Two Sample t Test: equal variances

We now consider an experimental design where we want to determine whether there is a difference between two groups within the population. For example, let’s suppose we want to test whether there is any difference between the effectiveness of a new drug for treating cancer. One approach is to create a random sample of 40 people, half of whom take the drug and half take a placebo. For this approach to give valid results it is important that people be assigned to each group at random. Such samples are independent.

When the population variances are known, hypothesis testing can be done using a normal distribution, as described in Comparing Two Means when Variances are Known. But population variances are not usually known. The approach we use instead is to pool sample variances and use the t distribution.

We consider three cases where the t distribution is used:

  • Equal variances
  • Unequal variances
  • Paired samples

We deal with the first of these cases in this section.

Theorem 1: Let  and ȳ be the sample means of two sets of data of size nx and ny respectively. If x and y are normal, or nx and ny are sufficiently large for the Central Limit Theorem to hold, and x and y have the same variance, then the random variable


has distribution T(nx + ny – 2) where


Observations, as defined above, can be viewed as a way to pool sx and sy, and so s2 is referred to as the pooled variance. Also note that the degrees of freedom of t is the value of the denominator of s2 in the formula given in Theorem 1.

Click here for a proof of Theorem 1.

Real Statistics Excel Functions: The following functions are provided in the Real Statistics Resource Pack.

VAR_POOLED(R1, R2) = pooled variance of the samples defined by ranges R1 and R2, i.e.  s2 of Theorem 1

STDEV_POOLED(R1, R2) = pooled standard deviation of the samples defined by ranges R1 and R2, i.e. s of Theorem 1

STDERR_POOLED(R1, R2, b) = pooled standard error of the samples defined by ranges R1 and R2. This is equal to the denominator of t in Theorem 1 if b = TRUE (default) and equal to the denominator of t in Theorem 1 of Two Sample t Test with Unequal Variances if b = FALSE. When the sample sizes are equal, b = TRUE or b = FALSE yields the same result.

Observation: Each of these functions ignores all empty and non-numeric cells.

Example 1: A marketing research firm tests the effectiveness of a new flavoring for a leading beverage using a sample of 20 people, half of whom taste the beverage with the old flavoring and the other half who taste the beverage with the new favoring. The people in the study are then given a questionnaire which evaluates how enjoyable the beverage was. The scores are as in Figure 1. Determine whether there is a significant difference between the perception of the two flavorings.

t test independent samples

Figure 1 – Data and box plot for Example 1

As we can see from the box plot in Figure 1 the data in each sample is reasonably symmetric and so we use the t test with the following null hypothesis:

H0: μ1μ2 = 0; i.e. there is no difference between the two flavorings

Since the sample variances are similar we decide that the population variances are also likely to be similar and so apply Theorem 1.


And so s =\sqrt{16.05} = 4.01. Now,


Since p-value = TDIST(t, df) = TDIST(2.18, 18) = .043 < .05 = α, we reject the null hypothesis, concluding that there is a significant difference between the two flavorings. In fact, the new flavoring is significantly more enjoyable.

The same result can be obtained by use of Excel’s Two-Sample Assuming Equal Variances data analysis tool, the results of which are as follows.

t test Excel tool

Figure 2 – Output from Excel’s data analysis tool

Observation: The Real Statistics Resource Pack also provides a data analysis tool which supports the two independent sample t test, but provides additional information not found in the standard Excel data analysis tool. Example 3 in Two Sample t Test: Unequal Variances gives an example of how to use this data analysis tool.

Example 2: To investigate the effect of a new hay fever drug on driving skills, a researcher studies 24 individuals with hay fever: 12 who have been taking the drug and 12 who have not. All participants then entered a simulator and were given a driving test which assigned a score to each driver as summarized in Figure 3.

t test histogram

Figure 3 – Sample data and histograms for Example 2

As in the previous example, we plan to use the t-test, but with a sample this small we first need to check to see that the data is normally distributed (or at least symmetric). This can be seen from the histograms. Also the variances are relatively similar (15.18 and 17.88) and so we can again use the t-Test: Two-Sample Assuming Equal Variances data analysis tool to test the following null hypothesis:

H0: μcontrol = μdrug

Excel t test tool

Figure 4 – Two sample data analysis results

Since tobs = .10 < 2.07 = tcrit (or p-value = .921 > .05 = α) we retain the null hypothesis; i.e. we are 95% confident that any difference between the two groups is due to chance.

Observation: The t-test is quite robust even when the underlying distributions are not normal provided the sample size is sufficiently large (usually over 25 or 30). The t-test can be valid even with smaller sample sizes, provided the samples have similar shape and are not too skewed.

Effect size

The Cohen effect size d can be calculated as in One Sample t Test, namely:


This is approximated by

Example 3: Find the effect size for study in Example 2.


This means that the control group has a driving score 4.1% of a standard deviation more than the group that is taking the hay fever medication.

157 Responses to Two Sample t Test: equal variances

  1. jamel850 says:

    Hello Charles,

    I’m having difficult trying to Find the variance of the difference of the ratings before and after the workshops of paired ratings. I use two paired samples so I perform T-Test Two Paired Samples. The actual answer is 204.81, but when I try running real stat I believe the closest thing to that answer is the STDEV 203.9515663. Is this the correct test to use or do I need to use something else to Find the variance of the differences?

    T Test: Two Paired Samples

    SUMMARY Alpha 0.05
    Groups Count Mean Std Dev Std Err
    22 29 72.86206897 47.95289663
    223 29 147.7931034 203.9515663
    Difference 29 -74.93103448 201.9973357 37.50996357

    T TEST
    p-value t-crit lower upper
    One Tail 0.027778076 1.701130934
    Two Tail 0.055556152 2.048407142 -151.7667117 1.904642784

    • Charles says:

      I would have to see the actual data to know what the correct variance is, but to calculate the variance of the difference between the paired data in ranges A1:A100 and B1:B100, you can use the array formula =VAR(A1:A100-B1:B100). Since this is an array formula you need to press Ctrl-Shft-Enter and not just Enter.

  2. juve caban-os says:

    Hi, I have a specific problem. What t-test im going to use? thank you.

