The t-distribution is a symmetric, bell-shaped probability distribution whose exact shape is controlled by a single parameter called the degrees of freedom. Its defining property is that it has heavier tails (more probability in the extremes) than the standard normal distribution, which makes it the correct distribution to use when you estimate a population mean from a small sample and the population standard deviation is unknown.
This guide focuses on the properties of the t-distribution — its shape, symmetry, tail behaviour, mean and variance, and how every one of these properties depends on the degrees of freedom. We then explain how the t-distribution differs from the normal distribution, when to use it, and work through a complete numerical example. For a broader walkthrough of t-tests and how the distribution is applied in practice, see our companion guide, everything you need to know about t-distributions.
What Is a T-Distribution?
The t-distribution — also called Student’s t-distribution — is a continuous probability distribution that arises when you estimate the mean of a normally distributed population using a small sample, where the population standard deviation is unknown and must be estimated from the data itself. It looks very similar to the normal distribution but accounts for the extra uncertainty introduced by estimating the standard deviation from a limited number of observations.
As the sample size grows, that extra uncertainty shrinks, and the t-distribution converges towards the standard normal distribution (mean 0, standard deviation 1). For large samples the two are almost indistinguishable.
“Student’s t-distribution… arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population’s standard deviation is unknown.” — Encyclopaedia Britannica, “Student’s t-test”
The distribution was published in 1908 by William Sealy Gosset, a chemist at the Guinness brewery in Dublin, under the pen name “Student” — which is why it is still known as Student’s t-distribution today.
Quick recap of the key terms
- Population: all members of the defined group you want to generalise your findings to. See population vs sample for the full distinction.
- Parameter: a measurable characteristic of the population, such as the population mean (μ) or population standard deviation (σ).
- Standard deviation: one of the most widely used measures of variability — the typical distance between each data point and the mean.
- Sample size (n): the number of observations selected for the study.
- Degrees of freedom (df): for a one-sample t-procedure, df = n − 1. This single number determines the exact shape of the t-distribution.
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Degrees of Freedom Explained
There is not one t-distribution but a whole family of them, and the member you use depends on the degrees of freedom (df). Degrees of freedom are the number of values in a calculation that are free to vary. For a one-sample t-procedure with sample size n, one degree of freedom is “used up” estimating the sample mean, so:
df = n − 1
For example, a sample of 10 observations has 9 degrees of freedom. The degrees of freedom matter because they set the exact shape of the curve:
- Low df (small sample): the curve is shorter at the centre and has noticeably fatter tails — more probability is pushed into the extremes to reflect the greater uncertainty.
- High df (large sample): the tails thin out, the peak rises, and the curve becomes almost identical to the standard normal distribution.
As df approaches infinity, the t-distribution becomes the standard normal distribution exactly. In practice, once df is around 30 or more, the difference is small enough that many textbooks treat the two as interchangeable.
Properties of the T-Distribution
These are the defining properties of the t-distribution. Each one is exact, and several of them depend directly on the degrees of freedom (ν, the Greek letter nu).
- It is bell-shaped and symmetric. Like the standard normal distribution, the t-distribution is symmetric about its centre, with a mean, median and mode all equal to 0 (the mean is only defined for df > 1).
- It is continuous and ranges over all real numbers. The t variable can take any value from −∞ to +∞.
- Its shape depends on the degrees of freedom. There is a different t-distribution for every value of df. This is the property that most distinguishes it from the single, fixed standard normal curve.
- It has heavier (fatter) tails than the normal distribution. More probability sits in the extremes, so extreme values are more likely than under a normal curve. This is why the curve is lower and flatter at the centre — a shape described as leptokurtic (excess kurtosis greater than 0), not platykurtic.
- Its variance is greater than 1 (for df > 2). The variance is given by Var(t) = ν / (ν − 2), which is defined only when ν > 2 and is always larger than the variance of 1 for the standard normal distribution. As ν increases, the variance shrinks towards 1.
