One Sample t Test Calculator, Free with Steps | StatsUnlock

A one sample t test calculator tests whether the mean of one sample differs from a known or hypothesized value μ₀, using t = (x̄ − μ₀) ÷ (s ÷ √n) with n − 1 degrees of freedom.

For 16 leaves with mean 52.44 mm and SD 4.19 mm tested against 50 mm, t(15) = 2.32 and p = 0.035, so the mean is significantly above 50 mm. Cohen's d is 0.58, a medium effect.

Rule of thumb: reject H₀ when p is below α (usually 0.05), which happens when |t| is larger than the critical value, about 2.13 at df = 15 and 2.04 at df = 30. Always report the 95% confidence interval and Cohen's d beside p.

A one sample t test tests whether the mean of a single continuous variable differs significantly from a known or hypothesized population mean. It assumes normality and requires only one group of independent observations. You use it when you have one set of measurements and one number to compare them with.

What it measures. The test asks a simple question: is the gap between your sample mean and the target value bigger than random sampling would usually produce? It answers by dividing that gap by the standard error of the mean. The standard error shrinks as n grows, so the same gap becomes more convincing with more data. The result, t, counts how many standard errors the sample mean sits away from μ₀.

A worked one-liner: 16 leaves average 52.4375 mm with SD 4.1947 mm. The standard error is 4.1947 ÷ 4 = 1.0487 mm. The gap from 50 mm is 2.4375 mm, so t = 2.4375 ÷ 1.0487 = 2.32 on 15 degrees of freedom, which gives p = 0.035.

Who uses it. Quality engineers test whether a production line still meets a label claim. Clinicians compare a patient group with a reference value such as 120 mmHg. Teachers compare a class with a national average. Ecologists test whether stream pH departs from neutral, or whether a population's mean body mass differs from a published value. In every case there is one sample and one benchmark.

A one sample t test vs a two sample t test. The one sample t test has one group and a fixed number. A two sample t test has two groups and compares their means with each other. A paired t test is a one sample t test run on the differences between paired measurements, tested against 0.

4.6 How to interpret one sample t test results, in order

Read the output in this order, and you will not be fooled by any single number. First look at the mean difference, x̄ − μ₀, in real units. That is the actual finding. Then look at the confidence interval for the mean. If μ₀ lies outside the interval, the two-tailed test is significant at the matching α. Then read t and p, which tell you how surprising the difference would be if H₀ were true. Finally, read Cohen's d, which says whether the difference is large enough to matter.

A good one-sentence reading of any result has all four parts: the direction, the size in units, the evidence, and the effect size. For the leaf example: "Leaves were 2.44 mm longer than the published 50 mm on average (95% CI 0.20 to 4.67 mm above), t(15) = 2.32, p = 0.035, d = 0.58."

4.7 What the p-value does and does not mean

The p-value is the probability of getting a t at least this far from zero if the true mean were exactly μ₀. A p of 0.035 means a gap this large would turn up in about 3.5% of samples drawn from a population whose mean really is 50 mm. It is not the probability that H₀ is true, and it is not the probability that your result is due to chance. Those two misreadings are the most common errors in published research (Greenland et al., 2016).

A p just under 0.05 is weak evidence, not proof. A p of 0.049 and a p of 0.051 describe almost identical data. Treat p as a continuous measure of evidence, and let the interval and the effect size carry the story (Wasserstein & Lazar, 2016).

4.8 Statistical significance vs practical significance

With a big enough sample, a trivially small difference becomes significant. In Example 6 below, 2,500 test scores average 100.75 against a norm of 100. The test gives p = 0.012, but d = 0.05, a negligible effect that no teacher would notice. The reverse also happens: a small pilot study can show a large, important difference that is not significant only because n is small. Always ask two questions: is it real (p and the CI), and does it matter (d and the difference in units)?