    The following Null hypothesis will be tested at 0.05 level of significance.
    1. There is no significant difference between the perceptions of the school heads and teachers on the validity of localized word problem activities in grade 10 mathematics.
    2. There is no significant difference between the perceptions of the school heads and teachers on the effect of the localized word problem activities on the learning of students.
    3. There is no significant difference between the perceptions of the school heads and teachers on the adequacy of localized word problem instructional materials.
    4. There is no significant difference between the perceptions of the school heads and
    teachers on the problems encountered in localizing word problem learning activities.

    • Charles says:

      Assuming these samples are independent, you need to look at the variances of the samples. If they are relatively similar, you can use the two sample t test with equal variance; otherwise you need to use the version with unequal variances. When in doubt, use the unequal variances version of the test.

  3. seles says:

    which test to used when I have to compare pull, push and factors affecting turnover intent i.e., 3 independent variables on dependent variable. I do not mind if it gives respective correlation. can I use t-test ?

    • Charles says:

      It sounds like an analysis that can be performed using One-way ANOVA.
      I can’t say for sure since you have only supplied a very limited amount of information.

      • seles says:

        Thank you for your reply. I’m doing a project to measure the push, pull, personal factors affecting turnover intention of employees. I select few departments in the organization.
        In this, I send out questionnaires of each factor where they rate using Likert scale.
        As, I said earlier the dependant variable is turnover intention and the independent variables are push, pull, personal factors.
        So, I hope it will help you understand about my concern.

  4. Khyati says:

    Hello Mr. Charles,
    I want to compare two drugs with 3 batches. I have only one result for each batch. The main purpose is to test equivalence of both the drugs. Which test should be appropriate?
    Thank you.

    • Charles says:

      From your description, I understand that you have 2 samples each with 3 elements. This are very small samples and so you can’t expect too much from any statistical test. Most likely, your best test would be the Mann-Whitney Test.

  5. rox says:

    Hiii. Id like to know the difference between two groups for one data…. what do I use? Tysm!! 🙂

    • Charles says:

      Sorry, but I don’t understand your question.
      Are you trying to determine whether there is a significant difference between the (population) means of two groups? If so, the t test described on the page that you are referencing could be the right choice…if the assumptions for that test are met.

  6. Richy says:

    Hi mr Charles.
    I want to compare two component but different brand. Point of check is thickness painting for its component. What should i use statistic tools for that? How many sample should i have. Thank you.

    • Charles says:

      It depends on what you want to test. If you want to test whether there is a significant difference between the paint thickness for the two brands, you can use a t test provided the assumptions are met. The sample size required is described on the website. You can also use the Real Statistics data analysis tool to calculate the required sample sizes.

  7. Takwa says:

    2 groups are tested 3 times each using different tests. Except for the language of presentation, the 2 groups are subject to the same procedure. Group 1 (n=23) used L1 to perform the tests, group 2 (n=28) used L2. How to call this design and what is the appropriate statistical test to compare the two groups’ means?
    thank you for your cooperation

    • Charles says:

      It sounds like a mixed ANOVA with one fixed factor (Language) and the other a repeated measures factor (Tests).

  8. deepu says:

    When we are doing t-test assuming equal variences if p-values for one tail and two tail are less than 0.05 but Tstatistic value is not greater than Tcritical value what does that mean
    1.can we accept null hypothesis.
    2.can we reject null hypothesis.

    • Charles says:

      If p-value < alpha (where alpha is .05), then we reject the null hypothesis. If t stat < t crit, then we reject the null hypothesis. Charles

  9. David O'Keeffe says:

    Or should I use a t-test assuming unequal variances, considering that the numbers are so different?

    Any help greatly appreciated.


  10. David o'Keeffe says:

    Hello Mr. Zaiontz,

    I am currently doing a project looking at the effect of a regulation on a small series of numbers, very small. I have 6 numbers pre-regulation, and 2 after. I need to statistically analyse the change in the numbers, which is quite a large effect in that the numbers for the 2 years after the introduction of the regulation are about half the level of the previous average.

    Nevertheless, I need to use statistical analysis.

    After some guesswork and basic research, I have chosen the difference between means test, with equal variance. I did it in excel, but because I need to show methodology and have my answers correspond to the equations, I also did it with the equations from example 1 above in the formula tab of excel.

    I thought these would be the same as the program in excel except the results are quite different.

    I have a few questions.

    Would you say that I am doing the correct test?

    If so, should I go by the numbers of the excel formula, or the equations above, and if I should trust in the excel function, do you know the equations which excel uses?

    Finally, in regards to the test in general, I am taking as the first group the 6 annualized figures before the change, and for the second group the 2 numbers after the change.

    Is this wrong? Should I get a mean of the previous numbers and then get a mean of the entire set of 8 numbers and then test for the difference?

    It’s due in a few days.

    I would greatly appreciate any help you could provide.

    Yours sincerely,

    Lost and confused.

    • Charles says:

      The formula for the t test presented is the one that Excel uses, and so you should get the same answer.
      It is difficult for me to determine whether you are using the correct test based on the information that you have provided. Are you saying that the sample prior to the change consists of 6 randomly chosen elements and the sample after the change consists of a different set of 2 randomly chosen elements?

  11. Marisse Urquico says:

    When alpha is not given in a case problem, shall I presume Excel PHStat’s default, 0.005? Please help. Thanks. Marisse

  12. kemmy says:

    Pls l have 20 test variables(likert scale type questions) and 2 independent variables. what is the best statistical tools
    to use. thanks

    • Charles says:

      This depends on what you want to test. It is not possible to give an answer without more information.

  13. Hannah says:

    I want to find out the difference in students attitude from 2 different types of school.
    The survey, which comprises of 21 questions uses 4-point likert scale.
    May I know how do I analyse it ?
    Is it suitable to use a t-test?
    I would appreciate if someone can help.

    • Charles says:

      If you create an overall score based on the responses to the 21 questions, then you can use a t test.
      If you want to analyze the differences for all the questions, then you might want to use ANOVA followed by comparisons, Tukey HSD, etc.

