An independent samples t test checks whether two unrelated groups have different population means, by dividing the gap between the two sample means by the standard error of that gap.

The result is a t statistic. Large positive or large negative values mean the two groups are far apart relative to the noise inside them. The p value is the tail area of the t distribution beyond that value, so it answers one narrow question: how surprising is a gap this big if the true means are identical? It says nothing about how big or how useful the difference is, which is why Cohen's d and the confidence interval sit beside it on every result card below.

Rule of thumb: if p is below 0.05 the means differ more than chance comfortably explains, and Cohen's d of 0.20 is small, 0.50 is medium and 0.80 is large. If the larger group variance is more than twice the smaller, report the Welch version, which this calculator selects automatically.

The independent samples t test compares the means of two separate groups of people, animals, plots or items. Each observation belongs to exactly one group, and no value in group 1 is paired with a particular value in group 2. A placebo arm against a treatment arm, boys against girls, burned plots against unburned plots: those are independent samples.

What it actually measures. The test builds a signal to noise ratio. The signal is the difference between the two sample means. The noise is the standard error of that difference, which grows when the groups are spread out and shrinks when the samples are large. The t statistic is signal divided by noise, so t rises when the gap widens, when the spread narrows, or when you collect more data.

A worked one liner. Group 1 has a mean of 55.0 with s = 6.0 and n = 20. Group 2 has a mean of 49.0 with s = 6.5 and n = 20. The pooled standard deviation is 6.26, the standard error of the difference is 1.98, so t = 6.0 ÷ 1.98 = 3.03 on 38 degrees of freedom, p = 0.0043, and Cohen's d = 0.96. The gap is both statistically clear and large.

1.1 Who uses an independent samples t test, and what for?

Researchers use it whenever two separate groups are compared on one continuous outcome: treatment against control, one teaching method against another, disturbed sites against undisturbed ones. It is the default first test in trials, education research, ecology and quality control, because two group comparisons are the most common design there is.

Clinical trials use it to compare an outcome between placebo and treatment arms. Education researchers use it to compare exam scores between two teaching methods. Ecologists use it to compare species counts between disturbed and undisturbed sites. Sports scientists use it to compare VO2 max between trained and untrained athletes. Manufacturing uses it to compare a quality measure across two production lines. In every case there are two groups, one continuous outcome, and no pairing between the groups.

1.2 What is an independent samples t test not used for?

It is not for paired measurements, not for three or more groups, not for categorical outcomes, and not for judging whether a difference matters. Each of those needs a different tool, and using the t test anyway changes the degrees of freedom, the error rate, or the meaning of the answer.

It is not a test of whether the distributions are identical, only whether the means differ. It is not valid for before and after measurements on the same subjects, which need the paired test. It is not the right tool for three or more groups, because running three t tests inflates the false positive rate from 5% to about 14%. And it is not a test of practical importance: that judgement comes from the effect size and the confidence interval.

1.3 What is the difference between an independent samples t test and a paired t test?

The independent test compares two separate groups of units, and the paired samples t test compares two measurements taken on the same units. Pairing is what changes the arithmetic: the paired test analyses the within subject differences, so it uses n minus 1 degrees of freedom instead of n₁ + n₂ minus 2, and it is more powerful whenever the pairing is real.

Naming note: the independent samples t test, the two sample t test, the unpaired t test and the between subjects t test are four names for the same procedure. SPSS calls it Independent-Samples T Test, R calls it t.test with two vectors, and Excel calls it t-Test: Two-Sample.

A t test hands you five numbers: t, df, p, an effect size and an interval. Reading them in the right order is what separates a defensible conclusion from a sentence a reviewer will strike out. Work down this section in order, because each part narrows what the previous one is allowed to claim.

4.1 What your t statistic is telling you

The t statistic is a ratio, not a measurement. A t of 3.0 means the observed gap between the means is three times as large as the typical sampling wobble in that gap. Values near zero mean the two groups are indistinguishable. The sign only records which group you subtracted from which, so a t of −3.0 is exactly as strong as a t of +3.0. Three things push t up: a wider gap between the means, tighter spread inside each group, and larger samples. That last one matters more than people expect, because the standard error falls with the square root of n, so quadrupling the sample halves the noise and doubles t for the same real difference.

4.2 What the p value does and does not say

The p value is the probability of a t statistic at least this extreme if the two population means were identical. That conditional clause is the whole meaning. It is not the probability that the null hypothesis is true, it is not the probability your result is a fluke, and 1 minus p is not the probability the effect is real. A p of 0.03 does not mean a 97% chance the treatment works.

Two further habits are worth building. First, do not treat 0.05 as a cliff: p = 0.049 and p = 0.051 are the same evidence, and rounding one into significance and the other out of it is an artefact of the threshold, not a finding. Second, report the exact value to three decimals rather than "p < 0.05", because the exact figure lets other researchers use your study in a meta analysis. Only write p < 0.001 when the value is genuinely smaller than that.

4.3 Reading the confidence interval, the number that carries the most information

The confidence interval is stated in the units you measured, which is what makes it more useful than the p value. An interval of 2.1 to 9.9 mmHg says the data are consistent with anything from a barely noticeable 2 point drop to a clinically important 10 point drop. An interval of −0.3 to 0.5 kg is a genuinely null finding: the study was precise enough to rule out anything that matters. An interval of −18 to 21 points is not a null finding at all, it is an uninformative one, and the honest conclusion is that the study was too small to decide.

The rule connecting the two: if the interval excludes the hypothesised difference, the two tailed p value is below alpha. They can never disagree, because they are computed from the same standard error and the same critical value.

