The t Distribution 2024

Introduction to t Distribution

The Student’s t distribution or simply t distribution is a probability distribution similar to the normal probability distribution with heavier tails. The t distribution produces values that fall far from the average compared to the normal distribution. The t distribution is an important statistical tool for making inferences about the population parameters when the population standard deviation is unknown.

The t-distribution is used when one needs to estimate the population parameters (such as mean) but the population standard deviation is unknown. When $n$ is small (less than 30), one must be careful in invoking the normal distribution for $\overline{X}$. The distribution of $\overline{X}$ depends on the shape of the population distribution. Therefore, no single inferential procedure can be expected to work for all kinds of population distributions.

One Sample t-Test Formula

If $X_1, X_2, \cdots, X_n$ is a random sample from a normal population with mean $\mu$ and standard deviation of $\sigma$, the sample mean $\overline{X}$ is exactly distributed as normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$ and $Z=\frac{\overline{X} – \mu}{\frac{\sigma}{\sqrt{n}}}$ is a standard normal variable. When $\sigma$ is unknown, the sample standard deviation is used, that is
$$t=\frac{\overline{X} – \mu}{\frac{s}{\sqrt{n}}},$$
which is analogous to the Z-statistic.

The Sampling Distribution for t

Consider samples of size $n$ drawn from a normal population with mean $\mu$ and for each sample, we compute $t$ using the sample $\overline{X}$ and sample standard deviation $S$ (or $s$), the sampling distribution for $t$ can be obtained

$$Y=\frac{k}{\left(1 + \frac{t^2}{n-1}\right)^{\frac{n}{2}} } = \frac{k}{\left(1+ \frac{t^2}{v} \right)^{\frac{v+1}{2} }},$$
where $k$ is a constant depending on $n$ such that the total area under the curve is one, and $v=n-1$ is called the number of degrees of freedom.

The t distributions are symmetric around zero but have thicker tails (more spread out) than the standard normal distribution. Note that with the large value of $n$, the t-distribution approaches the standard normal distribution.

Properties of the t Distribution

• The t distribution is bell-shaped, unimodal, and symmetrical around the mean of zero (like the standard normal distribution)
• The variance of the t-distribution is always greater than 1.
• The shape of the t-distribution changes as the number of degrees of freedom changes. So, we have a family of $t$ distributions.
• For small values of $n$, the distribution is considerably flatter around the center and more spread out than the normal distribution, but the t-distribution approaches the normal as the sample size increases without limit.
• The mean and variance of the t distribution are $\mu=0$ and $\sigma^2 = \frac{v}{v-2}$, where $v>2$.

Common Application of t Distribution

• t-tests are used to compared means between two groups
• t-test are used to compared if a sample mean is significantly different from a hypothesized population mean.
• t-values are used for constructing confidence intervals for population means when the population standard deviation is unknown.
• Used to test the significance of the correlation and regression coefficients.
• Used to construct confidence intervals of correlation and regression coefficients.
• Used to estimate the standard error of various statistical models.

Assumptions of the t Distribution

The t-distribution relies on the following assumptions:

• Independence: The observations in the sample must be independent of each other. This means that the value of one observation does not influence the value of another.
• Normality: The population from which the sample is drawn should be normally distributed. However, the t-distribution is relatively robust to violations of this assumption, especially for larger sample sizes.
• Homogeneity of Variance: If comparing two groups, the variances of the two populations should be equal. This assumption is important for accurate hypothesis testing.

Note that significant deviations from normality or unequal variances can affect the accuracy of the results. Therefore, it is always a good practice to check the assumptions before conducting a t-test and consider alternative non-parametric tests if the assumptions are not met.

Download Student’s t Distribution Table

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