Paired sample t test overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

Paired sample $t$ test	$z$ test for the difference between two proportions	Marginal Homogeneity test / Stuart-Maxwell test
Independent variable	Independent/grouping variable	Independent variable
2 paired groups	One categorical with 2 independent groups	2 paired groups
Dependent variable	Dependent variable	Dependent variable
One quantitative of interval or ratio level	One categorical with 2 independent groups	One categorical with $J$ independent groups ($J \geqslant 2$)
Null hypothesis	Null hypothesis	Null hypothesis
H₀: $\mu = \mu_0$ Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.	H₀: $\pi_1 = \pi_2$ Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2.	H₀: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group. Here $\pi_j$ is the population proportion in category $j.$
Alternative hypothesis	Alternative hypothesis	Alternative hypothesis
H₁ two sided: $\mu \neq \mu_0$ H₁ right sided: $\mu > \mu_0$ H₁ left sided: $\mu < \mu_0$	H₁ two sided: $\pi_1 \neq \pi_2$ H₁ right sided: $\pi_1 > \pi_2$ H₁ left sided: $\pi_1 < \pi_2$	H₁: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.
Assumptions	Assumptions	Assumptions
Difference scores are normally distributed in the population Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another	Sample size is large enough for $z$ to be approximately normally distributed. Rule of thumb: Significance test: number of successes and number of failures are each 5 or more in both sample groups Regular (large sample) 90%, 95%, or 99% confidence interval: number of successes and number of failures are each 10 or more in both sample groups Plus four 90%, 95%, or 99% confidence interval: sample sizes of both groups are 5 or more Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Test statistic	Test statistic	Test statistic
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$ Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.	$z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$ Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$	Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.
Sampling distribution of $t$ if H₀ were true	Sampling distribution of $z$ if H₀ were true	Sampling distribution of the test statistic if H₀ were true
$t$ distribution with $N - 1$ degrees of freedom	Approximately the standard normal distribution	Approximately the chi-squared distribution with $J - 1$ degrees of freedom
Significant?	Significant?	Significant?
Two sided: Check if $t$ observed in sample is at least as extreme as critical value $t^$ or Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $t$ observed in sample is equal to or larger than critical value $t^$ or Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$	Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$	If we denote the test statistic as $X^2$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
$C\%$ confidence interval for $\mu$	Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$	n.a.
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$ where the critical value $t^$ is the value under the $t_{N-1}$ 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 $\mu$ can also be used as significance test.	Regular (large sample): $(p_1 - p_2) \pm z^* \times \sqrt{\dfrac{p_1(1 - p_1)}{n_1} + \dfrac{p_2(1 - p_2)}{n_2}}$ where the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval) With plus four method: $(p_{1.plus} - p_{2.plus}) \pm z^* \times \sqrt{\dfrac{p_{1.plus}(1 - p_{1.plus})}{n_1 + 2} + \dfrac{p_{2.plus}(1 - p_{2.plus})}{n_2 + 2}}$ where $p_{1.plus} = \dfrac{X_1 + 1}{n_1 + 2}$, $p_{2.plus} = \dfrac{X_2 + 1}{n_2 + 2}$, and the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval)	-
Effect size	n.a.	n.a.
Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$	-	-
Visual representation	n.a.	n.a.
	-	-
Equivalent to	Equivalent to	n.a.
One sample $t$ test on the difference scores. Repeated measures ANOVA with one dichotomous within subjects factor.	When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels.	-
Example context	Example context	Example context
Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?	Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.	Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?
SPSS	SPSS	SPSS
Analyze > Compare Means > Paired-Samples T Test... Put the two paired variables in the boxes below Variable 1 and Variable 2	SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to: Analyze > Descriptive Statistics > Crosstabs... Put your independent (grouping) variable in the box below Row(s), and your dependent variable in the box below Column(s) Click the Statistics... button, and click on the square in front of Chi-square Continue and click OK	Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the Marginal Homogeneity test
Jamovi	Jamovi	n.a.
T-Tests > Paired Samples T-Test Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line Under Hypothesis, select your alternative hypothesis	Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to: Frequencies > Independent Samples - $\chi^2$ test of association Put your independent (grouping) variable in the box below Rows, and your dependent variable in the box below Columns	-
Practice questions	Practice questions	Practice questions

Paired sample t test - overview