The Open Educator - 9. Two Sample T-Test Unequal Variance (2023)

Comparing two Samples/Populations/Groups/Means/Values

Two-sample T-Test with unequal variance can be applied when (1) the samples are normally distributed, (2) the standard deviation of both populations are unknown and assume to be unequal, and the (3) sample is sufficiently large (over 30).

To compare the height of two male populations from the United States and Sweden, a sample of 30 males from each country is randomly selected and the measured heights are provided in Table 3.

(Video) Can I do a t-Test with Unequal Sample Sizes? – The Unbalanced Research Design

Table 6. Height data for US and Swedish male samples

The Open Educator - 9. Two Sample T-Test Unequal Variance (1)

The Open Educator - 9. Two Sample T-Test Unequal Variance (2)

As the population standard deviation is unknown, the data is assumed to be normally distributed and the sample size is large enough, the two-sample T-Test can be applied to analyze the data. The test statistics is calculated as in Equation 5.

(Video) Two Sample T Test Application, Data Collection, Analysis, Results Explained Using MS Excel & Minitab

The Open Educator - 9. Two Sample T-Test Unequal Variance (3)

Equation 5

[Any statistical software, including MS excel can perform the two-sample T-Tests. Therefore, the equation is only for reference only. Analysis output will be produced using software.]

MS Excel can be used for performing a two-sample T-Test.

Analysis using MS excel is provided in Figure 9.

The Open Educator - 9. Two Sample T-Test Unequal Variance (4)

Figure 9. Two Sample T-Test Equal Variance Analysis Results Using MS Excel

(Video) Two-Sample t Test in R (Independent Groups) with Example | R Tutorial 4.2 | MarinStatsLectures

Statistical Interpretation of the Results

We reject the null hypothesis because the p-value (0.0127) is smaller than the level of significance (0.05). [p-value is the observed probability of the null hypothesis to happen, which is calculated from the sample data using an appropriate method, two-sample T-Test for equal variance in this case]

Statistically, US male and Swedish male populations are significantly different with respect to the height. [rewrite the accepted hypothesis for an eighth grader without using the statistical jargon such as the p-value, level of significance, etc.]

The next question would be then who is taller or shorter. Both the sample and the population data show that the Swedish male population is taller than the US male population. However, the alternative hypothesis was written as “Not Equal.” Therefore, to test that the Swedish male population is taller than the US male population or the US male population is shorter than the Swedish male population. The hypothesis is written as below.

The Open Educator - 9. Two Sample T-Test Unequal Variance (5)

Now the alternative hypothesis become one-sided. As the one-sided probability is the half of the two-sided probability (p-value), we would still reject the null hypothesis. The new contextual conclusion would be “Statistically, the US male population is significantly shorter than the Swedish male population.” However, making this contextual conclusion for the original “not equal” alternative hypothesis would be wrong………….A common mistake.

(Video) T test Independent Unequal variances

(Video) How To... Perform an Independent Samples (Equal Variance) t Test in R #86

Paired T-Test (Matched Pair/Repeated Measure)


What is the two sample t test with unequal variance? ›

Two-sample T-Test with unequal variance can be applied when (1) the samples are normally distributed, (2) the standard deviation of both populations are unknown and assume to be unequal, and the (3) sample is sufficiently large (over 30).

What is the t-test statistic for unequal variances? ›

The test statistic for the unequal variance t-test (t′) is actually slightly simpler than that of the Student's t-test: t′=μ1−μ2√s21n1+s22n2. u=s22s21. In general, v calculated from Equation 4 will take a noninteger value; it is conventional to round down to the nearest integer before consulting standard t tables.

Can you do two sample t tests with unequal sample sizes? ›

The short answer: Yes, you can perform a t-test when the sample sizes are not equal. Equal sample sizes is not one of the assumptions made in a t-test. The real issues arise when the two samples do not have equal variances, which is one of the assumptions made in a t-test.

What does equal or unequal variance mean in t-test? ›

If the variances are equal then the equal and unequal variances versions of the t-test will yield similar results (even when the sample sizes are unequal), although the equal variances version will have slightly better statistical power.

What does a 2 sample t-test determine? ›

The two-sample t-test (Snedecor and Cochran, 1989) is used to determine if two population means are equal. A common application is to test if a new process or treatment is superior to a current process or treatment. There are several variations on this test. The data may either be paired or not paired.

What are the 2 types of two sample t-tests? ›

Dependent vs Independent Samples

There are two types of two-sample t-tests: dependent and independent. In an independent two-sample t-test (also known as an unpaired t-test), the samples in the two groups being compared are unrelated.

How do you know if data has equal or unequal variance? ›

An F-test (Snedecor and Cochran, 1983) is used to test if the variances of two populations are equal. This test can be a two-tailed test or a one-tailed test. The two-tailed version tests against the alternative that the variances are not equal.