  14. ethel cacayuran says:

    Hi!Iam checking the difference of the observed and practiced leadership of a certain organization. The observed leadership came from the 53 members who answered a survey and the practiced leadership came from the 18 officers who also answered the same survey though statements was written in the first person. I tried to test it in excel by using data analysis of unequal variance however I noticed that the df varies from one statement with another. Then I tried data analysis again and used t test for equal variance and the df stayed the same on all statements. What test should I really need to use?eThanks

    • Charles says:

      Whether to use the t test for equal variance or t test for unequal variance depends on the the variances. When in doubt use the t test for unequal variance.
      When you say that “the df varies from one statement with another”, I understand that you are doing a separate t test for each statement. If this is the case, then there is no problem having different values for the df for each test.
      Depending on the details of what you are analyzing, you might want to try one of the following approaches:
      1. Create an overall score based on all the statements (or some groups of statements) and do a t test on these scores
      2. Use ANOVA, or more likely MANOVA, where all the statements are tested simultaneously. This may cut down on the expermentwise error rate of performing a number of t tests.

  15. AG says:

    Hi Charles,

    I want to establish the effectiveness of an intervention. I have scores of students who have given both pre-test and post-test. However, there are some students who have given a pre-test and not the post-test and students who have given a post-test and not a pre-test. What method should I use that accounts for this anomaly.
    Note: A pre-test and post-test were also given to a control group of students who showed no improvement in scores. All other conditions between the intervention group and the control group can be assumed to be identical.

    • Charles says:

      For the types of analyses you describe in the first paragraph of your comment, you would use a paired t test (and not a two independent samples t test). You would eliminate any subject who were not given both the pre-test and post-test.

  16. mark says:


    I have data for schools scores and one variable tells me if the school had a treatment to improve their score. the treatment group is about 3000 and the not trated group is about 11000. Can i simple do the two sample t test equal variance or before i have to select a random sample of 3000 schools from the 11000?

    sorry for my english


    • Charles says:

      The sizes of the two samples don’t have to be equal. You can simply perform a two sample t test. If the variances are not too different you can use the equal variances version of the test; otherwise use the nonequal variances version of the test.

  17. Dr M says:

    I am working on a project where i have data from two different groups comparing the efficacy of a new procedure vs conventional method. 30 subjects in each group. i have a followup of them 6months down the line and the data for each group of subjects on 6months. how can i find the p value using t-test and which one- unpaired t-test assuming equal/unequal variances??
    also when can we assume variances equal or unequal-in one of your articles it was written that if one variance is upto 4times the other it can be taken as equal. is that true??
    kindly guide me as your help will be beneficial.

    • Charles says:

      Dr M,
      With the t test when in doubt it is probably best to assume that the variances are unequal. Even if the variances are equal, the p-value won’t be much different.
      If the variance of one sample is no more than 4 times (some say 3 times) the variance of another sample, then you can generally assume that the variances are sufficiently equal, is a rule of thumb. Generally it works, but it is not a hard and fast rule.

  18. Gitanjali Padhi says:

    i have administered intelligence test and achievement test . The total marks of both the test of individual students have been arranged in ascending order and rank them from higher to lower order . Each two adjacent scores will constitute a pair will be randomly assigned as control group and experimental group . In this manner 30 pairs have formed . The control group will be taught by traditional method of teaching and the experimental group will be taught by activity method of teaching . For significant difference which t test formula shall i used pl . suggest me . My id . Plea

    • Charles says:

      Since the 30 students in each group are different you should use the two independent sample t test. The pairing that you have done is not used.

  19. Ina says:

    Dear Charles,
    I would like to ask you which test I should use in this situation:
    I taught two groups of 30 students both (approximately the same level): one group was taught deductively, and the other one – inductively. Then they had a test. I am going to compare their means and then if I want to see if the difference in their scores is significant (or not), could I use a paired T_test? Or should i use a different test?

    Thank you very much,

  20. Nicole says:

    Hi Charles

    I want to ask which test I should use in this situation: I want to study the difference of classroom tasks with regard to motivation. Therefore, my independent variables are the different tasks (~23) and the dependent variable is motivation. Over a period of several lessons I had students (~30) fill out a short questionnaire marking on a scale from 1 to 5 how motivated they are when doing a specific task. I calculated the mean value for every task and now I want to test whether each task is significantly different from the average of all tasks. I learned that to do a two-sample t-test, the two populations must be independent, all subjects must have produced the same amount of data (though I’m not quite sure what that means…) and the variables must be ratio variables. In my study the different tasks are independent but they are always done by (almost) the same group of students, however, since not all students were present every single time and I also had to exclude some questionnaires because they weren’t complete, the number of subjects differs for each task. Can I use the two-sample t-test or is there another test that would be more suitable? I must say I’m quite confused and hope you can help me. Thank you!


    • Charles says:


      If you want to compare all 23 tasks, then you need to use a multivariate version of the t test, namely Hotelling’s T-square test. Since the two samples are not independent but involve the same students, you need to use the paired version of this test. See the following webpages for information about this test:
      Hotelling’s T-Square

      The problem then is that you need complete data. This means that you need to use one of the techniques for dealing with missing data. See the following for more information about this:
      Handling Missing Data


  21. Meg Learner says:

    I carried out a questionnaire exercise 4 years ago, attempting to replicate (as far as possible with only me as the resource) a questionnaire study done over a period of 3 years, with an initial population of 128,000, reducing to 2,000 by the end of the study by issuing a single online questionnaire to a very much smaller population based in a different part of the same country, whose population had not been part of the initial exercise. The results are all in SPSS but I also have them available in Excel. I carried out a descriptive analysis of the findings from my questionnaire and compared my results with the same questions from the original survey but am now being asked to compare my results in more depth with those from the original questionnaire. My statistical knowledge is sketchy 🙁 so I am not even sure whether the question I am going to ask here is a “good one”!). For both studies, only gender and age bands are known and I was wondering whether it would be possible to use the two sample t test to say whether the two populations are similar (or not) in their responses to individual questions in the two questionnaires. I have several books that I have consulted and online papers and I still can’t answer the question. Are you able to shed any light please? Thank you.

    • Charles says:


      It all depends on how you measure “the two populations are similar (or not) in their responses to individual questions in the two questionnaires”. I don’t have a definitive answer for you, but here are some quick ideas off the top of my head.

      1. Check to see whether the means are significantly different or similar. You have several factors: question, age and gender (possible hierarchical factors). You would then perform some form of ANOVA, depending on what specifically you decide to test. A t test could be used as a follow up test.