4.4 How big is the difference? Reading Cohen's d

Cohen's d expresses the gap in standard deviations, so it is comparable across studies that used different instruments. A d of 0.50 means the two group means sit half a standard deviation apart, which is roughly the distance between the 50th and the 69th percentile of the control group. Cohen's 0.20, 0.50 and 0.80 labels are conventions he himself described as arbitrary. Anchor them to your field: a d of 0.30 in a national education trial can matter more than a d of 1.20 in a tightly controlled lab, because the education effect applies to millions of pupils.

Hedges' g corrects d for the small sample bias that inflates it when n is under about 20 per group, so report g for small studies. Glass's delta divides by the control group standard deviation alone, which is the right choice when the treatment itself changes the variability.

4.5 Welch or Student: which version should you report?

The pooled Student test averages the two variances into one number, which is only legitimate if the populations really have the same spread. When that assumption fails, the pooled standard error is wrong, and the error is worst when the larger variance sits with the smaller group. Welch keeps the variances apart and pays for it with fractional degrees of freedom. The published simulation work is consistent: Welch loses almost nothing when variances are equal and protects the error rate when they are not, which is why several methods papers now recommend it as the unconditional default. Reporting Welch is never the thing a reviewer objects to.

Avoid the two stage habit of testing variances first and then choosing the t test based on that outcome. That conditional procedure has a real type I error rate above the nominal 5%, because you used the same data twice. Choose Welch in advance and the problem disappears.

4.6 Where the degrees of freedom come from

Degrees of freedom count the independent pieces of information left after estimating the means. Each group spends one on its own mean, so the pooled test has n₁ + n₂ − 2. The t distribution is wider than the normal curve at small df, which is exactly the penalty for having estimated the standard deviation from the data rather than knowing it. At df = 10 the two tailed critical value is 2.228; at df = 30 it is 2.042; at df = 100 it is 1.984, closing in on the normal 1.960. Welch's degrees of freedom come from the Welch Satterthwaite formula and are usually fractional, for example 34.62. That is not a rounding error, and you should report it to two decimals.

4.7 Statistical significance is not practical significance

With n = 5,000 per group, a difference of 0.4 points on a 100 point exam reaches p < 0.001 and has a d of about 0.06. It is real and it is useless. The reverse trap also exists: a pilot study of 12 per group with a d of 0.90 and p = 0.08 has found a potentially large effect and simply lacks the sample to confirm it. Calling that a null result is the more expensive mistake of the two, because it stops the follow up study from ever being run.

4.8 What your sample size did to this result

Observed power computed from the effect you happened to observe is not evidence about the design, because it is a direct function of the p value. Its only honest use is descriptive. The forward looking number is the required sample size: how many observations per group you would need to detect the effect you care about, decided before the next study. For alpha 0.05 two tailed and 80% power, you need roughly 394 per group for d = 0.20, 64 for d = 0.50, and 26 for d = 0.80.

4.9 A decision guide you can read in ten seconds

4.10 Common mistakes this calculator prevents

• Using the pooled test on unequal variances. The tool computes both versions every time and marks the one your setting reports.
• Wrong degrees of freedom. Welch df is computed to two decimals from the Welch Satterthwaite formula, not rounded down to the smaller group's n minus 1.
• Reading a p value from a coarse table. The exact tail area is computed from the incomplete beta function, so p is right to four decimals rather than bracketed between table columns.
• Reporting significance with no effect size. Cohen's d, Hedges' g and Glass's delta appear on every run, with the interpretation band named.
• Silent sample size errors. Blank cells, text and non numeric entries are counted and reported instead of quietly shrinking n.
• Switching to one tailed after seeing the data. The tails control sits above the Run button, and the report wording states which tail was used.
• Feeding paired data into the independent test. The assumption panel flags equal group sizes with a warning to check whether the values are actually matched.

4.11 If your result is not significant

Do three things before writing the discussion. Look at the confidence interval and state the largest difference the data still allow. Look at the effect size, because a moderate d with a wide interval is a power problem rather than a null finding. Then compute the sample size needed for the smallest difference that would change practice, and say it out loud in the paper. Absence of evidence and evidence of absence are different claims, and only the second one needs a narrow interval to support it.

4.12 Direction, sign and one tailed tests

This calculator always computes group 1 minus group 2, so a positive difference means group 1 scored higher. Use the swap button if you would rather frame it the other way, and the sign of t, the interval and the wording all follow. A one tailed test puts all of alpha in a single tail, which halves the p value and lowers the critical value from 1.96 to 1.645 at large df. That is a genuine gain in power for a genuinely directional prediction, and a serious problem if the direction was chosen after the data arrived. If a one tailed test is used, say in the methods that the direction was specified in advance.

When reporting independent samples t test results, a complete sentence carries six things: the test used, the two group means with their standard deviations, the t statistic with its degrees of freedom, the exact p value, the effect size, and the confidence interval for the difference. Miss any of them and a reviewer will ask for it. The five cards below fill in from your own run, so press Run t Test first and then copy the one that matches your outlet.

5.1 The Methods sentence, written before the results

State the design, the test, the software and the alpha level in the methods, not the results. A complete example: "Group means were compared with an independent samples t test using the Welch correction for unequal variances. Normality was screened with skewness and kurtosis, and homogeneity of variance with Levene's test. Effect sizes are reported as Cohen's d with 95% confidence intervals. Alpha was set at 0.05, two tailed. Analyses were run in the StatsUnlock independent samples t test calculator."

