unclear and weak research objectives in quantitative research ( Table 6 ) 16 and qualitative research ( Table 7 ) 17 , and how to transform these ambiguous research question(s) and hypothesis(es) into clear and good statements.
Variables | Unclear and weak statement (Statement 1) | Clear and good statement (Statement 2) | Points to avoid |
---|---|---|---|
Research question | Which is more effective between smoke moxibustion and smokeless moxibustion? | “Moreover, regarding smoke moxibustion versus smokeless moxibustion, it remains unclear which is more effective, safe, and acceptable to pregnant women, and whether there is any difference in the amount of heat generated.” | 1) Vague and unfocused questions |
2) Closed questions simply answerable by yes or no | |||
3) Questions requiring a simple choice | |||
Hypothesis | The smoke moxibustion group will have higher cephalic presentation. | “Hypothesis 1. The smoke moxibustion stick group (SM group) and smokeless moxibustion stick group (-SLM group) will have higher rates of cephalic presentation after treatment than the control group. | 1) Unverifiable hypotheses |
Hypothesis 2. The SM group and SLM group will have higher rates of cephalic presentation at birth than the control group. | 2) Incompletely stated groups of comparison | ||
Hypothesis 3. There will be no significant differences in the well-being of the mother and child among the three groups in terms of the following outcomes: premature birth, premature rupture of membranes (PROM) at < 37 weeks, Apgar score < 7 at 5 min, umbilical cord blood pH < 7.1, admission to neonatal intensive care unit (NICU), and intrauterine fetal death.” | 3) Insufficiently described variables or outcomes | ||
Research objective | To determine which is more effective between smoke moxibustion and smokeless moxibustion. | “The specific aims of this pilot study were (a) to compare the effects of smoke moxibustion and smokeless moxibustion treatments with the control group as a possible supplement to ECV for converting breech presentation to cephalic presentation and increasing adherence to the newly obtained cephalic position, and (b) to assess the effects of these treatments on the well-being of the mother and child.” | 1) Poor understanding of the research question and hypotheses |
2) Insufficient description of population, variables, or study outcomes |
a These statements were composed for comparison and illustrative purposes only.
b These statements are direct quotes from Higashihara and Horiuchi. 16
Variables | Unclear and weak statement (Statement 1) | Clear and good statement (Statement 2) | Points to avoid |
---|---|---|---|
Research question | Does disrespect and abuse (D&A) occur in childbirth in Tanzania? | How does disrespect and abuse (D&A) occur and what are the types of physical and psychological abuses observed in midwives’ actual care during facility-based childbirth in urban Tanzania? | 1) Ambiguous or oversimplistic questions |
2) Questions unverifiable by data collection and analysis | |||
Hypothesis | Disrespect and abuse (D&A) occur in childbirth in Tanzania. | Hypothesis 1: Several types of physical and psychological abuse by midwives in actual care occur during facility-based childbirth in urban Tanzania. | 1) Statements simply expressing facts |
Hypothesis 2: Weak nursing and midwifery management contribute to the D&A of women during facility-based childbirth in urban Tanzania. | 2) Insufficiently described concepts or variables | ||
Research objective | To describe disrespect and abuse (D&A) in childbirth in Tanzania. | “This study aimed to describe from actual observations the respectful and disrespectful care received by women from midwives during their labor period in two hospitals in urban Tanzania.” | 1) Statements unrelated to the research question and hypotheses |
2) Unattainable or unexplorable objectives |
a This statement is a direct quote from Shimoda et al. 17
The other statements were composed for comparison and illustrative purposes only.
To construct effective research questions and hypotheses, it is very important to 1) clarify the background and 2) identify the research problem at the outset of the research, within a specific timeframe. 9 Then, 3) review or conduct preliminary research to collect all available knowledge about the possible research questions by studying theories and previous studies. 18 Afterwards, 4) construct research questions to investigate the research problem. Identify variables to be accessed from the research questions 4 and make operational definitions of constructs from the research problem and questions. Thereafter, 5) construct specific deductive or inductive predictions in the form of hypotheses. 4 Finally, 6) state the study aims . This general flow for constructing effective research questions and hypotheses prior to conducting research is shown in Fig. 1 .
