Tests whether the mean of one sample differs from a known or hypothesized value — e.g., does average employee engagement differ from the scale midpoint of 3?
Compare GroupsUnivariatealso known as: Single-sample t-test
✓ When to use
You have one group and want to compare its mean against a fixed benchmark (scale midpoint, industry norm, target value).
The outcome is continuous (interval/ratio), such as an averaged multi-item scale score.
Typical uses: is mean satisfaction above the neutral point? Does average service time differ from the 10-minute standard?
✗ When NOT to use
You are comparing two separate groups — use the Independent-Samples t-Test.
The same group is measured twice — use the Paired t-Test.
The outcome is ordinal or clearly non-normal in a small sample — use the Wilcoxon Signed-Rank test against the benchmark.
The benchmark value itself was estimated from the same data — the comparison value must be external and fixed.
Data requirements
Dependent / outcome variable
One continuous variable (interval or ratio).
Independent / grouping variable
None — instead, a fixed comparison value (test value) chosen before analysis.
Design
One group, one measurement per participant; observations independent.
Sample size guidance
n ≥ 30 gives good robustness to non-normality; for smaller samples check normality. G*Power: detecting d = 0.50 at power .80 requires n ≈ 34.
Assumptions
Independence of observations.
The variable is approximately normally distributed (check Shapiro–Wilk, Q–Q plot); robust to moderate violations when n is large.
The variable is continuous; the test value is meaningful on the same scale.
Hypotheses
H₀ — The population mean equals the test value (μ = μ₀).
H₁ — The population mean differs from the test value (μ ≠ μ₀); one-tailed variants if direction was predicted a priori.
The concept
The test divides the difference between the sample mean and the test value by the standard error of the mean (SD/√n), yielding t with n − 1 degrees of freedom. A large |t| means the sample mean is many standard errors away from the benchmark — unlikely if the population mean truly equalled it.
Effect size is Cohen's d = (M − μ₀)/SD, interpreted on the usual 0.20/0.50/0.80 benchmarks. Always report the mean, SD, and the confidence interval of the difference so readers see how far from the benchmark the group actually is.
Worked example
An HR analyst measures employee engagement (7-item scale, 1–5) in a firm (n = 85, M = 3.41, SD = 0.72) and asks whether engagement differs from the neutral midpoint of 3.00.
Result: t(84) = 5.25, p < .001, d = 0.57 — engagement is significantly above the midpoint, a medium effect.
How to run it
# engagement: numeric vector
shapiro.test(df$engagement) # normality check
t.test(df$engagement, mu = 3) # test against midpoint 3
library(effectsize)
cohens_d(df$engagement, mu = 3) # effect size
from scipy import stats
import pingouin as pg
print(stats.shapiro(df["engagement"]))
print(pg.ttest(df["engagement"], 3)) # returns t, p, CI, Cohen's d
Analyze → Compare Means → One-Sample T Test.
Move the variable (e.g., Engagement) into 'Test Variable(s)'.
Enter the benchmark (e.g., 3) in 'Test Value'.
Click OK. Report t, df, Sig. (2-tailed), the mean difference and its 95% CI; SPSS 27+ prints Cohen's d.
Put scores in column A (say A2:A86) and compute =AVERAGE(A2:A86), =STDEV.S(A2:A86), =COUNT(A2:A86).
t statistic: =(mean − 3)/(SD/SQRT(n)) using your cells.
Two-tailed p-value: =T.DIST.2T(ABS(t), n−1).
Cohen's d: =(mean − 3)/SD.
Interpreting the output
t and df (n − 1) — report both.
p < α: the group mean differs significantly from the benchmark; look at the sign of the mean difference for direction.
95% CI of the difference — if it excludes 0, the result is significant; its width shows precision.
Cohen's d — practical size of the departure from the benchmark.
APA-style reporting
A one-sample t-test indicated that employee engagement (M = 3.41, SD = 0.72) was significantly higher than the scale midpoint of 3.00, t(84) = 5.25, p < .001, 95% CI [0.26, 0.57], d = 0.57.
Common mistakes
Choosing the test value after inspecting the data — the benchmark must be set a priori.
Comparing against a midpoint that is not meaningful for the construct (a scale midpoint is not automatically 'neutral').
Reporting significance without the mean, SD, and effect size.
Using it on a single ordinal item rather than a composite score.