4.9 Effect size bands for Cohen's d

Cohen's d for a one sample t test is (x̄ − μ₀) ÷ s. Cohen's (1992) bands are a starting point: about 0.2 is small, 0.5 medium, 0.8 large. They are not laws. In clinical blood pressure work a d of 0.3 can be important, while in a manufacturing process a d of 0.3 may be well inside tolerance. Hedges' g applies a small-sample correction and is slightly smaller than d when n is small. The calculator gives both, and a 95% interval for d based on the noncentral t distribution, so readers can see how precisely the effect is known.

4.10 One-tailed or two-tailed?

A two-tailed test asks whether the mean differs from μ₀ in either direction. A one-tailed test asks only about one direction, such as "greater than". For the same data, the one-tailed p is half the two-tailed p when the effect is in the predicted direction. That is exactly why a one-tailed test must be chosen before you see the data. Choosing the tail after looking doubles your real false-positive rate. If you cannot defend in advance why the other direction is impossible or irrelevant, use two tails.

4.11 Normality and robustness

The one sample t test assumes the population is roughly normal. In practice it is fairly robust: with 30 or more observations, mild skew barely changes the p-value, because the sampling distribution of the mean becomes close to normal (Lumley et al., 2002). Two problems still hurt at any size: heavy outliers and strong skew in small samples. That is why the calculator runs the Shapiro-Wilk test, reports skewness and flags outliers, and also runs a Wilcoxon signed-rank test as a check. If the t test and the Wilcoxon test disagree, trust neither blindly, and look at chart 3 and chart 4.

4.12 Power and sample size

Power is the chance the test detects a real difference of a given size. The calculator reports the power of your design to detect an effect as large as the one observed, and the n needed to reach 80% power for that effect. Treat the first number with care: power computed from the observed effect adds nothing to the p-value (a significant result always has observed power near or above 50%). The second number is more useful, because it tells you how big the next study should be. For planning, about 34 observations give 80% power to detect d = 0.5 at α = 0.05, two-tailed.

4.13 Several groups against one value

If you test several groups against the same μ₀, as in the stream pH sample with three sites, each test has its own 5% false-positive risk. With three tests, the chance of at least one false positive rises to about 14%. The calculator adds a Holm-adjusted p-value for each group so you can control this. Testing each group against μ₀ is also not the same as comparing the groups with each other. For that, use a one-way ANOVA.

4.14 Decision guide

4.15 Common mistakes this calculator prevents

• Using the population SD. The calculator always uses the sample SD with n − 1, the same as Excel's STDEV.S and R's sd().
• Wrong degrees of freedom. df is always n − 1 for a one sample t test, never n.
• Reporting p without a CI or effect size. Every run returns the interval for the mean, the interval for the difference and Cohen's d with its own interval.
• Halving p by hand. One-tailed p-values are computed directly from the t distribution in the right direction, so a result in the wrong direction is never reported as significant.
• Ignoring skew and outliers. Normality, skewness, outliers and a Wilcoxon cross-check are reported on every run.
• Silent data errors. Text, blanks and stray symbols are flagged, and the count n is shown so you can confirm what was used.

When reporting one sample t test results in APA format, give the test value, the mean and SD, t with its degrees of freedom, the exact p-value, the confidence interval and an effect size. The five templates below fill themselves with your numbers after you press Run.

▶ Run the analysis above to auto-fill all five examples with your results.

5.1 APA 7th edition results sentence

5.2 Thesis or dissertation paragraph

5.3 Plain-language summary

5.4 Structured abstract or conference poster

5.5 Replication or pre-registration statement

5.6 How to write up a one sample t test: tables, figures and the discussion

Methods wording. Name the test, the test value and where it came from, the tail, α, and how assumptions were checked. Example: "Mean leaf length was compared with the published species mean of 50 mm using a two-tailed one-sample t-test (α = .05). Normality was assessed with the Shapiro-Wilk test."

