# Two sample t-test: Definition, formula, and Examples

Two sample t-test is also called independent sample t-test or unpaired sample t-test. This tutorial explains what two sample t-test is, its formula, and examples of two sample t-test.

## Definition and Example

Two sample t-test is used to test whether means from two different groups of people or objects are significantly different.

The name of two sample t-test suggests that it involves two separate, independent samples.

You test how how men and women differ in their attitudes towards drinking coffee. Here, two samples of men and women are independent from each other.

AttitudesMen vs. AttitudesWomen

Another example, a restaurant chain has 20 stores in California, and has another 20 stores in New York state. They want to compare whether how the stores in California perform differently from stores in New York.

SalesCalifornia vs. SalesNew York

## Hypothesis for two sample t-test

The null hypothesis is that the underlying population means of two groups are the same:

H012

The alternative hypothesis is that the means are not equal:

H11≠μ2

## Formula of two sample t-test

$$\bar{x_1}$$ and $$\bar{x_2}$$ are the sample means, and $$s_1^2$$ and $$s_2^2$$ are the sample variances.

The following are formulas for two sample t-test, including situations of equal variances and unequal variances.

When we assume the variances of group 1 and group 2 are equal:

$t=\frac{\bar{x_1}-\bar{x_2}}{ \sqrt{s_p^2(1/n_1+1/n_2)}}$

Where,

$$s_p^2=\frac{((n_1-1)s_1^2)+((n_2-1)s_2^2)}{n_1+n_2-2}$$

When the variances of two groups are equal, the degree of freedom for the test statistic is df = n1+n2-2.

When we assume the variances of group 1 and group 2 are unequal:

$t=\frac{\bar{x_1}-\bar{x_2}}{ \sqrt{(s_1^2/n_1+s_2^2/n_2)}}$

When the variances of two groups are unequal, the degree of freedom for the test statistic is $$df=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})^2}{\frac{(s_1^2/n_1)^2}{n_1-1}+\frac{(s_2^2/n_2)^2}{n_2-1}}$$ . .

When n1 = n2

Note that, when n1 = n2 = n, the formula of equal variance and the formula of unequal variance are the same and they will generate the exactly same result for the t statistic. That is,

$t=\frac{\bar{x_1}-\bar{x_2}}{ \sqrt{ \frac{s_1^2+s_2^2}{n}}}$

However, when n1 = n2 = n, the degree of freedom for unequal variances is still not n1+n2-2. Instead, it is $$df=\frac{(n-1)(s_1^2+s_2^2)^2}{(s_1^2)^2+(s_2^2)^2}$$ .

## Manual Calculation Example

Suppose you want to test whether women and men differ in their attitudes toward a brand, and the attitude is measured on a 7-point scale (1= Not like at all, 7 = Like it a lot).

The following is the hypothetical data, one column for men’s attitudes and another one for women’s attitudes toward the brand.

The following is the means for men’s and women’s attitudes.

$$\bar{x_1}=\bar{x}_{men}=\frac{4+6+7+7+6+7}{6}=6.17$$

$$\bar{x_2}=\bar{x}_{women}=\frac{4+3+4+5+2+1}{6}=3.17$$

After knowing the means, we can calculate the sample variances for men and women respectively.

$$s_1^2=s_{men}^2=\frac{(4-6.17)^2+(6-6.17)^2+(7-6.17)^2+(7-6.17)^2+(6-6.17)^2+(7-6.17)^2}{6-1}=1.37$$

$$s_2^2=s_{women}^2=\frac{(4-3.17)^2+(3-3.17)^2+(4-3.17)^2+(5-3.17)^2+(2-3.17)^2+(1-3.17)^2}{6-1}=2.17$$

Then, we can calculate the pooled variance.

$$s_p^2=\frac{((n_1-1)s_1^2)+((n_2-1)s_2^2)}{n_1+n_2-2}=\frac{(6-1)1.37+(6-1)2.17}{6+6-2}=1.77$$

Then, we can get the t statistic.

$$t=\frac{\bar{x_1}-\bar{x_2}}{\sqrt{s_p^2(1/n_1+1/n_2)}}=\frac{6.17-3.17}{\sqrt{1.77(1/6+1/6)}}=3.90935$$

The degree of freedom is 10 if assuming equal variances. The critical t-value for p=0.05 and df=10 is 2.228. Thus, we can reject the null hypothesis. That is, we can conclude that men and women significantly differ in their attitudes. Further, the means suggest that men have significantly more favorable attitudes than women.