      2. You can use the Two Sample Kolmogorov-Smirnov test if you are interested in similarly of more than just the means. You probably need to simplify the problem in order to use this test.

      3. You can use one of the rating comparison measures (e.g. ICC). Again some sort of simplification would probably be needed.


  22. eli says:

    Hello Charles,
    Could you please give me your opinion about my data analysis. I have two populations, the one is 81 diferent municipalities use solar energy and shows energy savings and second population has another 81 different municipalities use hydro energy and data shows energy savings.
    I want to use two sample unpaired test no test mean difference between two groups. I am not sure about should i go with unknown, equal or inequal varience. I done descriptive data analysis and two samples variences are 45.59126984 57.12666667
    thank you in advance

    • Charles says:

      These variances are quite similar and so you should be able to use the equal variance test. In general, if you are not sure, then I suggest that you use the unequal variance test< if the variances are equal then the result will be similar anyway. Charles

  23. AH says:

    Hi Charles,

    I have a case where I have to apply both the Independent and Paired designs, and part of the tasks to conduct both designs are;
    – Based on the information you were given, determine which of the two experimental designs is more efficient, that is, which experimental design yields a higher power in detecting the 8% increase in the tires’ mean lifetime.
    -Investigate how the power of the tests in the two experimental designs vary according to the number of experimented tires, ????, and the errors variance, ????????2.

    And I’m confused in what I have to do, like does the “power” refer to the CI?


  24. daisy says:

    I have some inquiry to ask. I have been doing biomonitoring, where I get 5 sets of data for 5 stations, and I take three readings of each water parameters for each station. I had get the mean and standard deviation for the three readings. Do I have to find variance too?
    I am required to do t-test to analyze the data. But, I didn’t see where I should do the t-test since t-test is comparison between 2 populations and mine is 5 population. I hope that you can help to to understand this. Thank you.

    • Charles says:

      It sounds like you have been given an assignment where you need to apply the t test despite the fact that you have 5 samples. I can’t explain why someone might want you to do this. Also I can’t guess as to whether or not you are required to provide the variance for this assignment.

    • rida says:

      when u have met to cheack the equality of two population mean then u will always apply t- test its confirm but when u have involve the testing of more than two population mean then u always apply “ANOVA TEST”…………..the reason behind this that when u will apply t-test for the testing of more than 2 population mean than your level of significance alpha
      will increase which is not suitable and its consuming more time and it has tedious to calculate…………………

  25. Allam says:

    please Charles
    I want to know why we assume that two variance must be equal in case of t-distribution?
    and in case the two variances unknown and not equal why we use?

    • Charles says:

      We don’t need to assume that the variances are equal. If we have evidence to believe that the variances are equal then we can use the t test with equal variances. Otherwise, we should use the t test with unequal variances; in this case, if the variances are actual close to being equal, then the results from this test will be quite similar to those of the equal variances version of the t test.

      • Allam says:

        thanks Charles, but we we use different rules, one by sample variance and another by pooled variance.
        (S1^2)/n1 +(S2^2)/n2
        SP^2 (1/n1 +1/n2 )

        • Charles says:

          Note that if the variances are equal and the sample sizes are equal, then these formulas are the same.

  26. Nuno says:

    Dear Charles,
    I am analyzing the t-Student test to perform some statistical analysis on some data. My experiment is something like: I will record three movements with a Camera 1 and with a different Camera 2 and retrieve data from the position of the elbow. So what I want is to compare the position of the elbow measured with Camera 1 with the position measured with Camera 2, for each movement. So:

    Movement 1: Camera 1 VS Camera 2
    Movement 2: Camera 1 VS Camera 2
    Movement 3: Camera 1 VS Camera 2

    I was looking a lot for the equal or unequal variances method but probably for my case I should use a paired-sample T-test, what do you think?

    • Charles says:

      Dear Nuno,
      If the emphasis is on the camera and the elbow is just a measurement instrument, then this is probably a two independent sample t test. You may also want to use ANOVA so as to combine the three tests into one.

      • Nuno says:

        Thanks for the suggestion but I think I will not need to combine the tests.
        What I would like to know is, under the three cases of the independent sample t test, what should I use?
        Besides I also heard about a Wilcoxon test as an alternative to the t-Student, but when should one use that?

        • Charles says:

          Based on the limited information that i have about this situation, it seems like you want to use two independent sample t tests. If the variances are more or less equal, then you can use the equal variances version; otherwise the unequal variances version. If you are not sure, then I suggest that you use the unequal variances version since the results will be similar even if the variances are equal. I suggest that you take into account that the more tests you do the higher your type I error will be and so you should correct for this, as described on the webpage
          Experimentwise Error

  27. Benjamin Breer says:

    I believe I have found a computational error regarding how to determine the degrees of freedom. The section lists the formula utilized to ascertain the degrees of freedom by adding the two sample sizes together and then proceeding to subtract 2 from that total. However, it lists that 10 = 10 – 2 = 18. I believe this should be 10 + 10 – 2 = 18.

    • Charles says:

      Thanks for catching this typo. I have now corrected the error on the referenced webpage.
      I appreciate your help in making the site more accurate and easier to understand.

  28. anjali sharma says:

    it is very confusing regarding which t test should be use.
    my topic is comparison between public bank and private bank.A sample of 100 from public and 100 from private means a total 200 sample size.
    so plz suggest me which test can be appliede

  29. Vie says:

    I just wanna know if the formula for t-test foe mean is same as the formula for t-test for correlation coefficient? thanks

  30. roniel says:

    Hi Charles,

    I want to ask which appropriate test will be used in this situation, I want to study the difference of the competency of criminology students in developing latent fingerprint using fingerprint powder and cyanoacrylate with the same respondents. I only selected 35 students,they used both powder and cyanoacrylate. Each of them has 8 different items to test their competency. I have already computed the mean of every items. Can you help me? I think paired t-test will be used because of the same tespondents.,or t test equal variance?

    • Charles says:

      Since the same students are using each approach you need to use the paired t-test (provided the assumptions for the test are met).

