One-Sample t-Test

Test whether your sample mean differs significantly from a known or hypothesised population value.

The One-Sample t-Test determines whether the mean of a single sample differs significantly from a known or hypothesised population value (μ₀). It is the simplest form of the t-test family: rather than comparing two groups, it asks 'Is my sample mean consistent with a population mean of X?' Common applications include testing whether a class's average score equals the national benchmark, whether a sample's mean blood pressure matches the clinical norm, or whether process output meets a specification target.

The test produces a t-statistic that measures how many standard errors the sample mean is from the hypothesised value. Cohen's d for the one-sample case (the difference between the sample mean and μ₀, divided by the sample SD) quantifies the practical size of the deviation.

StatRyx runs Shapiro-Wilk normality checking, computes the t-statistic, two-tailed p-value, Cohen's d, and 95% confidence interval for the mean, all formatted in APA 7 style.

To interpret the result, look at three numbers: the t statistic and its p-value tell you whether the sample mean differs from the benchmark by more than chance would explain; the mean difference with its 95% confidence interval tells you by how much; and Cohen's d (mean difference divided by the SD) tells you whether that difference is small (0.2), medium (0.5) or large (0.8). A significant result with a tiny d usually means a large sample detected a trivial gap.

When to use the One-Sample t-Test

You have one continuous variable and want to test whether its mean equals a specific population value or benchmark
You are comparing a sample against a published norm, regulatory standard, or historical baseline
Your data are approximately normally distributed (or n > 30)
You want to check whether a scale's average differs from its neutral midpoint (e.g., 3 on a 1–5 Likert item)
You need a confidence interval for how far your sample mean sits from a target value

When not to use it

You are comparing two independent groups — use Independent Samples t-Test instead
You are comparing paired before/after measurements — use Paired Samples t-Test instead
Data are severely non-normal and n is small — use Wilcoxon Signed-Rank Test instead

Worked example

n	M	SD	μ₀	t	df	p	d
60	74.3	9.8	70	3.40	59	.001	0.44

How to report it (APA 7)

A one-sample t-test indicated that the sample mean (M = 74.3, SD = 9.8) was significantly higher than the national benchmark of 70, t(59) = 3.40, p = .001, d = 0.44, 95% CI [71.8, 76.8].

Frequently asked questions

Is the one-sample t-test free on StatRyx?

Yes. You can run it without an account (3 free analyses), and a free account adds 10 more — a one-time allowance, no payment needed. Every result includes the APA 7 write-up. Paid plans (Student from $9/month) add more analyses and Word, PDF and Excel export.

What if I don't know the population mean?

You specify the hypothesised value (μ₀) yourself — it could be a published norm, a regulatory threshold, or a theoretical value like zero.

What sample size do I need?

For detecting a medium effect (d = 0.5) at 80% power and α = .05, you need approximately 34 participants. With n > 30 the test is robust to moderate non-normality.

How do I report a one-sample t-test in APA 7?

Give the sample mean and SD, the test value, then t(df), the exact p-value and Cohen's d: "Scores (M = 74.3, SD = 9.8) were higher than the benchmark of 70, t(59) = 3.40, p = .001, d = 0.44." Degrees of freedom are n − 1.

Can I compare a Likert scale mean to the scale midpoint?

Yes — that is one of the most common uses. Compute the scale score per respondent first (the mean of its items), then test it against the midpoint. For a single ordinal item with few response options, the Wilcoxon signed-rank test against the midpoint is the safer choice.

Run a One-Sample t-Test on your data →