Tables. A results table for a one sample t test usually has the columns n, M, SD, μ₀, mean difference, 95% CI, t, df, p and d. Use one row per variable or group. Keep decimals consistent down each column.

Figures. The clearest figure is chart 1: the raw points, the mean with its CI as an error bar, and a dashed line at μ₀. State in the caption that the error bar is a 95% confidence interval, not an SD.

Discussion sentence. Interpret the size, not just the significance: "Leaves in this population were about 2.4 mm (5%) longer than the published mean, a medium-sized difference that may reflect the wetter growing season."

5.7 Reporting checklist

• ✓ Test value μ₀ and its source
• ✓ n, mean and SD
• ✓ t, df and exact p, with the tail stated
• ✓ 95% CI for the mean or the mean difference
• ✓ Cohen's d (or Hedges' g for small n)
• ✓ How normality and outliers were checked

These are the one sample t test formula and every formula behind the results above. Each one uses the leaf example: n = 16, x̄ = 52.4375 mm, s = 4.1947 mm, μ₀ = 50 mm.

This flowchart shows how to calculate a one sample t test, step by step, from a raw column of numbers to a decision. Every box uses the same worked example as the Excel, R and Python sections: 16 leaf lengths in mm, tested against the published species mean of 50 mm.

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return t = 2.3243, df = 15 and p = 0.0346 for this dataset.

1. Enter your data. The default tab takes comma-separated numbers, exactly as the placeholder shows: 52, 48, 55, 61, 47, .... Spaces, tabs, semicolons and new lines also work, so a column pasted from Excel is fine.
2. Name the group. The group name above each box is editable. Type something meaningful, such as Leaf length, site 2, and it appears in the tables, charts and reports.
3. Add more groups if needed. Press Add group to test several groups against the same value. Each group gets its own test and a Holm-adjusted p-value.
4. Or upload a file. On the Upload tab, choose a CSV or Excel file and click the columns you want. Each column you click becomes its own group.
5. Or use summary statistics. If you only have n, the mean and the SD, open the Summary Statistics tab and type them in.
6. Set the hypothesized mean μ₀. This is the benchmark: a norm, a label claim, a target or a published mean.
7. Choose the alternative hypothesis and α. Use two-tailed unless you decided on a direction before collecting data.
8. Press Run One Sample t Test. Nothing is computed until you do. Changing a setting clears old results, so you never read stale numbers.
9. Read the results in order. Mean difference first, then the confidence interval, then t and p, then Cohen's d and the assumption checks.
10. Copy and export. Section 5 fills five reporting templates with your numbers, and the buttons download a text report or a print-ready PDF.

Excel has no single function called "one sample t test", and that is what trips most people up. T.TEST only compares two ranges. You can still do a one sample t test in Excel in two ways: build it from AVERAGE, STDEV.S and T.DIST.2T, which shows every step, or use a short trick with T.TEST and a column of μ₀ values. Both give exactly the same answer as R, Python and this calculator. The screenshots below use the leaf data in A2:A17 and μ₀ = 50.

9.1 The functions you need

9.2 Step by step, how to do a one sample t test in Excel

9.3 The awkward cases

One-tailed p. Use =T.DIST.RT(C7,C8) when H₁ says "greater". It returns 0.0173 here. For "less", use =T.DIST(C7,C8,TRUE), which returns 0.9827 here, correctly showing no evidence for a smaller mean. Halving the two-tailed p by hand gives 0.0173 in both cases, which is wrong for the "less" direction.

Analysis ToolPak. Excel's Data Analysis ToolPak has no one-sample option, but "t-Test: Paired Two Sample for Means" with a column of μ₀ values gives the same t, df and p. Turn it on first: File → Options → Add-ins → Manage Excel Add-ins → Go → tick Analysis ToolPak. Set "Hypothesized Mean Difference" to 0.

Older Excel. Excel 2007 and earlier use TDIST(ABS(t),df,2) and TINV(0.05,df). They still work in newer versions for compatibility.