5.2 APA 7th edition results sentence

5.3 Thesis or dissertation paragraph

5.4 Plain language version for a report or press summary

5.5 Structured abstract or conference poster line

5.6 Replication and pre-registration wording

5.7 Table and figure conventions

A results table for a two group comparison has one row per group and columns for n, M, SD, and optionally the 95% CI of each mean. The test statistic goes in the table note or in a separate row spanning the table, never as a lone asterisk. Figures should show the data, not just the means: a dot plot or violin with the mean and its interval marked communicates far more than two bars, and bar charts of two means with standard error whiskers are actively discouraged by several journals now.

5.8 Final reporting checklist

• Test named, including Welch or pooled, and the tails stated.
• n, M and SD given for both groups.
• t with its df to the right number of decimals.
• Exact p value, not just an inequality.
• Effect size with its type named, plus its confidence interval where available.
• Confidence interval for the difference in means, in measurement units.
• Assumption checks reported, with what was done if one failed.
• Direction of the difference stated in words.
• Practical meaning discussed separately from statistical significance.

Seven formulas do all the work. Card 1 is the test itself, and each card after it opens up one piece of the card above.

Worked example to try right now. Leave the first sample dataset loaded, set alpha to 0.05 and tails to two tailed, then press Run t Test. Placebo has a mean of 141.23 mmHg and Treatment 132.43 mmHg. The variance ratio is only 1.05, so Auto reports the pooled Student test: t = 3.801 on 58 degrees of freedom, p = 0.00035, Cohen's d = 0.98, and the 95% interval for the difference runs from 4.17 to 13.43 mmHg. That is a clear, large and clinically meaningful drop in blood pressure.

Excel can do the whole test with one function, and that is both its strength and the reason people get it wrong. T.TEST returns a p value and nothing else: no t statistic, no degrees of freedom, no confidence interval, no effect size. This section shows the one liner first, then builds the same result piece by piece so you can report it properly.

8.1 The functions you need

Worked example used throughout sections 8, 9, 10 and 12: Traditional teaching (n = 10, M = 71.00, SD = 4.35) against Flipped classroom (n = 10, M = 77.40, SD = 4.65).

8.2 Step by step

8.3 The awkward cases in Excel

• One tailed tests. Change the third argument to 1. Excel reports the tail in the direction the data actually went, so it will not warn you if your prediction was the other way. Check the sign of the mean difference yourself.
• Welch degrees of freedom. T.TEST with type 3 hides them. Compute them yourself: =(E4/10+F4/10)^2/((E4/10)^2/9+(F4/10)^2/9) with the variances in E4 and F4, which gives 17.92 here.
• Summary data only. If you have only means, SDs and n, T.TEST cannot help at all because it needs ranges. Build t from the formulas in step 4, or use the Mean and SD Only tab in the calculator above.
• Unequal column lengths. Leave the extra cells empty rather than filling them with zeros. A zero is a real data point to Excel and will drag the mean down.
• Effect size confidence intervals. There is no practical Excel route to a CI for Cohen's d, because it needs the noncentral t distribution. Use the calculator above or R's effectsize package.

8.4 Why your Excel answer might differ from R or Python

8.5 Charting the result in Excel

Select both data columns and insert a clustered column chart of the two means, then add error bars: Chart Design, Add Chart Element, Error Bars, More Options, Custom, and enter the standard error for each group. Two formatting changes make it publication ready: set the vertical axis to start at a value that does not exaggerate the gap, and delete the gridlines and the chart border. Better still, insert a scatter plot with a small random horizontal jitter so every observation is visible, because two bars hide the overlap that decides whether the difference matters.

8.6 Excel errors you will probably hit

In R the one liner is t.test(group1, group2), and it is rarely the whole thing you should report. R defaults to Welch, gives you no effect size, and will not check your assumptions unless you ask. The script below does the complete job in base R with no packages at all.

9.1 The complete script

# ------------------------------------------------------------------
# Independent samples t test, complete script, base R only
# StatsUnlock - statsunlock.com
# Change only the two data lines below, or uncomment the read.csv block.
# ------------------------------------------------------------------
set.seed(42)

traditional <- c(72, 65, 78, 70, 68, 74, 66, 71, 69, 77)
flipped     <- c(76, 71, 83, 74, 79, 85, 72, 78, 75, 81)

# --- reading your own file instead -------------------------------
# dat <- read.csv("my_data.csv")            # long format: value, group
# traditional <- dat$value[dat$group == "Traditional"]
# flipped     <- dat$value[dat$group == "Flipped"]
# traditional <- traditional[!is.na(traditional)]   # drop missing values
# flipped     <- flipped[!is.na(flipped)]

g1 <- "Traditional"
g2 <- "Flipped"
alpha <- 0.05

n1 <- length(traditional); n2 <- length(flipped)
m1 <- mean(traditional);   m2 <- mean(flipped)
s1 <- sd(traditional);     s2 <- sd(flipped)

# --- the two tests ----------------------------------------------
student <- t.test(traditional, flipped, var.equal = TRUE)
welch   <- t.test(traditional, flipped, var.equal = FALSE)   # R's default

# --- assumption checks ------------------------------------------
vratio <- max(s1, s2)^2 / min(s1, s2)^2
ftest  <- var.test(traditional, flipped)
sw1    <- shapiro.test(traditional)
sw2    <- shapiro.test(flipped)

# --- effect sizes -----------------------------------------------
sp <- sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2))
d  <- (m1 - m2) / sp
g  <- d * (1 - 3 / (4 * (n1 + n2) - 9))          # Hedges' g
band <- if (abs(d) < 0.2) "negligible" else
        if (abs(d) < 0.5) "small" else
        if (abs(d) < 0.8) "medium" else "large"

use <- if (vratio > 2 || ftest$p.value < 0.05) welch else student
label <- if (identical(use, welch)) "Welch (unequal variances)" else "Student (pooled variance)"