  31. Aderemi muhydeen says:

    Pls if a question is saying dat we av mean A as 700 and mean B as 695,, sample size A is 10 while sample size B is 15 … ….. . Sample variance A 40 and sample variance B is 40 as well. Which test statistics is applicable to solve dis question . is it t test or z test.. .. … . pls reply or yu can drop in my mail

    • Charles says:

      If your objective is to determine whether there is a significant difference between the population means for A and B, then you should use the t test (assuming the assumptions for the test are met). The results from the z test will generally be similar, but the t test is preferred.

  32. Wendy Foo says:

    Hi, I have a research on variation of CT number for one sample at 60 kVp and 120 kVp. Each kVp has 10 values. Which type of t-test should I perform? If I want to conduct my research for more kVp, can I still analyse the data using t-test? My last question, if 60 kVp data is normally distributed but 120 kVp data is not normally distributed which test should I proceed?

    Many thanks in advance.

    • Charles says:

      I don’t know what a CT number is, but in general if you are trying to compare two samples, you can use the t test. If you have more than two samples, you can use ANOVA.
      If one of the samples is not normal, then you can still use these tests provided the data is not too far from normal, in fact the tests should work pretty well if the data is reasonably symmetric.

      • Wendy Foo says:

        The research is performed on the same sample with different kVp, I want to compare the result of two different kVp, does it mean I need to use Two Paired Samples?

        • Charles says:

          If you have a sample of n subjects and you measure two different kVp values for each subject in the sample, then you don’t have two independent samples (since the two values are derived from the same subject). In this case you need to use a paired sample test (e.g. the paired t test) and not the two independent sample t test.

  33. Dan says:

    Hi, I’m in need of some help.

    I’m looking to test the difference between two datasets for solar panels. One of the datasets is the recorded output in kWh and the other is the modelled output I’ve calculated (in kWh). The datasets are 12 numbers, one for each monthly output. My issue is that the month of the year has a large effect on the output, so is it ok to do a T-Test? Or will I get results that aren’t related to one another but more the effect of the month of the year, as they correlate in its effect. I’d also like to know which T-Test I’d need to choose for this. Any help would be great thanks.

    • Charles says:


      Since you have time-series data, you need to account for potential autocorrelation and seasonality. A t test assumes that each sample is random, which won’t be the case when there is autocorrelation. Instead you may need to use other techniques. I am in the process of showing how to perform such analyses and hope to have this ready in a few weeks.

      You could test for autocorrelation. If there isn’t a significant level of autocorrelation, you could use the standard t test. One such test for autocorrelation (restricted to first-order autocorrelation) is described on the webpage
      Durbin-Watson Test

      The following is a paper which shows how to modify a t test when there is autocorrelation. I have not read this paper, so I won’t comment further on it.


  34. Reayldon says:

    Hi, Charles

    I came across one question which the standard deviations for two variables are 1.22 and 1.64 respectively. Also, the variances for these two are 1.49 and 2.70,

    So which t-test should I use? equal variance or unequal variance?

    • Charles says:

      Generally as long as one variance is not more than 3 or 4 times the other you can use the equal variance test. But when in doubt you should always use the unequal variance test, since if the variances are really equal then the two tests will yield almost the same result. In this case, I would expect the two tests to have very similar p-values.

  35. Sailent Simaremare says:

    Hi Charles, i have difficulty to distinguish the t test,
    when should i use the two sample t-test and paired sample t-test,

    • Charles says:

      Use the two sample t test with two independent samples (e.g. 50 school teachers and 45 students).
      Use the paired sample t test with a sample of paired values where each member of the pair is not independent of the other member of the pair (e.g. 40 students with one score in the morning and the other score in the evening).

  36. sebantee ganguly says:

    ttest result with my datas are like 2.38689E-05, what does it mean ??

    • Charles says:

      The result is in scientific notation. 2.38689E-05 is the same as 2.38689 x 10^(-05), i.e. 0.0000238689, which is a very small number.

      • sebantee ganguly says:

        so, when the result of my t test is like 2.38689x 10^(-05), what should be the inference ? the null hypothesis may be rejected …right ? is the result statistically significant ?

        • Charles says:

          This is a very small number, and so the null hypothesis may be rejected; the result is statistically significant.

          • Jomar Nepomuceno says:

            Isn’t it that when the computed t value is smaller than the critical value, the remark should be not significant?

          • Charles says:

            It depends on which one tail test you are referring to. For a left tail test, If t < t-crit (left) then you have a significant result, while for the right tail t > t-crit (right) yields a significant result.

  37. amit says:

    want to know these things:
    1. I am planning for one intervention :
    I will be adminsterg pre-test to both expermtl grp and contrl group
    then intervention- exptl grp

    then post test to both exp and control group.
    to check normality I will be performing independent t-test Pre-test of exptl and pre-test of control.
    then again post-test of expl and post test of control I want to perform independent t-test.

    also will be performing paired t-test on pre-test and post -test of exptal group.

    similarly will be performing paired t-test on pre-test andpost -test of control group after intervention.

    Also ANOVA after doing incase we get siginfincet difference in 1,2,and 3 varaibles
    then want to go for t -test between 1 and 2
    t-test betwee, 1 and 3
    t test between 2 and 3 ………………..I will be using independent t-test am I correct
    please guide.

    Also I am confused what 2 use a. independent t -test for equal variances
    b.independent t -test for unequal variances
    can I use independent t -test for unequal variances in excel?

    • Charles says:

      Regarding the follow up tests to ANOVA, you need to be careful about the experiment-wise error.You should read the Planned Comparisons and Unplanned Comparisons webpages.

      Regrading t test for independent samples, you can always use the test for unequal variances. Even when the variances are equal the unequal variance test gives pretty good results.


      • Vernika says:

        Hello Sir,

        I am also working on a similar problem as Amit.
        However after drawing the histogram of the variables, it doesnt seem that the variables are normally distributed.

        What should be done ?
        Should I use Welch and unequal variances ?

        • Charles says:

          The t test is pretty robust to violations of normality, and so usually I would just run the t test. If the data is quite different from normal, then I would use a non-parametric test — e.g. Wilcoxon signed-ranks test.
          The situation is similar for ANOVA. If the data departs a lot from normality then you could use Kruskal-Wallis if the homogeneity of variances assumption holds.

          • Vernika says:

            Thankyou Sir.

