Direct comparison
t-Distribution vs Normal Distribution
T-distribution vs normal: both bell-shaped, but t has heavier tails set by df. See 95% critical values from 12.706 down to 1.960, and when to use each.
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How do t-distribution, Normal distribution compare side by side?
The table below compares t-distribution, Normal distribution across 11 procurement-relevant dimensions, from shape and symmetry through source.
Side-by-side comparison
| Dimension | t-distribution | Normal distribution |
|---|---|---|
| Shape and symmetry | Symmetric and bell-shaped. NIST: the plots for different df "all have a similar shape." | Symmetric and bell-shaped. Skewness 0. |
| Parameters | One shape parameter, nu (degrees of freedom). In a testing context it is a standardized distribution with no location or scale parameter. | Location mu and scale sigma. mu = 0 and sigma = 1 is the standard normal distribution. |
| Tails | Heavier tails. NIST: the difference between t distributions and the normal is "in the heaviness of the tails." | Lighter tails than the t-distribution. Kurtosis 3. |
| Mean | 0 (also the median and mode). Undefined when nu = 1. | The location parameter mu (also the median). |
| Standard deviation | sqrt(nu / (nu - 2)). Undefined when nu is 1 or 2. | The scale parameter sigma. |
| Skewness and kurtosis | Skewness 0 (undefined for nu of 3 or less, though the distribution is symmetric in all cases). Kurtosis 3(nu - 2)/(nu - 4), undefined for nu of 4 or less. | Skewness 0 and kurtosis 3 always. |
| Critical value, two-sided 95% | 12.706 (df 1), 4.303 (df 2), 2.571 (df 5), 2.228 (df 10), 2.042 (df 30), 1.984 (df 100). | 1.960. This is also the infinity-df row of the NIST t table. |
| Relationship | Approaches the normal as nu becomes large; the approximation is "quite good" for nu above 30. With nu = 1 it is the Cauchy distribution. | The distribution the t-distribution approaches as nu becomes large. |
| Typical role in inference | NIST: used for the critical regions of hypothesis tests and for confidence intervals. Its confidence interval for a mean uses the sample standard deviation s and a t percentile with N - 1 df. | NIST: probably the most important distribution in statistics; many classical statistical tests assume the data follow it. |
| When to pick each | Pick t when the interval or test relies on a sample standard deviation and the sample is modest, so the heavier tails and the larger critical value matter. | Pick the normal when the distribution of the data or the statistic is the normal (known scale), or when df is large and the 95% critical values 1.984 and 1.960 are close enough for your purpose. |
| Source | NIST/SEMATECH e-Handbook of Statistical Methods: t Distribution (https://www.itl.nist.gov/div898/handbook/eda/section3/eda3664.htm), t critical values (https://www.itl.nist.gov/div898/handbook/eda/section3/eda3672.htm) and Confidence limits for the mean (https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm), all read 2026-10-29. | NIST/SEMATECH e-Handbook of Statistical Methods: Normal Distribution (https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm), read 2026-10-29. |
Common questions
Common questions about t-distribution vs Normal distribution
When should I use the t-distribution instead of the normal distribution?
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Use the t-distribution when a confidence interval or test uses a standard deviation estimated from the sample. NIST's confidence interval for a mean uses the sample standard deviation s with a t percentile at N - 1 degrees of freedom. Use the normal when the scale is given or when df is large enough that the two critical values are nearly the same.
At what degrees of freedom does the t-distribution become the normal distribution?
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It never equals the normal at a finite df; it approaches it as df grows. NIST says the approximation is quite good for df above 30, which is a description, not a hard cutoff. In the NIST table the two-sided 95% critical value is 2.042 at 30 df, 1.984 at 100 df and 1.960 in the infinity row.
Why does the t-distribution have heavier tails?
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NIST describes the difference between t distributions as lying in the heaviness of their tails, set by the degrees of freedom. Fewer df means heavier tails: at 1 df the t-distribution is the Cauchy distribution, whose mean is undefined, and the standard deviation is undefined at 1 or 2 df.
Is the t-distribution ever the same as the normal distribution?
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Only in the limit of very large df. At finite df the standard deviation sqrt(nu / (nu - 2)) exceeds 1, whereas the standard normal has sigma = 1, and the critical values differ: for example 2.228 at 10 df against 1.960.
Does the t-distribution require normal data?
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The distribution itself is defined by its density, not by your data. NIST notes that many classical statistical tests assume the data follow a normal distribution, and those tests are where the t-distribution is used. See the CASRAI guide to normality of distribution for how to assess that assumption.