What does unequal variance mean in statistics? ›

The conservative choice is to use the "Unequal Variances" column, meaning that the data sets are not pooled. This doesn't require you to make assumptions that you can't really be sure of, and it almost never makes much of a change in your results.

How do you know if data has unequal variance? ›

If the p-value that corresponds to the test statistic is less than some significance level (like 0.05), then we have sufficient evidence to say that the samples do not have equal variances.

Does unequal sample size mean unequal variance? ›

Unequal sample sizes can lead to: Unequal variances between samples, which affects the assumption of equal variances in tests like ANOVA. Having both unequal sample sizes and variances dramatically affects statistical power and Type I error rates (Rusticus & Lovato, 2014). A general loss of power.

Should I use equal or unequal variance t-test? ›

The Two-Sample assuming Equal Variances test is used when you know (either through the question or you have analyzed the variance in the data) that the variances are the same. The Two-Sample assuming UNequal Variances test is used when either: You know the variances are not the same.

How do you compare two samples with different sizes? ›

Use a permuation test.

Randomly shuffle the values between the two groups, maintaining the original sample size. What fraction of those shuffled data sets have a difference between means as large (or larger) than observed. That is the P value.

What is the t-test assumption of equal variance? ›

Equal Variance Assumption in t-tests

A two sample t-test is used to test whether or not the means of two populations are equal. The test makes the assumption that the variances are equal between the two groups.

How do you calculate t-test with unequal variances in Excel? ›

From the menu bar, select Tools > Data Analysis > t-test > Two Sample Assuming Unequal Variances. Type in the input range for each of the diets into the dialogue box provided, then select the output range and click OK. Make sure the labels box is checked if you include the column headings (preferred).

What does equality of variance tell us? ›

If the variances of two random variables are equal, that means on average, the values it can take, are spread out equally from their respective means.

What is the t-value in a 2 sample t-test? ›

The 2-sample t-test takes your sample data from two groups and boils it down to the t-value. The process is very similar to the 1-sample t-test, and you can still use the analogy of the signal-to-noise ratio. Unlike the paired t-test, the 2-sample t-test requires independent groups for each sample.

How do you interpret t-test results? ›

Interpreting the results isn't very complicated. All you have to do is compare the p-value to an alpha significance level. If the value turns out to be smaller than the alpha level, then you can safely reject the hypothesis. In this scenario, since the alternative hypothesis will be true, the data will be significant.

What does the t-test test tell you? ›

The t test estimates the true difference between two group means using the ratio of the difference in group means over the pooled standard error of both groups. You can calculate it manually using a formula, or use statistical analysis software.

What is the difference between 2 sample t-test and paired t-test? ›

Two-sample t-test is used when the data of two samples are statistically independent, while the paired t-test is used when data is in the form of matched pairs.

What does the t-test for the difference of 2 means test? ›

Formula: where and are the means of the two samples, Δ is the hypothesized difference between the population means (0 if testing for equal means), s 1 and s 2are the standard deviations of the two samples, and n 1and n 2are the sizes of the two samples.

What are the differences between the two types of t tests? ›

The difference between paired and unpaired t-tests depends on the nature of the variable if we are comparing two related variables paired t-test is used, while for comparing unrelated variables or independent variables unpaired t-test is used.

How do you compare variance between two groups? ›

In order to compare multiple groups at once, we can look at the ANOVA, or Analysis of Variance. Unlike the t-test, it compares the variance within each sample relative to the variance between the samples.

How do you test for unequal variance in R? ›

1. F-test in R. The F test statistic can be obtained by calculating the ratio of the two variances. In the F test, the ratio deviates more from 1 then stronger the evidence of unequal variances.

What does it mean if one variance is higher than the other? ›

A small variance indicates that the data points tend to be very close to the mean, and to each other. A high variance indicates that the data points are very spread out from the mean, and from one another. Variance is the average of the squared distances from each point to the mean.

What does variance in t-test mean? ›

t-Test with unequal variances

It is the difference between the averages of the two groups. The denominator is an estimate of the overall standard error of the difference between means. It is based on the separate standard error for each group.

Why is it important to have equal variances? ›

In short, homogeneity of variance is key because otherwise you just don't know if the independent variables you have selected within your multiple regression model are statistically significant.

How do you determine which variance is greater? ›

It is calculated by taking the average of squared deviations from the mean. Variance tells you the degree of spread in your data set. The more spread the data, the larger the variance is in relation to the mean.

Which test is preferred when sample sizes and variances are unequal between groups? ›

Welch's t-test also known as unequal variances t-test is used when you want to test whether the means of two population are equal. This test is generally applied when the there is a difference between the variations of two populations and also when their sample sizes are unequal.