- It converges to the standard normal distribution. As the degrees of freedom (and hence the sample size) increase, the t-distribution becomes indistinguishable from the standard normal distribution. A sample is conventionally considered “large” when n ≥ 30.
| Property | T-Distribution |
|---|---|
| Shape | Symmetric, bell-shaped |
| Centre (mean/median/mode) | 0 (mean defined for df > 1) |
| Range | −∞ to +∞ |
| Controlling parameter | Degrees of freedom (df = n − 1) |
| Tails | Heavier than the normal distribution |
| Kurtosis | Leptokurtic (excess kurtosis > 0) |
| Variance | ν / (ν − 2), for ν > 2 (always > 1) |
| Large-sample behaviour | Approaches the standard normal distribution |
T-Distribution vs Normal Distribution
The t-distribution and the standard normal distribution share the same symmetric, bell-shaped form centred on 0, but they differ in three important ways. The chart below contrasts a t-distribution (with low degrees of freedom) against the standard normal curve.
- Tails: the t-distribution has heavier tails, so it assigns more probability to extreme outcomes. The normal distribution has lighter tails.
- Peak: the t-distribution is lower and flatter at the centre; the normal distribution is taller and more peaked.
- Number of curves: there is only one standard normal distribution, but a whole family of t-distributions — one for each value of the degrees of freedom.
| Feature | T-Distribution | Standard Normal Distribution |
|---|---|---|
| Shape parameter | Degrees of freedom | None (fixed curve) |
| Tails | Heavier | Lighter |
| Peak height | Lower / flatter | Higher / sharper |
| Variance | > 1 (shrinks to 1 as df grows) | Exactly 1 |
| Use when σ is… | Unknown (estimated from sample) | Known |
| Best for sample size | Small (n < 30) | Large (n ≥ 30) |
When to Use the T-Distribution
Use the t-distribution rather than the normal distribution when all of these conditions hold:
- You are making inferences about a population mean (for example, building a confidence interval or running a hypothesis test).
- The population standard deviation (σ) is unknown and is estimated from the sample standard deviation (s).
- The sample size is small (typically n < 30), or the population can reasonably be assumed to be approximately normal.
If the population standard deviation is known, or the sample is large, the standard normal (z) distribution is appropriate instead. For a step-by-step decision aid covering all the common tests, see our guide on which statistical test you should use. The t-distribution underpins several of the most common procedures in inferential statistics:
- One-sample t-test — comparing a sample mean to a known or hypothesised value.
- Independent two-sample t-test — comparing the means of two separate groups.
- Paired (dependent) t-test — comparing before-and-after measurements on the same subjects.
- Confidence intervals for a mean and regression coefficients — where t-values are used to test significance and find p-values.
The full mechanics of each of these tests, including the formulas for the two-sample and paired cases, are covered in our companion article, everything you need to know about t-distributions.
Worked Example: A One-Sample T-Test
The t-statistic for a one-sample test measures how many estimated standard errors the sample mean lies from the hypothesised population mean. It is calculated as:
t = (x̄ − μ) / (s / √n)
where x̄ is the sample mean, μ is the hypothesised population mean, s is the sample standard deviation, and n is the sample size. The denominator, s / √n, is the standard error of the mean.
Step 1 — Degrees of freedom: df = n − 1 = 16 − 1 = 15.
Step 2 — Standard error: s / √n = 8 / √16 = 8 / 4 = 2 ml.
Step 3 — Test statistic: t = (246 − 250) / 2 = −4 / 2 = −2.00.
Step 4 — Critical value: for a two-tailed test at α = 0.05 with 15 degrees of freedom, the critical t-value from the t-table is approximately ±2.131.
Step 5 — Decision: the calculated |t| = 2.00 is less than the critical value 2.131, so we fail to reject the null hypothesis. At the 5% level there is not quite enough evidence to conclude the cups are under-filled.
Notice the role of the heavier tails: had we (incorrectly) used the normal distribution, the critical value would have been only ±1.96, and we would have rejected the null hypothesis. The t-distribution’s fatter tails guard against over-confident conclusions from small samples.
The t-score itself (here, −2.00) is simply the number of estimated standard errors the sample mean lies from the hypothesised mean. T-scores are used to find the bounds of a confidence interval and to obtain p-values for t-tests and regression coefficients. You can look one up in a t-table or with any t-distribution calculator.
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