# --- printed summary --------------------------------------------
cat("\n================ INDEPENDENT SAMPLES t TEST ================\n")
cat(sprintf("%-12s n = %2d   M = %8.4f   SD = %7.4f\n", g1, n1, m1, s1))
cat(sprintf("%-12s n = %2d   M = %8.4f   SD = %7.4f\n", g2, n2, m2, s2))
cat(sprintf("Mean difference (%s - %s) = %.4f\n", g1, g2, m1 - m2))
cat("------------------------------------------------------------\n")
cat(sprintf("Student : t(%.0f) = %.4f, p = %.6f\n", student$parameter, student$statistic, student$p.value))
cat(sprintf("Welch   : t(%.2f) = %.4f, p = %.6f\n", welch$parameter, welch$statistic, welch$p.value))
cat(sprintf("Reported: %s\n", label))
cat(sprintf("95%% CI for the difference: [%.4f, %.4f]\n", use$conf.int[1], use$conf.int[2]))
cat(sprintf("Cohen's d = %.4f (%s), Hedges' g = %.4f\n", d, band, g))
cat("------------------------------------------------------------\n")
cat(sprintf("Variance ratio = %.2f, F test p = %.4f\n", vratio, ftest$p.value))
cat(sprintf("Shapiro-Wilk %s p = %.4f, %s p = %.4f\n", g1, sw1$p.value, g2, sw2$p.value))
cat("------------------------------------------------------------\n")
verdict <- if (use$p.value < alpha) {
  sprintf("p = %.6f is below %.2f and d = %.2f is %s, so %s differs from %s by %.2f points.",
          use$p.value, alpha, d, band, g1, g2, m1 - m2)
} else {
  sprintf("p = %.6f is not below %.2f, so the difference of %.2f points is not statistically clear. The interval still allows anything from %.2f to %.2f.",
          use$p.value, alpha, m1 - m2, use$conf.int[1], use$conf.int[2])
}
cat("VERDICT:", verdict, "\n")
cat("Generated by STATS UNLOCK - statsunlock.com\n")

# --- one publication ready figure at 300 dpi --------------------
png("ttest_plot.png", width = 2400, height = 1100, res = 300)
par(mfrow = c(1, 2), mar = c(4.2, 4.2, 2.6, 1))

vals <- list(traditional, flipped)
boxplot(vals, names = c(g1, g2), col = c("#dbeafe", "#dcfce7"),
        border = c("#1d4ed8", "#15803d"), ylab = "Exam score (%)",
        main = "Group comparison")
for (i in 1:2) {
  points(jitter(rep(i, length(vals[[i]])), amount = 0.09), vals[[i]],
         pch = 21, bg = "#ffffff", col = "#374151", cex = 0.9)
  points(i, mean(vals[[i]]), pch = 18, col = "#c2410c", cex = 1.8)
}
legend("topleft", legend = "orange diamond = mean", bty = "n", cex = 0.75)

xs <- seq(-4.5, 4.5, length.out = 400)
plot(xs, dt(xs, use$parameter), type = "l", lwd = 2, col = "#334155",
     xlab = "t", ylab = "Density", main = sprintf("t distribution, df = %.2f", use$parameter))
tc <- qt(1 - alpha / 2, use$parameter)
polygon(c(xs[xs >= tc], max(xs)), c(dt(xs[xs >= tc], use$parameter), 0), col = "#fee2e2", border = NA)
polygon(c(min(xs), xs[xs <= -tc]), c(0, dt(xs[xs <= -tc], use$parameter)), col = "#fee2e2", border = NA)
abline(v = use$statistic, col = "#16a34a", lwd = 2.5)
abline(v = c(-tc, tc), col = "#b91c1c", lwd = 1.4, lty = 2)
text(use$statistic, max(dt(xs, use$parameter)) * 0.92,
     sprintf("t = %.2f", use$statistic), col = "#166534", pos = if (use$statistic < 0) 4 else 2, cex = 0.85)
dev.off()
cat("Figure saved as ttest_plot.png (300 dpi)\n")

Output for the worked example: Student t(18) = -3.1806, p = 0.005178; Welch t(17.92) = -3.1806, p = 0.005200; Cohen's d = -1.4224; 95% CI [-10.6274, -2.1726].

9.2 What each part does

9.3 What the figure shows

The left panel is a box plot of both groups with every observation jittered on top and the mean marked as an orange diamond. It shows the overlap that a bar chart would hide. The right panel is the t distribution at the reported degrees of freedom, with the rejection region shaded red and your t statistic drawn as a green line. When the green line sits inside the red region, the result is significant at your alpha, which is the same decision the p value made in words.

9.4 Common R problems and fixes

9.5 Useful one liners

t.test(value ~ group, data = dat)                    # formula form, long data
t.test(x, y, var.equal = TRUE)                       # pooled Student version
t.test(x, y, alternative = "greater")                # one tailed, x greater than y
t.test(x, y, conf.level = 0.99)                      # 99 percent interval
round(unlist(t.test(x, y)[c("statistic","parameter","p.value")]), 4)
effectsize::cohens_d(x, y)                           # d with a confidence interval
wilcox.test(x, y)                                    # Mann-Whitney fallback
power.t.test(delta = 0.5, sd = 1, power = 0.8)       # n per group for d = 0.5

SciPy does the test in one call, stats.ttest_ind(a, b), and it returns only the statistic and the p value. No degrees of freedom, no confidence interval, no effect size. The script below fills all three gaps and follows the same shape as the R script above, so you can read them side by side.

Install once with pip install numpy scipy matplotlib pandas. In Jupyter, drop the matplotlib.use("Agg") line and add plt.show() at the end.