Research questions are used more frequently in qualitative research than objectives or hypotheses. 3 These questions seek to discover, understand, explore or describe experiences by asking “What” or “How.” The questions are open-ended to elicit a description rather than to relate variables or compare groups. The questions are continually reviewed, reformulated, and changed during the qualitative study. 3 Research questions are also used more frequently in survey projects than hypotheses in experiments in quantitative research to compare variables and their relationships.
Hypotheses are constructed based on the variables identified and as an if-then statement, following the template, ‘If a specific action is taken, then a certain outcome is expected.’ At this stage, some ideas regarding expectations from the research to be conducted must be drawn. 18 Then, the variables to be manipulated (independent) and influenced (dependent) are defined. 4 Thereafter, the hypothesis is stated and refined, and reproducible data tailored to the hypothesis are identified, collected, and analyzed. 4 The hypotheses must be testable and specific, 18 and should describe the variables and their relationships, the specific group being studied, and the predicted research outcome. 18 Hypotheses construction involves a testable proposition to be deduced from theory, and independent and dependent variables to be separated and measured separately. 3 Therefore, good hypotheses must be based on good research questions constructed at the start of a study or trial. 12
In summary, research questions are constructed after establishing the background of the study. Hypotheses are then developed based on the research questions. Thus, it is crucial to have excellent research questions to generate superior hypotheses. In turn, these would determine the research objectives and the design of the study, and ultimately, the outcome of the research. 12 Algorithms for building research questions and hypotheses are shown in Fig. 2 for quantitative research and in Fig. 3 for qualitative research.
Research questions and hypotheses are crucial components to any type of research, whether quantitative or qualitative. These questions should be developed at the very beginning of the study. Excellent research questions lead to superior hypotheses, which, like a compass, set the direction of research, and can often determine the successful conduct of the study. Many research studies have floundered because the development of research questions and subsequent hypotheses was not given the thought and meticulous attention needed. The development of research questions and hypotheses is an iterative process based on extensive knowledge of the literature and insightful grasp of the knowledge gap. Focused, concise, and specific research questions provide a strong foundation for constructing hypotheses which serve as formal predictions about the research outcomes. Research questions and hypotheses are crucial elements of research that should not be overlooked. They should be carefully thought of and constructed when planning research. This avoids unethical studies and poor outcomes by defining well-founded objectives that determine the design, course, and outcome of the study.
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Methodology
Published on June 12, 2020 by Pritha Bhandari . Revised on June 22, 2023.
Quantitative research is the process of collecting and analyzing numerical data. It can be used to find patterns and averages, make predictions, test causal relationships, and generalize results to wider populations.
Quantitative research is the opposite of qualitative research , which involves collecting and analyzing non-numerical data (e.g., text, video, or audio).
Quantitative research is widely used in the natural and social sciences: biology, chemistry, psychology, economics, sociology, marketing, etc.
Quantitative research methods, quantitative data analysis, advantages of quantitative research, disadvantages of quantitative research, other interesting articles, frequently asked questions about quantitative research.
You can use quantitative research methods for descriptive, correlational or experimental research.
Correlational and experimental research can both be used to formally test hypotheses , or predictions, using statistics. The results may be generalized to broader populations based on the sampling method used.
To collect quantitative data, you will often need to use operational definitions that translate abstract concepts (e.g., mood) into observable and quantifiable measures (e.g., self-ratings of feelings and energy levels).