Which test is preferred when sample sizes and variances are unequal between groups *? ›

Take home message of this post: We should use Welch's t-test by default, instead of Student's t-test, because Welch's t-test performs better than Student's t-test whenever sample sizes and variances are unequal between groups, and gives the same result when sample sizes and variances are equal.

How does variance change with sample size? ›

As a sample size increases, sample variance (variation between observations) increases but the variance of the sample mean (standard error) decreases and hence precision increases.

When can the T variance be used? ›

The t-distribution is used when data are approximately normally distributed, which means the data follow a bell shape but the population variance is unknown. The variance in a t-distribution is estimated based on the degrees of freedom of the data set (total number of observations minus 1).

Can you use ANOVA with unequal variances? ›

Unfortunately, simulation studies find that this assumption is a strict requirement. If your groups have unequal variances, your results can be incorrect if you use the classic test. On the other hand, Welch's ANOVA isn't sensitive to unequal variances.

Why should you use an analysis of variance rather than a series of t-tests? ›

The Student's t test is used to compare the means between two groups, whereas ANOVA is used to compare the means among three or more groups.

Is it better statistically to have unequal sample sizes or equal sample sizes? ›

It can be shown that the greater the differences in sample sizes between the groups, the lower the statistical power of an ANOVA. This is why researchers typically want equal sample sizes so that they have higher power and thus a greater probability of detecting true differences.

Can two groups have the same mean but different variance? ›

Answer and Explanation: Yes, two sets of data can have the same mean but not the same variance. It implies that the center value of the two sets is the same, but the spread or the dispersion of the data values around that center value is different.

Why does the sample size have to be greater than 2 samples? ›

Larger samples more closely approximate the population. Because the primary goal of inferential statistics is to generalize from a sample to a population, it is less of an inference if the sample size is large.

What is the t-test formula in Excel? ›


The formula uses the following arguments: Array1 (It is a required argument) – The first data set. Array2 (It is a required argument) – The second data set. Tails (It is a required argument) – Specifies if it is a one-tailed or two-tailed test.

What Excel tool is used for a two sample test for equality of variances? ›

In Excel, click Data Analysis on the Data tab. From the Data Analysis popup, choose t-Test: Two-Sample Assuming Equal Variances.

How do I use unequal in Excel? ›

Excel's "does not equal" operator is simple: a pair of brackets pointing away from each other, like so: "<>". Whenever Excel sees this symbol in your formulas, it will assess whether the two statements on opposite sides of these brackets are equal to one another.

What does it mean if variance is significant? ›

Essentially, if the “between” variance is much larger than the “within” variance, the factor is considered statistically significant. Recall, ANOVA seeks to determine a difference in means at each level of a factor. If the factor level impacts the mean, then that factor is statistically significant.

Do you assume equal or unequal variances for variants in t tests? ›

As a rule of thumb, if the ratio of the larger variance to the smaller variance is less than 4 then we can assume the variances are approximately equal and use the Student's t-test.

What is the variance of the difference in the two sample means? ›

When the population variances are known, the difference of the means has a normal distribution. The variance of the difference is the sum of the variances divided by the sample sizes.

How do you determine if variances are equal or unequal in Excel? ›

(The test for equality of variances is an F-test.) In Excel, select Tools/ Data Analysis / F-Test Two Sample for Variance. In the F-Test Two Sample for Variance dialog box: For the Input Range for Variable 1, highlight the seven values of Score in group 1 (values from 20 to 27.5).

How can you check the assumption of equal variances? ›

Levene's test ( Levene 1960) is used to test if k samples have equal variances. Equal variances across samples is called homogeneity of variance. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples. The Levene test can be used to verify that assumption.

How to test if two sample means are significantly different? ›

A t-test is an inferential statistic used to determine if there is a significant difference between the means of two groups and how they are related. T-tests are used when the data sets follow a normal distribution and have unknown variances, like the data set recorded from flipping a coin 100 times.

Why can unequal sample sizes be a problem for ANOVA? ›

The main practical issue in one-way ANOVA is that unequal sample sizes affect the robustness of the equal variance assumption. ANOVA is considered robust to moderate departures from this assumption. But that's not true when the sample sizes are very different.

Why is unequal n usually a bad thing in an ANOVA? ›

The problem with unequal n is that it causes confounding. The difference between weighted and unweighted means is a difference critical for understanding how to deal with the confounding resulting from unequal n. Weighted and unweighted means will be explained using the data shown in Table 4.

How do you know if an ANOVA is balanced or unbalanced? ›

An ANOVA has a balanced design if the sample sizes are equal across all treatment combinations. Conversely, an ANOVA has an unbalanced design if the sample sizes are not equal across all treatment combinations.


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