10.1 The complete script

# ------------------------------------------------------------------
# Independent samples t test, complete script
# StatsUnlock - statsunlock.com
# Needs: numpy, scipy, matplotlib     pip install numpy scipy matplotlib
# Change only the two data lines below, or uncomment the pandas block.
# ------------------------------------------------------------------
import numpy as np
from scipy import stats
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt

rng = np.random.default_rng(42)          # seeded so the jitter is reproducible

traditional = np.array([72, 65, 78, 70, 68, 74, 66, 71, 69, 77], dtype=float)
flipped     = np.array([76, 71, 83, 74, 79, 85, 72, 78, 75, 81], dtype=float)

# --- reading your own file instead --------------------------------
# import pandas as pd
# dat = pd.read_csv("my_data.csv")                     # long format: value, group
# traditional = dat.loc[dat["group"] == "Traditional", "value"].to_numpy(float)
# flipped     = dat.loc[dat["group"] == "Flipped", "value"].to_numpy(float)
# traditional = traditional[~np.isnan(traditional)]    # drop missing values
# flipped     = flipped[~np.isnan(flipped)]

g1, g2 = "Traditional", "Flipped"
alpha = 0.05

n1, n2 = traditional.size, flipped.size
m1, m2 = traditional.mean(), flipped.mean()
s1 = traditional.std(ddof=1)             # ddof=1 matches R's sd() and Excel's STDEV.S
s2 = flipped.std(ddof=1)
diff = m1 - m2

# --- the two tests ------------------------------------------------
t_stu, p_stu = stats.ttest_ind(traditional, flipped, equal_var=True)    # SciPy default
t_wel, p_wel = stats.ttest_ind(traditional, flipped, equal_var=False)   # Welch

df_stu = n1 + n2 - 2
df_wel = (s1**2/n1 + s2**2/n2)**2 / ((s1**2/n1)**2/(n1-1) + (s2**2/n2)**2/(n2-1))

sp = np.sqrt(((n1-1)*s1**2 + (n2-1)*s2**2) / df_stu)
se_stu = sp * np.sqrt(1/n1 + 1/n2)
se_wel = np.sqrt(s1**2/n1 + s2**2/n2)

# --- assumption checks --------------------------------------------
vratio = max(s1, s2)**2 / min(s1, s2)**2
f_p = 2 * min(stats.f.cdf(s1**2/s2**2, n1-1, n2-1), 1 - stats.f.cdf(s1**2/s2**2, n1-1, n2-1))
lev = stats.levene(traditional, flipped, center="median")   # Brown-Forsythe form
sw1 = stats.shapiro(traditional)
sw2 = stats.shapiro(flipped)

use_welch = (vratio > 2) or (lev.pvalue < 0.05)
t, p, df, se = (t_wel, p_wel, df_wel, se_wel) if use_welch else (t_stu, p_stu, df_stu, se_stu)
label = "Welch (unequal variances)" if use_welch else "Student (pooled variance)"

tcrit = stats.t.ppf(1 - alpha/2, df)
ci = (diff - tcrit*se, diff + tcrit*se)

# --- effect sizes -------------------------------------------------
d = diff / sp
g = d * (1 - 3 / (4*(n1+n2) - 9))
band = ("negligible" if abs(d) < 0.2 else "small" if abs(d) < 0.5
        else "medium" if abs(d) < 0.8 else "large")

# --- printed summary ----------------------------------------------
print("\n================ INDEPENDENT SAMPLES t TEST ================")
print(f"{g1:<12} n = {n1:2d}   M = {m1:8.4f}   SD = {s1:7.4f}")
print(f"{g2:<12} n = {n2:2d}   M = {m2:8.4f}   SD = {s2:7.4f}")
print(f"Mean difference ({g1} - {g2}) = {diff:.4f}")
print("------------------------------------------------------------")
print(f"Student : t({df_stu:.0f}) = {t_stu:.4f}, p = {p_stu:.6f}")
print(f"Welch   : t({df_wel:.2f}) = {t_wel:.4f}, p = {p_wel:.6f}")
print(f"Reported: {label}")
print(f"95% CI for the difference: [{ci[0]:.4f}, {ci[1]:.4f}]")
print(f"Cohen's d = {d:.4f} ({band}), Hedges' g = {g:.4f}")
print("------------------------------------------------------------")
print(f"Variance ratio = {vratio:.2f}, F test p = {f_p:.4f}, Levene p = {lev.pvalue:.4f}")
print(f"Shapiro-Wilk {g1} p = {sw1.pvalue:.4f}, {g2} p = {sw2.pvalue:.4f}")
print("------------------------------------------------------------")
if p < alpha:
    print(f"VERDICT: p = {p:.6f} is below {alpha}, and d = {d:.2f} is {band}. "
          f"{g1} differs from {g2} by {diff:.2f} points, 95% CI [{ci[0]:.2f}, {ci[1]:.2f}].")
else:
    print(f"VERDICT: p = {p:.6f} is not below {alpha}. The {diff:.2f} point difference is not "
          f"statistically clear, and the data still allow anything from {ci[0]:.2f} to {ci[1]:.2f}.")
print("Generated by STATS UNLOCK - statsunlock.com")