The classic mistake: =T.DIST(C7,C8,TRUE) is the left-tail cumulative probability, not a p-value. For t = 2.32 it returns 0.983, which some people misreport as p = 0.98.

9.4 Why your Excel answer might differ from R or Python

For a one sample t test, Excel, R's t.test, SciPy's ttest_1samp and this calculator all use the same Student formula with df = n − 1, so there is no Welch versus Student disagreement as there is with two-sample tests. All four return t = 2.3243, df = 15, p = 0.0346 for the leaf data.

9.5 Charting the result in Excel

1. Put the label "Leaf length" in E2 and the mean (=C3) in F2.
2. Select E2:F2 and choose Insert → Column chart → Clustered Column.
3. Click the chart, press +, tick Error Bars → More Options → Custom → Specify Value, and point both directions at a cell holding the CI half-width, =C10*C5.
4. Add μ₀ as a horizontal line: right-click → Select Data → Add a series holding 50, then Change Chart Type → Combo → Line.
5. Label the caption "Error bar = 95% confidence interval". Without that label, readers assume it is the SD.

Chart 1 in Section 3 draws the raw points, the mean with its CI and the μ₀ line automatically, which is usually the better figure for a paper.

9.6 Excel errors you will probably hit

A one sample t test in R is one line: t.test(x, mu = 50). But that line gives no effect size and no assumption check, and reviewers ask for both. The script below is a complete one sample t test in R: it runs the test, adds Cohen's d, Hedges' g, a Shapiro-Wilk test and a Wilcoxon cross-check, prints a plain-English verdict and saves one 300 dpi figure. It uses base R only, so there is nothing to install.

10.1 The complete script

R, base only, saves one_sample_t_test.png

# ============================================================
# One sample t test with one publication-ready figure
# StatsUnlock - https://statsunlock.com/one-sample-t-test-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your data and settings -----------------------------
# Replace with your own values, or read a CSV:
#   x <- read.csv("data.csv")$leaf_length
x <- c(52, 48, 55, 61, 47, 50, 58, 53,
       49, 56, 51, 54, 46, 57, 50, 52)
x <- x[!is.na(x)]          # drop missing values

mu0         <- 50           # hypothesized mean
alpha       <- 0.05
alternative <- "two.sided"  # or "greater" / "less"
label       <- "Leaf length"
units       <- "mm"

# ---- 2. The test -------------------------------------------
n  <- length(x)
m  <- mean(x)
s  <- sd(x)                 # sample SD, n - 1 denominator
se <- s / sqrt(n)

res <- t.test(x, mu = mu0, alternative = alternative,
              conf.level = 1 - alpha)

t_stat <- unname(res$statistic)
df     <- unname(res$parameter)
p      <- res$p.value
ci     <- res$conf.int

# ---- 3. Effect size and assumption checks ------------------
d  <- (m - mu0) / s                      # Cohen's d
g  <- d * (1 - 3 / (4 * df - 1))         # Hedges' g
sw <- shapiro.test(x)                    # normality
wx <- suppressWarnings(wilcox.test(x, mu = mu0, exact = FALSE,
                                   correct = TRUE))

size <- ifelse(abs(d) < 0.2, "negligible",
        ifelse(abs(d) < 0.5, "small",
        ifelse(abs(d) < 0.8, "medium", "large")))

