Two sample $t$ test - equal variances assumed

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Contents

When to use?

Deciding which statistical method to use to analyze your data can be a challenging task. Whether a statistical method is appropriate for your data is partly determined by the measurement level of your variables. The two sample $t$ test - equal variances assumed requires the following variable types:

Variable types required for the two sample $t$ test - equal variances assumed :
Independent/grouping variable:
One categorical with 2 independent groups
Dependent variable:
One quantitative of interval or ratio level

Note that theoretically, it is always possible to 'downgrade' the measurement level of a variable. For instance, a test that can be performed on a variable of ordinal measurement level can also be performed on a variable of interval measurement level, in which case the interval variable is downgraded to an ordinal variable. However, downgrading the measurement level of variables is generally a bad idea since it means you are throwing away important information in your data (an exception is the downgrade from ratio to interval level, which is generally irrelevant in data analysis).

If you are not sure which method you should use, you might like the assistance of our method selection tool or our method selection table.

Null hypothesis

The two sample $t$ test - equal variances assumed tests the following null hypothesis (H0):

H0: $\mu_1 = \mu_2$

$\mu_1$ is the population mean for group 1, $\mu_2$ is the population mean for group 2
Alternative hypothesis

The two sample $t$ test - equal variances assumed tests the above null hypothesis against the following alternative hypothesis (H1 or Ha):

H1 two sided: $\mu_1 \neq \mu_2$
H1 right sided: $\mu_1 > \mu_2$
H1 left sided: $\mu_1 < \mu_2$
Assumptions

Statistical tests always make assumptions about the sampling procedure that's been used to obtain the sample data. So called parametric tests also make assumptions about how data are distributed in the population. Non-parametric tests are more 'robust' and make no or less strict assumptions about population distributions, but are generally less powerful. Violation of assumptions may render the outcome of statistical tests useless, although violation of some assumptions (e.g. independence assumptions) are generally more problematic than violation of other assumptions (e.g. normality assumptions in combination with large samples).

The two sample $t$ test - equal variances assumed makes the following assumptions:

Test statistic

The two sample $t$ test - equal variances assumed is based on the following test statistic:

$t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
$\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis.

The denominator $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ is removed from 0.

Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.
Pooled standard deviation

$s_p = \sqrt{\dfrac{(n_1 - 1) \times s^2_1 + (n_2 - 1) \times s^2_2}{n_1 + n_2 - 2}}$
Sampling distribution

Sampling distribution of $t$ if H0 were true:

$t$ distribution with $n_1 + n_2 - 2$ degrees of freedom
Significant?

This is how you find out if your test result is significant:

Two sided: Right sided: Left sided:
$C\%$ confidence interval for $\mu_1 - \mu_2$

$(\bar{y}_1 - \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2 - 2}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20)

The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test.
Effect size

Cohen's $d$:
Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1 - \bar{y}_2}{s_p}$$ Indicates how many standard deviations $s_p$ the two sample means are removed from each other
Visual representation

Two sample t test - equal variances assumed
Equivalent to

The two sample $t$ test - equal variances assumed is equivalent to:

One way ANOVA with an independent variable with 2 levels ($I$ = 2):
OLS regression with one categorical independent variable with 2 levels:
Example context

The two sample $t$ test - equal variances assumed could for instance be used to answer the question:

Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women.
SPSS

How to perform the two sample $t$ test - equal variances assumed in SPSS:

Analyze > Compare Means > Independent-Samples T Test...
Jamovi

How to perform the two sample $t$ test - equal variances assumed in jamovi:

T-Tests > Independent Samples T-Test