# --- one publication ready figure at 300 dpi ----------------------
fig, ax = plt.subplots(1, 2, figsize=(9.2, 3.9))

bp = ax[0].boxplot([traditional, flipped], tick_labels=[g1, g2], patch_artist=True, widths=0.55)
for patch, col in zip(bp["boxes"], ["#dbeafe", "#dcfce7"]):
    patch.set_facecolor(col)
for i, v in enumerate([traditional, flipped], start=1):
    ax[0].scatter(i + rng.uniform(-0.09, 0.09, v.size), v, s=26,
                  facecolor="white", edgecolor="#374151", zorder=3)
    ax[0].scatter(i, v.mean(), marker="D", s=55, color="#c2410c", zorder=4)
ax[0].set_ylabel("Exam score (%)")
ax[0].set_title("Group comparison (orange diamond = mean)", fontsize=9)

xs = np.linspace(-4.5, 4.5, 400)
ax[1].plot(xs, stats.t.pdf(xs, df), color="#334155", lw=2)
ax[1].fill_between(xs, 0, stats.t.pdf(xs, df), where=np.abs(xs) >= tcrit, color="#fee2e2")
ax[1].axvline(t, color="#16a34a", lw=2.2, label=f"t = {t:.2f}")
ax[1].axvline(tcrit, color="#b91c1c", lw=1.2, ls="--")
ax[1].axvline(-tcrit, color="#b91c1c", lw=1.2, ls="--", label=f"critical = {tcrit:.2f}")
ax[1].set_xlabel("t"); ax[1].set_ylabel("Density")
ax[1].set_title(f"t distribution, df = {df:.2f}", fontsize=9)
ax[1].legend(fontsize=8, frameon=False)

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

Output for the worked example, identical to R and Excel to four decimals: Student t(18) = -3.1806, p = 0.005178; Welch t(17.92) = -3.1806, p = 0.005200; Cohen's d = -1.4224; Levene p = 0.7199.

10.2 What each part does

10.3 What the figure shows

Left, the two groups as box plots with every point jittered over them and the mean as an orange diamond, so the reader sees the raw spread rather than a summary. Right, the t distribution at the reported degrees of freedom with the rejection region shaded and your t statistic marked in green. The two panels together answer both questions a reviewer asks: how different are the groups, and how surprising is that difference.

10.4 Common Python problems and fixes

10.5 Useful one liners

stats.ttest_ind(a, b, equal_var=False)                 # Welch
stats.ttest_ind(a, b, alternative="greater")           # one tailed
stats.ttest_ind(a, b, nan_policy="omit")               # ignore missing values
stats.mannwhitneyu(a, b)                               # non parametric fallback
stats.bootstrap((a, b), lambda x, y: x.mean() - y.mean(), n_resamples=9999)
import pingouin as pg; pg.ttest(a, b, correction=True) # t, df, p, CI, d and BF in one table
from statsmodels.stats.power import TTestIndPower
TTestIndPower().solve_power(effect_size=0.5, power=0.8, alpha=0.05)   # n per group

Run it with python ttest.py from a terminal. In Jupyter, remove the Agg backend line and add plt.show() so the figure appears inline as well as on disk.

Eight lookup tables you can use without running anything. All critical values are computed from the t distribution rather than copied from a textbook, so they are exact to three decimals.

11.1 Critical t values, two tailed

What to conclude: if your absolute t exceeds the value in the alpha 0.05 column at your df, the result is significant at the 5% level. Notice how the values shrink toward the normal 1.960 as df grows, which is the whole reason the t distribution exists.

11.2 Critical t values, one tailed

What to conclude: a one tailed test needs a smaller t to reach the same alpha, which is exactly why switching to it after seeing the data is treated as p hacking. Use it only for a direction you registered in advance.

11.3 Effect size interpretation bands

What to conclude: Cohen described these cut points as arbitrary conventions for use when nothing better exists. In education a d of 0.40 is often treated as the threshold of practical value, and in medicine the minimum clinically important difference is set in measurement units, not in standard deviations. Source: Cohen (1988), with the field specific thresholds in Lakens (2013).

11.4 Required sample size per group, two tailed alpha 0.05

What to conclude: these are per group, so double them for the total. Halving the effect size you want to detect roughly quadruples the sample you need, which is why chasing small effects is so expensive. Values computed from the noncentral t distribution.

11.5 Power you actually have at a given sample size

What to conclude: a study with 20 per group has a 34% chance of detecting a medium effect that really exists. That is why so many small studies report null results, and why a non significant p from a small sample is weak evidence of nothing.

11.6 Assumption check thresholds

What to conclude: Watch means report the check and keep going. Act means switch method or explain the choice in the paper. Thresholds follow Kim (2013) for skewness and kurtosis and the Welch guidance in Delacre et al. (2017).

11.7 Reading the p value in words

What to conclude: these bands describe evidence, not truth, and none of them says anything about the size of the difference. Pair every one of them with a confidence interval.

11.8 Student against Welch, side by side

What to conclude: the damage from the pooled test is worst when the smaller group carries the larger variance. Reporting Welch every time removes the problem and costs almost no power when the variances really are equal.

Eight worked examples, spanning the range of results a two sample t test actually produces. Every number below was computed with the same engine that powers the calculator above, and every figure is drawn from the real data.

Design this test assumes: a between subjects design with two independent groups, one continuous outcome measured once per unit, and allocation to groups that is either random or based on a pre-existing characteristic the outcome does not cause.