Research method | How to use | Example |
---|---|---|
Control or manipulate an to measure its effect on a dependent variable. | To test whether an intervention can reduce procrastination in college students, you give equal-sized groups either a procrastination intervention or a comparable task. You compare self-ratings of procrastination behaviors between the groups after the intervention. | |
Ask questions of a group of people in-person, over-the-phone or online. | You distribute with rating scales to first-year international college students to investigate their experiences of culture shock. | |
(Systematic) observation | Identify a behavior or occurrence of interest and monitor it in its natural setting. | To study college classroom participation, you sit in on classes to observe them, counting and recording the prevalence of active and passive behaviors by students from different backgrounds. |
Secondary research | Collect data that has been gathered for other purposes e.g., national surveys or historical records. | To assess whether attitudes towards climate change have changed since the 1980s, you collect relevant questionnaire data from widely available . |
Note that quantitative research is at risk for certain research biases , including information bias , omitted variable bias , sampling bias , or selection bias . Be sure that you’re aware of potential biases as you collect and analyze your data to prevent them from impacting your work too much.
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Once data is collected, you may need to process it before it can be analyzed. For example, survey and test data may need to be transformed from words to numbers. Then, you can use statistical analysis to answer your research questions .
Descriptive statistics will give you a summary of your data and include measures of averages and variability. You can also use graphs, scatter plots and frequency tables to visualize your data and check for any trends or outliers.
Using inferential statistics , you can make predictions or generalizations based on your data. You can test your hypothesis or use your sample data to estimate the population parameter .
First, you use descriptive statistics to get a summary of the data. You find the mean (average) and the mode (most frequent rating) of procrastination of the two groups, and plot the data to see if there are any outliers.
You can also assess the reliability and validity of your data collection methods to indicate how consistently and accurately your methods actually measured what you wanted them to.
Quantitative research is often used to standardize data collection and generalize findings . Strengths of this approach include:
Repeating the study is possible because of standardized data collection protocols and tangible definitions of abstract concepts.
The study can be reproduced in other cultural settings, times or with different groups of participants. Results can be compared statistically.
Data from large samples can be processed and analyzed using reliable and consistent procedures through quantitative data analysis.
Using formalized and established hypothesis testing procedures means that you have to carefully consider and report your research variables, predictions, data collection and testing methods before coming to a conclusion.
Despite the benefits of quantitative research, it is sometimes inadequate in explaining complex research topics. Its limitations include:
Using precise and restrictive operational definitions may inadequately represent complex concepts. For example, the concept of mood may be represented with just a number in quantitative research, but explained with elaboration in qualitative research.
Predetermined variables and measurement procedures can mean that you ignore other relevant observations.
Despite standardized procedures, structural biases can still affect quantitative research. Missing data , imprecise measurements or inappropriate sampling methods are biases that can lead to the wrong conclusions.
Quantitative research often uses unnatural settings like laboratories or fails to consider historical and cultural contexts that may affect data collection and results.
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Research bias
Quantitative research deals with numbers and statistics, while qualitative research deals with words and meanings.
Quantitative methods allow you to systematically measure variables and test hypotheses . Qualitative methods allow you to explore concepts and experiences in more detail.
In mixed methods research , you use both qualitative and quantitative data collection and analysis methods to answer your research question .
Data collection is the systematic process by which observations or measurements are gathered in research. It is used in many different contexts by academics, governments, businesses, and other organizations.
Operationalization means turning abstract conceptual ideas into measurable observations.
For example, the concept of social anxiety isn’t directly observable, but it can be operationally defined in terms of self-rating scores, behavioral avoidance of crowded places, or physical anxiety symptoms in social situations.
Before collecting data , it’s important to consider how you will operationalize the variables that you want to measure.
Reliability and validity are both about how well a method measures something:
If you are doing experimental research, you also have to consider the internal and external validity of your experiment.
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.
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A t test is a statistical test used to compare the means of two groups. The type of t test you use depends on what you want to find out.
What is a T Test? A t test is a statistical hypothesis test that assesses sample means to draw conclusions about population means. Frequently, analysts use a t test to determine whether the population means for two groups are different. For example, it can determine whether the difference between the treatment and control group means is statistically significant.
T-test was first described by William Sealy Gosset in 1908, when he published his article under the pseudonym 'student' while working for a brewery.[1] In simple terms, a Student's t-test is a ratio that quantifies how significant the difference is between the 'means' of two groups while taking their variance or distribution into account.
T test definition. Types of t test. Step by step examples for solving problems using graph, Student's t-test tables and calculators.