# ---- 4. Printed summary ------------------------------------
cat(sprintf("One sample t test: %s vs mu0 = %g %s\n", label, mu0, units))
cat(sprintf("n = %d, mean = %.4f, SD = %.4f, SE = %.4f\n", n, m, s, se))
cat(sprintf("t(%d) = %.4f, p = %.4f\n", df, t_stat, p))
cat(sprintf("%g%% CI for the mean: %.4f to %.4f\n",
            100 * (1 - alpha), ci[1], ci[2]))
cat(sprintf("Cohen's d = %.4f, Hedges' g = %.4f (%s)\n", d, g, size))
cat(sprintf("Shapiro-Wilk W = %.4f, p = %.4f\n",
            sw$statistic, sw$p.value))
cat(sprintf("Wilcoxon signed-rank check: p = %.4f\n", wx$p.value))

if (p < alpha) {
  cat(sprintf("Verdict: p is below %g, so the mean differs from %g. The effect is %s (d = %.2f).\n",
              alpha, mu0, size, d))
} else {
  cat(sprintf("Verdict: p is not below %g, so there is no clear evidence the mean differs from %g.\n",
              alpha, mu0))
}

# ---- 5. One figure, 300 dpi --------------------------------
set.seed(42)
png("one_sample_t_test.png", width = 2400, height = 1200, res = 300)
par(mfrow = c(1, 2), mar = c(4, 4.2, 2.5, 1))

jit <- runif(n, 0.75, 1.25)
plot(jit, x, xlim = c(0.4, 1.8), xaxt = "n", pch = 19,
     col = "#2563eb", xlab = "", ylab = paste0(label, " (", units, ")"),
     main = "Data, mean and mu0", ylim = range(c(x, mu0, ci)))
abline(h = mu0, lty = 2, col = "#c2410c", lwd = 2)
segments(1.5, ci[1], 1.5, ci[2], col = "#16a34a", lwd = 3)
points(1.5, m, pch = 18, cex = 2, col = "#16a34a")
legend("topleft", bty = "n", cex = 0.7,
       legend = c("mean with CI", "mu0"),
       col = c("#16a34a", "#c2410c"), lty = c(1, 2), lwd = 2)

qqnorm(x, main = "Normal Q-Q plot", pch = 19, col = "#9333ea")
qqline(x, col = "#0d9488", lwd = 2)
invisible(dev.off())
cat("Figure saved: one_sample_t_test.png\n")

Output for the leaf data: n = 16, mean = 52.4375, SD = 4.1947, t(15) = 2.3243, p = 0.0346, 95% CI 50.2023 to 54.6727, d = 0.5811, g = 0.5515, Shapiro-Wilk W = 0.9789, p = 0.9541.

10.2 What each part does

10.3 What the figure shows

The left panel shows each leaf as a blue point, the mean as a green diamond with its 95% CI, and μ₀ = 50 mm as a dashed orange line. The green interval sits entirely above the dashed line, which is the picture of a significant two-tailed result. The right panel is a normal Q-Q plot. The points lie close to the line, so normality is reasonable.

10.4 Common R problems and fixes

10.5 Useful one-liners

In Python the one sample t test is scipy.stats.ttest_1samp(x, popmean=50). The script below adds what SciPy leaves out: Cohen's d, Hedges' g, a Shapiro-Wilk check, a Wilcoxon cross-check, a plain-English verdict and one 300 dpi figure. It prints exactly the same numbers as the R script and this calculator.

Install with pip install numpy scipy matplotlib if you do not already have them.

11.1 The complete script

Python 3, saves one_sample_t_test.png

# ============================================================
# One sample t test with one publication-ready figure
# StatsUnlock - https://statsunlock.com/one-sample-t-test-calculator/
# ============================================================
import numpy as np
from scipy import stats
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt

# ---- 1. Your data and settings -----------------------------
# Replace with your own values, or read a CSV:
#   import pandas as pd
#   x = pd.read_csv("data.csv")["leaf_length"].to_numpy(dtype=float)
x = np.array([52, 48, 55, 61, 47, 50, 58, 53,
              49, 56, 51, 54, 46, 57, 50, 52], dtype=float)
x = x[~np.isnan(x)]              # drop missing values

mu0 = 50                         # hypothesized mean
alpha = 0.05
alternative = "two-sided"        # or "greater" / "less"
label, units = "Leaf length", "mm"

