# T-Distribution – Definition and How to Use It

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The t distribution in statistics is an alternative to the normal distribution used when sample sizes are small. It allows for the estimation of confidence intervals and the determination of critical values. Discover the representation of t distribution and how you can utilize it in research for estimating population parameters for small sample sizes.

## T Distribution – In a Nutshell

• T distribution, also known as the student’s t distribution, is a probability distribution used to determine confidence levels.
• It has a bell shape graphical representation and a heavier tail than a normal distribution.
• Despite the symmetric shape similar to the z distribution, the t distribution tends to produce values far from the mean.
• T-tests estimate significance and find the matching p-value in a statistical test. T distribution is also used in regression1

## Definition: T Distribution

Statistical studies used limited sample sizes in the past, and research needed various approaches to gather more sample information. Research studies with small sample sizes rely on t distribution to make educated guesses on the population.

T distribution is a standard distribution used for small sample sizes. You use t distribution when you need to analyze the mean, when the standard deviation is unknown, and especially when the sample size is smaller than 30.

The distributed data usually forms a bell shape on a graph with fewer observations on the tails compared to the mean. Since it is a conservative type of standard normal distribution, it has a heavier tail that gives it a lower probability at the center.

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## The T Distribution

This distribution was developed by William Sealy Gosset in 1908 to be used as a continuous probability distribution in small sample sizes.2 Back then, the z distribution was available for testing mean, but they required larger sample sizes.

The distribution was designed to factor in the uncertainty associated with small sample sizes. Hence, it describes the variability of distances between a sample mean and population mean since the standard deviation is unknown. T distribution has one parameter, the degrees of freedom based on the data set.

Example: Z Distribution vs. T Distribution

The t and z distribution representations on a graph are similar but have a few differences due to standard deviation.

Like the normal distribution graph, the t distribution has a smooth and symmetric shape; you can fold it in half at the mean.

T and z distributions have a mean of zero, but the degree of freedom in the t distribution makes it have a heavier tail.

## T Distribution vs. Z Distribution

When the degree of freedom, the total observations minus one, increases, the t distribution is almost identical to the normal distribution.3 At a degree of freedom of 30, the t distribution graph becomes similar to the standard normal distribution. Hence, as the sample size increases, you can use the z distribution instead of the t distribution. Some of the differences between the z distribution and t distribution include:

 T distribution Z distribution Defined by the mean, degree of freedom and standard deviation Defined only by standard deviation and mean Has a heavier tail, and the data is far from the mean Data is centered around the mean The standard deviation value is unknown Standard deviation is known Used with small sample sizes Used with large sample sizes

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## T Distribution – T Scores

A t-score or t-value represents the standard deviations from the mean in the t distribution table. The t-score is a test statistic that shows how far an observation is from the mean on a t distribution table. You can find the t-score from the t-table or calculate it using an online t-value calculator. You use the t-scores to find the following:

• The p-value in the test statistic and use it or regression and t-tests.
• The upper and lower ranges of the confidence intervals when your data is almost normally distributed.

### Confidence intervals

Researchers use t-scores to create the upper and lower limits of confidence intervals. The t-value used to generate the lower and upper ranges of the prediction interval is called the critical value noted by t or t*.

Example:

When you sample 20 students from different courses to estimate the average standardized test scores in the same topic, you use t-scores to determine differences in the groups.

Using the two-tailed t-test, you can come up with an estimate and a confidence interval based on the estimate.

If the confidence interval doesn’t cross zero or is far from zero, there is no difference between the test scores between the two groups.

### P-values

When studying a sample, your goal is to determine how far your data is from the research null hypothesis using the test statistics. The statistical tests usually go a step further to determine the likelihood of the data similarities using the p-value.

Since the test statistic for regression and t-tests is t-score, you can identify the p-values in a t-table using the degrees of freedom and p-value. When the t-score produces a p-value lower than the statistical significance range, it is called the critical value.

Example:

In the sample of 20 students, the t-tests will generate a t-score of 12.79. Hence, the two groups have a difference of 12.79 standard deviations from the mean in the null hypothesis.

Since the degree of freedom is 38 (n-1 for both groups), you can look up the p-value, which gives you the same conclusion as the confidence interval. That means you will need help finding the difference between the groups.

## FAQs

#### What is a t-distribution?

A t-distribution is a normal distribution used for small sample sizes that don’t have a known variance value. It is used to find the p-value and confidence interval when data is normally distributed or in regression analysis.

#### What is a test statistic?

It is a value generated from statistical tests that describes how far or close your observations are to the null hypothesis. The test statistic will tell you how different a group is from the rest of the population.

#### What is the difference between the z and t-distribution?

The t-distribution uses a smaller sample size than the z-distribution, and you need to increase the sample size or attain the same level of statistical significance.

## Sources

1 Ali, Parveen, and Ahtisham Younas. “Understanding and Interpreting Regression Analysis.” Evidence-Based Nursing. Royal College of Nursing. October 1, 2021. https://ebn.bmj.com/content/24/4/116.

2 Frost, Jim. “T Distribution: Definition & Uses.” StatisticsByJim. June 13, 2022. https://statisticsbyjim.com/probability/t-distribution/.

3 Brereton, Richard G. “The T‐Distribution and Its Relationship to the Normal Distribution.” Journal of Chemometrics. June 7, 2015. https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/cem.2713.

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