            My problem is only a post test analysis. I have the data for both the treatment and control group after intervention so is it possible that apart from the ttest, I get the results of my impact variables due to an intervention with the help of regression alaysis also creating a treatment dummy? Can I do such a regression by controlling other factors or I can just compare the means of the variable using the t test ?

          • Charles says:

            Sorry Vernika, but I still don’t understand your question. E.g. I don’t know what you mean by “impact variables”, nor what it means for these variables to be “due to an intervention”, nor why you decided to use “regression analysis” or needed to create “a treatment dummy” (whatever that is).

  38. Tobee says:

    I am trying to write my thesis on the comparison between patient and doctor of response to therapy in psoriatic arthritis. I am setting the doctor’s responses as the control and the overraters and underraters are being compared. I have set the doctor’s responses as control. How would I set up a T-test in excel? I already know it will be a 2 tailed t-test but am unsure the type I should do? For example what am I to enter into excel to get the deciding P-value?

    I already have this in excel:
    =TTEST(array1,array2,2,type?) what would type be according to my case?

    If I could get some help as soon as possible that would be great thanks.

    • Charles says:

      From your description, it sounds like you have three groups: control, overraters and underraters, in which case you would need ANOVA and not just a t test.

      In any case, if you have just two groups and these groups are independent of each other then you can use type = 3. You could use type = 2 if you knew that the variances of the two groups were similar, but there really isn’t much to lose by not assuming this since the p-value of the type = 2 and type = 3 tests would almost be identical in this case.

      If the the two groups were not independent then you would need to use type = 1.


  39. aween says:

    hello, this was really useful but could you please help me i do not know which type to use. i have two groups A and B and both groups did a PRE and POST test. the tests were the same in both groups

    • Charles says:

      I am not sure which types you are referring to. Equal vs unequal variances? single sample vs two independent samples vs paired samples? Please explain.

      • Rajesh says:

        hello sir
        what is difference between equal variance and uneual variance in t-test

        • Charles says:

          Hello Rajesh,
          You use the equal variance version of the t test when the variances of the two samples are reasonably similar and you use the unequal variance version of the test when the variances are not similar. The good news is that you can always use the unequal variance version of the test since when the variances are almost equal the unequal variance version and the equal variance version of the t test produce pretty much the same result.

  40. isaac says:

    hi, good article! but i have a question: when using a two sample t test (independent), when should you assume equal variances?

    • Charles says:

      To be on the safe side you can always use the unequal variances test. If the variances are close to equal the equal variances test will be quite similar to the unequal variances test.

  41. Jessica says:


    The t-stat number was negative at -5.97472 and I conducted a one tailed test so the P is 8.54E-07. I am not sure why the t-stat was negative. The MD was 10.7 between pre and post.

  42. Jessica says:


    I am in graduate school trying to write my research paper. I am struggling with the statistics I must include. I used a quasi experimental one group pretest posttest design. The pretest mean was 72.9 and the SD was 22.019. The posttest mean was 83.6 and the SD was 20.71. I am having trouble deciding which t-test to use? Do I use a one tailed t-test paired two sample for means?

    • Charles says:

      It sounds like you have one sample and got a measurement for each member of the group at two moments in time (pretest and posttest). If this correct, then you should use a paired samples t test (assuming the assumptions for a t test are satisfied).

      • Jessica says:


        I read this on your website…Another approach is to take a sample of 20 people and have each person drink a glass of wine and take a memory test, and then have the same people drink a glass of beer and again take a memory test; finally we compare the results. This is the approach used with paired samples.
        I am just confused by statistics in general. I guess my brain does not think that way.
        I used an online calculator to do the one tailed t-test paired for two sample means and it gave me T value 1.899552, P value 0.031234 and result is significant at p<0.05. I do not understand what this means. Is there any way to put this in every day language?

        • Charles says:


          A lot of people have trouble understanding the basic concept of hypothesis testing. It does take some getting used to.

          In your example, you are trying to test whether the paired samples come from populations whose means are the same (or at least equal enough so that any difference may be due to random processes). The way the testing works is that you assume this (null) hypothesis is true (i.e. the two samples do come from populations with the same mean). The testing shows that (based on the null hypothesis being true) the probability that the sample data exhibit the obtained test statistic of 1.899552 is only 3.1234%, which is not so high. Since this probability is less than 5%, we are doubtful that the null hypothesis is really true, and so we reject this hypothesis and so conclude there is likely to be a significant difference between the means.

          You can get more information on the webpage Hypothesis Testing.


          • Jessica says:

            I appreciate your help. The more I read, the more confused I seem to get. The figures I gave you were from an online calculator but when I put the numbers in Excel and did the t-test, I got different numbers with no explanation about if it is statistically significant or not.
            Again, thank you for trying to help me.

          • Jessica says:


            Here I go again….
            These are the results I got when I did a one tailed t-test paired: two sample for means.
            What does this mean? Why is there a negative number when the posttest scores were higher than the pretest? Is this statistically significant or not?

            t-Test: Paired Two Sample for Means

            Variable 1 Variable 2
            Mean 72.9 83.6
            Variance 501.5413793 443.6965517
            Observations 30 30
            Pearson Correlation 0.899895057
            Hypothesized Mean Difference 0
            df 29
            t Stat -5.974725464
            P(T<=t) one-tail 8.54E-07
            t Critical one-tail 1.699127027
            P(T<=t) two-tail 1.70719E-06
            t Critical two-tail 2.045229642

            I hope you can clarify this for me.

          • Charles says:

            A p-value less than .05 is usually considered to be statistically significant. For the two-tailed test you show a value of 1.70719E-06. This equivalent to the decimal number .00000170719, which is clearly less than .05. This statistically significant.
            I don’t see any negative number, unless you are referring to the exponent in 1.70719E-06. This not a negative number. A negative exponent just moves the decimal point to the right (as described above).