# ---- 2. The test -------------------------------------------
n = len(x)
m = x.mean()
s = x.std(ddof=1)                # sample SD, matches Excel STDEV.S
se = s / np.sqrt(n)

res = stats.ttest_1samp(x, popmean=mu0, alternative=alternative)
t_stat, p = res.statistic, res.pvalue
df = n - 1
ci = res.confidence_interval(confidence_level=1 - alpha)

# ---- 3. Effect size and assumption checks ------------------
d = (m - mu0) / s                        # Cohen's d
g = d * (1 - 3 / (4 * df - 1))           # Hedges' g
sw = stats.shapiro(x)                    # normality
wx = stats.wilcoxon(np.round(x - mu0, 10), method="approx", correction=True)

if abs(d) < 0.2:
    size = "negligible"
elif abs(d) < 0.5:
    size = "small"
elif abs(d) < 0.8:
    size = "medium"
else:
    size = "large"

# ---- 4. Printed summary ------------------------------------
print(f"One sample t test: {label} vs mu0 = {mu0} {units}")
print(f"n = {n}, mean = {m:.4f}, SD = {s:.4f}, SE = {se:.4f}")
print(f"t({df}) = {t_stat:.4f}, p = {p:.4f}")
print(f"{100 * (1 - alpha):g}% CI for the mean: {ci.low:.4f} to {ci.high:.4f}")
print(f"Cohen's d = {d:.4f}, Hedges' g = {g:.4f} ({size})")
print(f"Shapiro-Wilk W = {sw.statistic:.4f}, p = {sw.pvalue:.4f}")
print(f"Wilcoxon signed-rank check: p = {wx.pvalue:.4f}")
if p < alpha:
    print(f"Verdict: p is below {alpha}, so the mean differs from {mu0}. "
          f"The effect is {size} (d = {d:.2f}).")
else:
    print(f"Verdict: p is not below {alpha}, so there is no clear evidence "
          f"the mean differs from {mu0}.")

# ---- 5. One figure, 300 dpi --------------------------------
rng = np.random.default_rng(42)
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(8, 4))

jit = rng.uniform(0.75, 1.25, n)
ax1.scatter(jit, x, color="#2563eb", zorder=3)
ax1.axhline(mu0, ls="--", color="#c2410c", lw=2, label="mu0")
ax1.errorbar(1.5, m, yerr=[[m - ci.low], [ci.high - m]], fmt="D",
             color="#16a34a", capsize=6, lw=3, label="mean with CI")
ax1.set_xlim(0.4, 1.8)
ax1.set_xticks([])
ax1.set_ylabel(f"{label} ({units})")
ax1.set_title("Data, mean and mu0")
ax1.legend(frameon=False, fontsize=8)

stats.probplot(x, dist="norm", plot=ax2)
ax2.get_lines()[0].set_color("#9333ea")
ax2.get_lines()[1].set_color("#0d9488")
ax2.set_title("Normal Q-Q plot")

fig.tight_layout()
fig.savefig("one_sample_t_test.png", dpi=300, bbox_inches="tight")
print("Figure saved: one_sample_t_test.png")

Output for the leaf data, checked: t(15) = 2.3243, p = 0.0346, 95% CI 50.2023 to 54.6727, d = 0.5811, g = 0.5515, Shapiro-Wilk W = 0.9789, p = 0.9541, Wilcoxon p = 0.0476.

11.2 What each part does

11.3 What the figure shows

The same two panels as the R figure: the raw values with the mean and its 95% CI against the dashed μ₀ line, and a normal Q-Q plot. Because both scripts seed their jitter, re-running gives an identical picture.

11.4 Common Python problems and fixes

11.5 Useful one-liners

12.1 Critical values of t for a one sample t test

What to conclude: if |t| is larger than the value in your df row and α column, reject H₀. For df = 15 and a two-tailed α = 0.05 the cut-off is 2.131, so the leaf example (t = 2.324) is significant. When your df is missing, use the next smaller df, which is the conservative choice. Values computed from the t distribution (Student, 1908).