T-Test: A t-test is an analysis of two populations means through the use of statistical examination; a t-test with two samples is commonly used with small sample sizes, testing the difference ...
The characteristics of the data dictate the appropriate type of t test to run. All t tests are used as standalone analyses for very simple experiments and research questions as well as to perform individual tests within more complicated statistical models such as linear regression. In this guide, we'll lay out everything you need to know about t tests, including providing a simple workflow ...
This procedure is an inferential statistical hypothesis test, meaning it uses samples to draw conclusions about populations. The independent samples t test is also known as the two sample t test. This test assesses two groups. For an example of an independent t test, do students who learn using Method A have a different mean score than those ...
A t-test is a tool for evaluating the means of one or two populations using hypothesis testing. Learn about types of t-tests, t-test assumptions and how to perform a t-test.
In this review, we'll look at significance testing, using mostly the t -test as a guide. As you read educational research, you'll encounter t -test and ANOVA statistics frequently. Part I reviews the basics of significance testing as related to the null hypothesis and p values. Part II shows you how to conduct a t -test, using an online calculator. Part III deal s with interpreting t -test ...
T-tests are statistical hypothesis tests that you use to analyze one or two sample means. Depending on the t-test that you use, you can compare a sample mean to a hypothesized value, the means of two independent samples, or the difference between paired samples. In this post, I show you how t-tests use t-values and t-distributions to calculate probabilities and test hypotheses.
What Are t-Values? T-tests are called t-tests because the test results are all based on t-values. T-values are an example of what statisticians call test statistics. A test statistic is a standardized value that is calculated from sample data during a hypothesis test.
The t-test gauges whether the means of two groups are statistically different from each other using ratio: difference between group means/variability of groups.
Your choice of statistical test depends on the types of variables you're dealing with and whether your data meets certain assumptions.
The t test is one type of inferential statistics. It is used to determine whether there is a significant difference between the means of two groups. With all inferential statistics, we assume the dependent variable fits a normal distribution. When we assume a normal distribution exists, we can identify the probability of a particular outcome.
During the last months, I've probably run the t-test dozens of times but recently I realized that I did not fully understand some concepts such as why it is not possible to accept the null hypothesis or where the numbers in the t-tables come from. After doing some research, I found that several articles provide those answers but not so many gather all of the information together.
The unequal variance t test tends to be less powerful than the usual t test if the variances are in fact the same, since it uses fewer assumptions. However, it should not be used indiscriminantly because, if the standard deviations are different, how can we interpret a nonsignificant difference in means, for example?
9.1: The t-statistic The z-statistic was a useful way to link the material and ease us into the new way to looking at data, but it isn't a very common test because it relies on knowing the populations standard deviation, σ, which is rarely going to be the case.
Chapter 6The t-test and Basic Inference PrinciplesThe t-test is used as an examp. e of the basic principles of statistical inference.One of the simplest situations for which we might design an experiment is the case of a nominal two-level expla. atory variable and a quantitative.
Learn how to use t-tests and hypothesis testing to compare sample means and draw statistical inferences. A simple and intuitive explanation with examples.
An independent-group t test can be carried out for a comparison of means between two independent groups, with a paired t test for paired data. As the t test is a parametric test, samples should meet certain preconditions, such as normality, equal variances and independence.
Which t-test should I use? Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means. If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group ...
Quantitative data is data which can be expressed numerically to indicate a quantity, amount, or measurement. not all numbers constitute quantitative data (e.g. tax file number!) distinct from qualitative data. Quantitative data collection involves measurement of variables. A variable is a characteristic of a unit being observed that may assume ...
It is crucial to have knowledge of both quantitative and qualitative research 2 as both types of research involve writing research questions and hypotheses. 7 However, these crucial elements of research are sometimes overlooked; if not overlooked, then framed without the forethought and meticulous attention it needs. Planning and careful consideration are needed when developing quantitative or ...
Quantitative research is the process of collecting and analyzing numerical data. It can be used to find patterns and averages, make predictions, test causal relationships, and generalize results to wider populations.