13.1 Steps for collecting data an independent samples t test can accept

1. Define the population and the sampling unit before anything else. The unit is whatever you will count as one row: one patient, one plot, one leaf, one transaction. Every later assumption is about these units.
2. Decide the smallest difference worth detecting. Express it in measurement units first, for example 5 mmHg or 3 exam points, then convert to a d by dividing by the expected standard deviation.
3. Compute the sample size for that difference. Use table 11.4 or the number this calculator reports. Recruit at least that many per group, plus an allowance for dropout.
4. Assign units to the two groups. Randomise where you can, using a random number generator rather than alternation. Where groups are pre-existing, such as burned and unburned sites, record every variable that might differ between them.
5. Keep the groups genuinely independent. One unit contributes one value to one group. Two measurements from the same patient, two leaves from the same tree, or two plots inside the same field are not independent, and they need a paired test or a mixed model instead.
6. Standardise the measurement. Same instrument, same time of day, same observer protocol, same units for both groups. Calibrate before each session and record the calibration.
7. Blind the measurement where possible. The person recording the outcome should not know the group allocation, because expectation shifts borderline readings.
8. Record raw values, never categories. Write 137 mmHg, not "high". You can always bin later, but you cannot recover the number once it is thrown away.
9. Record missing data explicitly. Leave the cell blank and note the reason in a separate column. Never enter zero for missing, and never enter 999.
10. Log every exclusion as it happens. Note which unit was removed, when, and why, before you look at the outcome.
11. Enter the data in long or wide format consistently. Two columns of values for this calculator, or a value column plus a group column for R and Python.
12. Screen before analysing. Plot both groups, check the ranges, look for impossible values and duplicated rows, then run the test once.

13.2 The data layout this calculator expects

Wide format is what the Paste tab and the Upload column picker use: one column per group, and the two columns may be different lengths. Long format is what R's formula interface and pandas prefer. Both describe the same twenty students.

13.3 Minimum sample size

Technically the test runs with two values per group. Practically, fewer than 10 per group gives an effect size estimate too unstable to report, and fewer than 30 per group means normality actually matters. Aim for at least 30 per group unless the expected effect is large, and never plan a study without checking table 11.4 first.

13.4 Pre-registration line

Write this before collecting anything: "Group means will be compared with an independent samples t test using Welch's correction, alpha 0.05, two tailed. The smallest effect size of interest is d = 0.50, requiring 64 participants per group for 80% power. If either group shows skewness beyond ±2, a Mann-Whitney U test will be reported instead."

13.5 Mistakes that make data unusable

• Measuring the same unit twice and treating the two values as independent.
• Recording a percentage change instead of the raw value, which hides the baseline.
• Mixing units between groups, for example pounds in one arm and kilograms in the other.
• Rounding to whole numbers when the effect is smaller than 1 unit.
• Collecting until the p value looks good, which invalidates the test entirely.
• Losing the link between a value and its group label during data entry.

This free independent samples t test calculator is designed for anyone comparing exactly two unrelated groups on one continuous outcome, from a student checking a homework answer to a researcher preparing a manuscript table.

14.1 Use it when

• You have two groups and each observation belongs to only one of them.
• The outcome is continuous: a score, a time, a length, a concentration, a count large enough to behave continuously.
• Observations within each group are independent of each other.
• The data are roughly symmetric, or each group has about 30 or more observations.
• You want the effect size and confidence interval as well as the p value.
• You have only published means, standard deviations and sample sizes and need to reconstruct the test.

14.2 Do not use it when

• The two sets of values are paired, matched or repeated on the same subjects. Use the paired t test.
• There are three or more groups. Use one way ANOVA, then a post hoc test.
• The outcome is a proportion or a count of successes. Use a chi-square test or a proportion test.
• The outcome is ordinal with few categories, such as a five point scale with most answers on two values. Use Mann-Whitney.
• The samples are small and clearly skewed. Use Mann-Whitney or a bootstrap.
• You need to adjust for a covariate such as baseline score or age. Use ANCOVA or regression.
• Observations are clustered, for example pupils inside classes. Use a mixed effects model.

14.3 Four real situations

14.4 Decision tree

16.1 Assumptions

1. Independence of observations. Requires that each value comes from a different unit and no value influences another. Check by reviewing the design, not the data, because no statistic can detect a broken design. If it fails, the standard error is too small, p values are too low, and false positives multiply. Use a paired test or a mixed model instead.
2. Independence of the two groups. Requires that no unit appears in both groups. Check the identifiers for duplicates. If it fails, the test ignores the pairing and loses power, so the p value is too high.
3. Continuous outcome measured on an interval or ratio scale. Requires that differences between values mean the same thing everywhere on the scale. Check the measurement definition. If it fails, on a coarse ordinal scale for example, the mean is not meaningful and Mann-Whitney is the honest choice.
4. Approximate normality within each group. Requires the sampling distribution of each mean to be near normal, which is far weaker than requiring normal raw data. Check with skewness, kurtosis and a plot. If it fails with n under 30, p values are mildly off, usually conservative for symmetric heavy tails and liberal for strong skew.
5. Equal variances, for the pooled Student version only. Requires similar spread in the two populations. Check the variance ratio and Levene's test. If it fails, the pooled test is liberal when the smaller group has the larger variance and conservative when the larger group does. Welch removes the assumption entirely.
6. No influential outliers. Requires that no single value dominates a mean or a variance. Check with a box plot and the 1.5 × IQR fences. If it fails, both the mean and the standard deviation are distorted, usually toward a smaller t. Report the analysis with and without the point.
7. Random or at least representative sampling. Requires the groups to represent the populations you want to talk about. Check the recruitment description. If it fails, the test is still arithmetically valid but the conclusion does not generalise beyond the sample.

16.2 Limitations

1. It compares means only. Two groups can have identical means and completely different distributions. If the shape is the question, compare medians, variances or the full distribution with a Kolmogorov-Smirnov test.
2. It handles exactly two groups. For three or more, use one-way ANOVA, because repeated pairwise t tests inflate the false positive rate.
3. It cannot adjust for covariates. If baseline differences exist between groups, use ANCOVA or a regression model with the covariate included.
4. It assumes one measurement per unit. Repeated measures over time need a mixed effects model or a repeated measures ANOVA.
5. The p value depends on sample size. Any non zero difference becomes significant with enough data, which is why the effect size and interval are reported beside it.
6. Effect size estimates are unstable in small samples. Below about 20 per group, d has a wide interval of its own. Report Hedges' g and be cautious about comparing it with published effects.
7. It says nothing about causation. Random assignment is what supports a causal claim, not the test. With observational groups, a significant t test is evidence of association only.
8. Observed power is not informative. Power computed after the fact from the observed effect merely restates the p value. Plan power in advance instead.