12.2 Cohen's d for a one sample t test: interpretation bands

What to conclude: these are Cohen's (1992) general guides, not rules. Judge the size against what matters in your field, and always report the difference in real units too (Lakens, 2013; Sullivan & Feinn, 2012).

12.3 Sample size needed for a one sample t test

What to conclude: halving the effect you want to detect roughly quadruples the n you need. Values are exact, computed from the noncentral t distribution for a two-tailed test, and agree with G*Power (Faul et al., 2007).

12.4 Power of a one sample t test at common sample sizes

What to conclude: with n = 10 you will miss a medium effect about 7 times in 10. A non-significant result from a small sample is weak evidence that there is no effect (two-tailed, α = 0.05).

12.5 How much normality matters at each sample size

What to conclude: the larger the sample, the less the shape of the raw data matters for the mean, but outliers matter at every n (Lumley et al., 2002; Ghasemi & Zahediasl, 2012).

12.6 Which one-sample test should you use?

What to conclude: the one sample t test is the default for a mean, but it is not the only option. Match the test to the data type and its shape.

12.7 How to describe a p-value in words

What to conclude: these words describe evidence, not truth. Several statisticians have argued for α = 0.005 for new discoveries (Benjamin et al., 2018). Whatever you pick, pick it before the analysis.

Eight one sample t test examples, each with the full numbers, a figure, what it means and how to write it. Load the matching sample dataset in Section 2 to reproduce any of them.

Study design: a one sample t test assumes a single random sample of independent units, each measured once on an interval or ratio scale, compared with a test value μ₀ that was fixed before the data were seen.

1. Fix μ₀ for your one sample t test before collecting data. Write down the value and its source: a regulation, a label, a published mean or a design target. A μ₀ chosen after seeing the data is not a test.
2. Decide the alternative hypothesis in advance. Two-tailed is the default. Use one-tailed only if a difference in the other direction would be impossible or would not change any decision.
3. Plan the sample size for the one sample t test. Decide the smallest difference that matters, convert it to d, and read the n from Table 12.3. For d = 0.5 at 80% power you need 34 units.
4. Define the unit of observation. One leaf, one patient, one box. Repeated readings on the same unit are not independent; average them first.
5. Sample at random from the population you want to describe. A convenience sample answers a question about a different population.
6. Fix the instrument, units and precision. Calibrate once, record the method, and do not switch instruments partway through.
7. Record missing values as blanks, never zeros.
8. Write the outlier rule before you look. For example: "values beyond 3 SD will be checked against field notes and kept unless a recording error is found."
9. State the fallback test. If normality clearly fails in a small sample, you will report the Wilcoxon signed-rank test.
10. Enter data in one column per group, exactly as the layout below shows, ready to paste or upload.

One column per group, one row per unit, numbers only below the header row. Unequal group sizes are fine.

Minimum sample: the test runs with n = 2, but below about 10 it has little power and the normality assumption cannot be checked. Aim for at least 15, and use Table 12.3 for a real target.

Pre-registration: record μ₀ and its source, the tail, α, the planned n, the outlier rule and the fallback test before collecting data.

Common collection mistakes:

• Choosing μ₀ or the tail after seeing the mean.
• Counting repeat measurements of one unit as separate observations.
• Recording missing data as 0.
• Stopping data collection as soon as p drops below 0.05.
• Mixing units, for example some values in cm and some in mm.

This free one sample t test calculator is designed for one question: does the mean of one group differ from one fixed value?

Use it when:

• ✓ You have one sample of a continuous measurement.
• ✓ You have a single benchmark value μ₀ that does not come from the same data.
• ✓ The population standard deviation is unknown, which is almost always the case.
• ✓ The observations are independent of each other.
• ✓ The data are roughly symmetric, or n is 30 or more without heavy outliers.

