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.

## 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.

## 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.

*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 |

*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***.

### 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.

## FAQs

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.

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.

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.