The independent samples t test answers one clean question: are two unrelated groups far enough apart, relative to their own variability, that chance alone is an uncomfortable explanation? This calculator answers it with the exact tail area rather than a table lookup, reports both the pooled and the Welch versions every time, and puts an effect size and a confidence interval next to the p value so the result cannot be read as more, or less, than it is.

17.1 What was found

17.2 What it means in practice

17.3 Cautions

17.4 Next steps

17.5 The three sentences worth remembering

First, a p value below 0.05 tells you a difference is detectable, not that it is important, so read Cohen's d immediately afterwards. Second, the confidence interval is the most useful number on the page, because it is in your own measurement units and it shows what the data actually rule out. Third, Welch is the safer default, and choosing it before you look at the variances is better practice than testing the variances and then deciding.

Used carefully, the independent samples t test calculator above turns two columns of numbers into a defensible, fully reported comparison in about a minute, complete with assumption checks, four charts, and wording you can paste straight into a manuscript.

If this calculator produced numbers that appear in your thesis, report or paper, cite it in the methods section alongside the software you used for everything else.

Six StatsUnlock calculators that pair naturally with a two sample t test. Each one covers a case this page cannot.

• Paired samples t test calculator for before and after measurements on the same subjects, where the values are linked case by case.
• One-way ANOVA calculator for three or more groups, which avoids the false positive inflation of repeated t tests.
• Mann-Whitney U test calculator, the non parametric alternative when the samples are small and clearly skewed.
• One-sample t test calculator for comparing a single group mean against a known benchmark value.
• F test for variance equality calculator for checking the equal variance assumption on its own, in more detail than the panel above.
• Confidence interval calculator for building an interval around a single mean or proportion.

Twenty peer reviewed sources behind the methods, thresholds and recommendations on this page. Every entry links to its DOI.

1. Student. (1908). The probable error of a mean. Biometrika, 6(1), 1-25. doi:10.2307/2331554
2. Welch, B. L. (1947). The generalization of Student's problem when several different population variances are involved. Biometrika, 34(1-2), 28-35. doi:10.1093/biomet/34.1-2.28
3. Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin, 2(6), 110-114. doi:10.2307/3002019
4. Ruxton, G. D. (2006). The unequal variance t-test is an underused alternative to Student's t-test and the Mann-Whitney U test. Behavioral Ecology, 17(4), 688-690. doi:10.1093/beheco/ark016
5. Delacre, M., Lakens, D., & Leys, C. (2017). Why psychologists should by default use Welch's t-test instead of Student's t-test. International Review of Social Psychology, 30(1), 92-101. doi:10.5334/irsp.82
6. Zimmerman, D. W. (2004). A note on preliminary tests of equality of variances. British Journal of Mathematical and Statistical Psychology, 57(1), 173-181. doi:10.1348/000711004849222
7. Levene, H. (1960). Robust tests for equality of variances. In Contributions to Probability and Statistics (pp. 278-292). Reprinted analysis in Journal of the American Statistical Association, 69, 364-367. doi:10.1080/01621459.1974.10482955
8. Brown, M. B., & Forsythe, A. B. (1974). Robust tests for the equality of variances. Journal of the American Statistical Association, 69(346), 364-367. doi:10.1080/01621459.1974.10482955
9. Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3-4), 591-611. doi:10.1093/biomet/52.3-4.591
10. D'Agostino, R. B., & Pearson, E. S. (1973). Tests for departure from normality. Biometrika, 60(3), 613-622. doi:10.1093/biomet/60.3.613
11. Hedges, L. V. (1981). Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107-128. doi:10.3102/10769986006002107
12. Cumming, G. (2014). The new statistics: Why and how. Psychological Science, 25(1), 7-29. doi:10.1177/0956797613504966
13. Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science. Frontiers in Psychology, 4, 863. doi:10.3389/fpsyg.2013.00863
14. Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129-133. doi:10.1080/00031305.2016.1154108
15. Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337-350. doi:10.1007/s10654-016-0149-3
16. Fagerland, M. W. (2012). t-tests, non-parametric tests, and large studies: A paradox of statistical practice? BMC Medical Research Methodology, 12, 78. doi:10.1186/1471-2288-12-78
17. Rasch, D., Kubinger, K. D., & Moder, K. (2011). The two-sample t test: Pre-testing its assumptions does not pay off. Statistical Papers, 52(1), 219-231. doi:10.1007/s00362-009-0224-x
18. Sawilowsky, S. S., & Blair, R. C. (1992). A more realistic look at the robustness and type II error properties of the t test to departures from population normality. Psychological Bulletin, 111(2), 352-360. doi:10.1037/0033-2909.111.2.352
19. Button, K. S., Ioannidis, J. P. A., Mokrysz, C., Nosek, B. A., Flint, J., Robinson, E. S. J., & Munafo, M. R. (2013). Power failure: Why small sample size undermines the reliability of neuroscience. Nature Reviews Neuroscience, 14(5), 365-376. doi:10.1038/nrn3475
20. Amrhein, V., Greenland, S., & McShane, B. (2019). Scientists rise up against statistical significance. Nature, 567(7748), 305-307. doi:10.1038/d41586-019-00857-9

Further reading on effect size conventions: Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum. Book, no DOI, listed separately because the twenty numbered entries above are journal articles only.