Do not use it when:

• ✗ You want to compare two groups with each other. Use an independent samples t test.
• ✗ You measured the same units twice. Use a paired t test (a one sample t test on the differences).
• ✗ The data are strongly skewed or ordinal with a small n. Use the Wilcoxon signed-rank test.
• ✗ The outcome is a yes/no proportion. Use a one proportion z test.
• ✗ You know the true population σ. Use a one sample z test.
• ✗ You want to show the mean is close to μ₀. Use an equivalence test (TOST).

15.1 Four real-world examples

• Quality control: do bottles contain the 750 ml on the label?
• Medicine: is mean fasting glucose in a clinic above the 100 mg/dL reference?
• Education: does a school's mean score differ from the national average of 70?
• Ecology: does mean stream pH below a mill differ from neutral 7.0?

15.2 Decision tree

Assumptions of a one sample t test

1. Independence. Each observation is a separate unit, not influenced by the others. Check: the study design; there is no statistical test that fixes this. If it fails: clustered or repeated data make the SE too small, so p is too small and false positives rise.
2. Continuous outcome. The variable is measured on an interval or ratio scale. Check: what the numbers mean. If it fails: with ordinal ratings the mean may not be meaningful; results can be misleading in either direction.
3. Approximate normality. The population is roughly normal, or n is large enough for the mean to be. Check: Shapiro-Wilk, skewness, chart 3 and chart 4. If it fails: with skew in small samples, one-tailed p-values can be too small in one tail and too large in the other.
4. No extreme outliers. Check: the outlier row in the assumption table and chart 1. If it fails: outliers inflate the SD, so t shrinks and real differences are missed, as in Example 7.
5. Random sampling. The sample represents the population of interest. Check: how units were selected. If it fails: the result is biased in an unknown direction, and no statistic can show it.
6. A fixed, pre-specified μ₀. Check: that the value came from outside the data. If it fails: choosing μ₀ after seeing the data biases p downwards.

Limitations

• It tests the mean only. For the median, use the Wilcoxon signed-rank test or the sign test.
• A non-significant result does not prove the mean equals μ₀. To show closeness, use an equivalence test (TOST; Lakens, 2017).
• It handles one group against a value. To compare groups with each other, use an independent samples t test or ANOVA.
• It assumes a simple random sample. Survey designs with weights or clusters need design-based methods.
• Testing many groups inflates false positives. Use the Holm-adjusted p shown in the multi-group table.
• The Cohen's d interval assumes normality; with heavy skew, a bootstrap interval is more honest.

What was found

A one sample t test calculator answers one clear question: is the mean of my sample different from a fixed value? It does it by dividing the gap between the sample mean and μ₀ by the standard error, and reading the result on a t distribution with n − 1 degrees of freedom. In the worked example, 16 leaves averaging 52.44 mm gave t(15) = 2.32 and p = 0.035 against the published 50 mm.

What it means in practice

The p-value says whether the difference is likely to be real. The confidence interval says how large it could be, and Cohen's d says whether it is large enough to matter. A good report has all three, plus n, the mean and the SD. A significant result with a negligible d, as in Example 6, should be reported as real but trivial. A non-significant result with a large d, as in Example 5, usually means the study was too small.

Cautions

The test is only as good as its assumptions. Independence matters most, and no test can check it for you. Normality matters in small samples, and outliers matter at every size. When the t test and the Wilcoxon signed-rank test disagree, look at the data before you trust either one. And fix μ₀, the tail and α before you look at the result.

Next steps

Paste your own data into the calculator, read the confidence interval before the p-value, check the four charts, and copy the APA sentence from Section 5. If you need to compare two groups instead, use the paired or independent samples t test. Come back to this one sample t test calculator whenever you need to test a mean against a standard, a target or a published value.

The following peer-reviewed references support the methods used in this one sample t test calculator: the t distribution, normality testing, effect sizes and their intervals, power, and good practice in reporting p-values.

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