  43. Vu Nguyen says:

    Dear Charles,

    Could you help to interpret the below result from pairs sample t-test for a group of inspector before & after applied kappa statistic:
    . # of inspectors: 23

    t-Test: Paired Two Sample for Means

    Variable 1 Variable 2
    Mean 0.002449451 0.001223162
    Variance 6.37156E-06 8.40796E-07
    Observations 23 23
    Pearson Correlation 0.252600165
    Hypothesized Mean Difference 0
    df 22
    t Stat 2.392373093
    P(T<=t) one-tail 0.012857038
    t Critical one-tail 1.717144374
    P(T<=t) two-tail 0.025714076
    t Critical two-tail 2.073873068

    The result showed that, p value = 0.025<0.05, so we reject the null hypothesis.
    It means that, there is a significant mean different between two period of times (before and after applied kappa) ?

    Please help to advise me to fully interpret the result.

    Best regards,
    Vu Nguyen

    • Charles says:

      Yes, if p value = 0.025<0.05, you would reject the null hypothesis, which means that there is a significant mean different between two period of times (before and after).

  44. Laudy Peters says:

    can you give a formula of T-Test for related measures if the S as X and S1 as Y?? Cause i don’t really understand about this. I’m a student of university and i have a subject about apply statistic.

  45. aparna says:

    I am taking a survey on expected and percieved service levels and then conduct a t test, to know the difference is significant or not. So the participants have to be different for expected and percieved? and do i have to assume equal variance or unequal variance

    Thank you

    • Charles says:

      You can create two samples with different participants. If the sample variances are quite similar then you can use the equal variance test. Otherwise you should use the unequal variance test.

      It is likely (depending on the exact nature of the surveys) that you can create one sample and test expected and perceived service levels with all the participants in the single sample. In this case you would use the paired t test.


  46. debashri boruah says:

    I have 2 different samples from 2 different states. I want to perform a statistical test to detect p-value. In case of comparing the same disease results of two different samples,which test we can use?

    • Charles says:

      If the assumptions for the t test are satisfied you can use the two sample t test. If the variances of the two samples are similar then you can use the t sample t test with equal variances. This test is pretty forgiving about what “similar” means. E.g. generally even if one sample has variance 4 and the other has variance 8, the equal variances test will usually work quite well. If the variances are very different (say one has variance 4 and the other has variance 40) then you should use the two sample t test with unequal variances. When in doubt use the two sample t test with unequal variances.

  47. Ken says:

    Can I use t test if the actual distribution not normal?

    • Charles says:

      The t test assumes that the data is normal, but it is pretty forgiving of violations of normality. The test should work quite well unless the data is very skewed, in which case the Mann-Whitney test may be a better solution.

  48. George Szmn says:

    I just want to make sure that I am doing this correctly.
    N1 126 Salaries for mlb team that gained a playoff spot btwn 2000 and 2014
    N2 324 Salaries for mlb team that didnt gain a spot.
    compared the means using t test non paired gave me statistically differente
    I assumed that variances were equal Ftest 1.54 prob 0.999

    can you correct me if i´m wrong

    • Charles says:

      George Szmn,
      Unless for some strange reason the two samples are highly skewed (not likely to be a problem), the approach you used looks correct.

  49. Melody says:


    How can i prove that the p-value of 2 sample t-test with equal variance is the same as the p-value of a one-way Anova with 2 samples?

    Thank you

  50. Rebecca says:


    I am trying to find out if there is a significant improvement between a groups mental wellbeing scores at 2 different points of time (ie. before intervention and after intervention).

    Rather than having each individuals score to use for calculations, I have the mean score for each question on the questionnaire before intervention and the mean score after intervention.

    For example my data table looks like this

    Question Average Initial Score Average Final Score
    Feeling useful 2.5 4.6
    Feeling relaxed 2.5 4.6
    Feeling loved 2.5 4.6
    Feeling cheerful 2.5 4.6

    Using excel, would I use a t-test, two tailed, and type 1?

    • Charles says:

      You should use the detailed data not the sums or average scores (since you also need to know the sample sizes and variances). This will be a paired samples (aka repeated measures) t test (which if I remember correctly is type 1). Generally you would use a 2 tailed test (unless you are very confident that the scores won’t go down after intervention).

  51. Jess says:


    Are you still able to perform a Ttest when there are different numbers of participants for each component?


  52. aldiz says:

    I also have another question. I want to perform a t test for means, I have a sample (with a mean and variance) and a reference value. The classic formula for the t statistic uses means and variances. I know how to compare it by using the t test formula. But what if my reference value has a standard deviation? Do I use the formula and assume my reference is a sample too and use n = 1 for my “reference sample”? For example, I have a sample with mean 3,4 and variance 1,2, and I want to compare it to a reference value of 3, and I assume that my reference value has a variance of 2…

    • Charles says:

      I don’t completely understand your question, especially since I don’t know what you mean by “reference value”. Based on how you describe the problem, the approach you suggest might be appropriate, although it is more likely that you need to use a sample size for the second “sample” equal to the sample size of the first sample.

      • aldiz says:

        Ok, for example : I have a distribution of 50 temperatures, with the mean and standard deviation. Then, I want to compare it with a reference value. For example, in that time of the day, in a particuliar place the real temperature should be 20 degrees, with a standard deviation of ± 5 degrees. This reference value is not a distribution, I only have one value. In the formula, the result depends on n, but if I take n = 1 for the reference, the test will be very easy to pass, on the other side if I take n = 50 for both, its not really true beceause I only have one value for the reference…

    • Charles says:

      As you can see from Theorem 1 of the referenced webpage, for each sample you need to know the sample mean, sample variance and sample size of each sample to conduct a t test. Without this information, you won’t be able to conduct the test.

  53. aldiz says:

    Hi, I have a specific problem. I have 2 different distributions (samples) i.e. in measure in 2 different conditions. I want to perform a statistical test to detect a mean difference of at least 5 between the 2 samples. My H0 hypothesis is : M1 – M2 = 5, my H1 hypothesis is M1-M2 < 5 (because I want to prove that the difference between the 2 means is less than 5 units). How can I do that with Excel?

    • Charles says:

      You can use Excel’s t test data analysis tool as described on the referenced webpage. Insert 5 in the Hypothesized Mean Difference field and look at the one tail results.

      • aldiz says:

        Ok, I cant have directly the p value or statistic of that in one excel cell (I want the excel sheet to be already ready and I just plug the data)?

