One sample t test for the mean overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons 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

One sample $t$ test for the mean	Spearman's rho	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Independent variable	Variable 1
None	One of ordinal level
Dependent variable	Variable 2
One quantitative of interval or ratio level	One of ordinal level
Null hypothesis	Null hypothesis
H₀: $\mu = \mu_0$ Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.	H₀: $\rho_s = 0$ Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H₀: there is no monotonic relationship between the two variables in the population.
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: $\rho_s \neq 0$ H₁ right sided: $\rho_s > 0$ H₁ left sided: $\rho_s < 0$
Assumptions	Assumptions
Scores are normally distributed in the population Sample is a simple random sample from the population. That is, 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 Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.
Test statistic	Test statistic
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$ Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size. 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 $\mu_0$.	$t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.
Sampling distribution of $t$ if H₀ were true	Sampling distribution of $t$ if H₀ were true
$t$ distribution with $N - 1$ degrees of freedom	Approximately the $t$ distribution with $N - 2$ degrees of freedom
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 $t$ 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 $\alpha$
$C\%$ confidence interval for $\mu$	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.	-
Effect size	n.a.
Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$	-
Visual representation	n.a.
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Example context	Example context
Is the average mental health score of office workers different from $\mu_0 = 50$?	Is there a monotonic relationship between physical health and mental health?
SPSS	SPSS
Analyze > Compare Means > One-Sample T Test... Put your variable in the box below Test Variable(s) Fill in the value for $\mu_0$ in the box next to Test Value	Analyze > Correlate > Bivariate... Put your two variables in the box below Variables Under Correlation Coefficients, select Spearman
Jamovi	Jamovi
T-Tests > One Sample T-Test Put your variable in the box below Dependent Variables Under Hypothesis, fill in the value for $\mu_0$ in the box next to Test Value, and select your alternative hypothesis	Regression > Correlation Matrix Put your two variables in the white box at the right Under Correlation Coefficients, select Spearman Under Hypothesis, select your alternative hypothesis
Practice questions	Practice questions

One sample t test for the mean - overview