    • Charles says:

      If I understand correctly, you want to conduct a one-tailed two independent sample t test. In Excel this can be done using the Two-Sample Assuming Equal Variances data analysis tool. Click on the Data ribbon and choose the Data Analysis tab. Select Two-Sample Assuming Equal Variances data analysis tool (assuming the variance are similar; otherwise choose the Two-Sample Assuming Unequal Variances data analysis tool). Fill in the dialog box with the input ranges and fill in the Hypothesized Mean Difference field with the value 5.

  54. jaynos cortes says:

    Hello. Can anybody help me with my problem? I have a research and initially had 3 treatments, so I proposed ANOVA as my statistical tool in determining the significant difference. But along the experiment it happened that one of the 3 treatments did not respond to my procedure. So, my results have only 2 treatments to determine the difference. Is it applicable to shift to T-test instead of ANOVA since I only have to get the significant difference of two treatments because the third treatment has no variance?

    • Charles says:

      Yes you would use a t-test. You could use ANOVA, but in this case you would not include the treatment with no results. The results will be the same as for the t test, but the t test is probably easier to calculate and interpret. When you report your results, you should explain that you had planned to analyze three treatments and explain why you got no data for this treatment.

  55. Pingback: About Significant Test | Lin Chen's Blog

  56. Kennedy says:

    I typed a comment last evening asking for help trying to decide how to handle a statistics problem. I need to form a hypothesis where two groups of students at a Private Catholic School one group who lives in North Bethesda (120 observations) are said to receive higher grades on the SATs. Those students living in South Bethesda (132 observations) received lower test scores on the SAT. Sample mean is 86 and 87 respectively with 8.1 and 7.3 populations variances and an .01 level of significance. Is there evidence that students living in South Bethesda may be getting lower grades on the SATs?

    • Charles says:


      I sent a response yesterday. I am resending it below:

      Since you have the population variances you can use a two sample test using the normal distribution, as described in Theorem 1 of Comparing Two Means.

      The null hypothesis is mean1 >= mean2 (these are population means). The test statistic is z = (m1-m2)/stdev, where m1-m2 = 88-87 = 1 (sample means) and stdev = sqrt(var) where var = v1^2/n1 + v2^2/n2 = 6.2^2/132 + 7.0^2/125. If NORMSDIST(z) > .99 then you reject the hypothesis that the the workers receive the same pay. This is a one tailed test. If you want a two-tailed test you need to replace .99 by .995.

      If instead of the population variance you had the sample variances you would use Theorem 1 of Two Sample t Test instead.


  57. Nikita says:

    What do you do if the number of cases are not equal? How would you calculate the df then?

    • Charles says:

      The formulas and functions on the referenced webpage don’t require that the number of cases be the same. Just use the formulas and functions shown on the webpage.

  58. Dilshodjon says:

    Hi Charles,
    In the first example, can I know the meanings of data entered in Old and New columns?
    For example, does the greater the number explain the greater the enjoyment?
    What scale could be used?
    Clarification could be helpful.
    Thank you in advance for the post and the answer as well.

    • Charles says:

      Presumably the higher the score the more enjoyable the beverage (although for the t test it really doesn’t matter whether a higher score represents more or less enjoyment). Scale also really doesn’t matter, but let’s assume that each point represents the more enjoyment option in a series of True/False questions.

  59. Swarup says:

    a number of samples 170, in 2012 mean 12366 std dev 3891 and in year 2011 mean 12549 std dev 4232 . corelation-coeff .92879. Can it be concluded that the value of stock holding decreased? how to solve this problem using excel? i mean i am facing trouble to determine std err

    • Charles says:

      You are referencing the webpae regarding two sample t-test with independent samples, but the correlation coeffcient of .92879 shows that these samples are not likely to be independent. Perhaps you are trying to run a two paired samples test — e.g. where the stock holdings are in say 170 different factories, comparing 2011 with 2012. In this case you might be able to use the fact that var(x-y) = var(x) + var(y) – cov(x, y) and cov(x,y) = corr(x,y) * sqrt(var(x))*sqrt(var(y)). This is the best I can do with the information you have provided. I hope that it is helpful.

  60. Noemi says:

    Assuming I have 2 treatments and three trials per treatment. How should I solve the statistical problem of this?

    • Charles says:

      Are you saying that you have two treatments and a sample of 3 for each treatment? You aren’t going to do much with such a small sample, but it is probably best to use the Mann-Whitney test instead of a t-test in this case.

  61. marzieh says:

    Dear Charles
    First , many thanks for sharing such valuable knowledge . I have a question regarding this page . In example 1 , while I understand that the null hypotheis is rejected because of P value of tow-tail is lower than 0.05, I can not understand this sentence that you have added at the end of example :”In fact, the new flavoring is significantly more enjoyable.” I do not know how should I recognize whether the dependent variable in sample one is increased or decreased significantly compared to the sample 2. All my appreciation for any advice as I am stucked in it

    • Charles says:

      Dear Marzieh,
      Before you collect data and conduct the test you don’t know whether the mean of the sample will be in the right tail, left tail or neither. After the test you have evidence as to whether or not the sample mean is in one of the tails and if so which tail. It is on this basis that I drew a conclusion. Of course this conclusion won’t always be correct, but the evidence points in the direction indicated.

  62. Jo G. says:

    I am using a t test to compare before and after weights using a diet. n=78. In excel, I am not sure whether to use 1, 2, or 3, under type in the formula box. Can you explain clearly the differences? Thanks.

    • Charles says:


      You use type = 1 when the two samples are not independent. E.g. (1) when the first sample contains men and the second contains their wives or, as in your case (2) the first sample contains each person’s weight before dieting while the second sample contains their weight after dieting. In example (2) the same person is being sampled and so the samples can’t be independent.

      Type 2 and 3 are used when the two samples are independent. E.g. 20 people are selected at random and half are randomly put in group 1 and half are randomly put in group 2. The difference between type 2 and type 3 relates to the variances of the populations from which the samples are drawn. If the variances are equal then use type = 2, while if the variances are unequal use type = 3. In reality the variance don’t have to be identical to use type = 2. Even if they are close you will usually get good results You usually judge two populations to have equal variances if the two samples have variance that are not too different; in fact even if one sample has a variance which is 4 times the other, the results will be pretty good even if you use type = 2.

      For your situation, it looks like you want to use type = 1.


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