# Student’s t-distribution

Student’s t-distribution is a probability distribution used in statistics to estimate population parameters when the sample size is small and the population variance is unknown.
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Updated: Jun 5, 2024

## 3 key takeaways

• The t-distribution is similar to the normal distribution but has heavier tails.
• It is used primarily in hypothesis testing and constructing confidence intervals for small sample sizes.
• The shape of the t-distribution depends on the degrees of freedom, becoming more like the normal distribution as the degrees of freedom increase.

## What is Student’s t-distribution?

Student’s t-distribution, often simply called the t-distribution, is a type of probability distribution that is symmetric and bell-shaped, much like the normal distribution. However, it has heavier tails, which means it is more prone to producing values that fall far from its mean. This characteristic makes the t-distribution particularly useful for small sample sizes or when the population variance is unknown.

The t-distribution was first introduced by William Sealy Gosset under the pseudonym “Student” while he was working at the Guinness Brewery. He developed it as a way to handle the problem of small sample sizes in quality control of the brewing process.

## Key properties of the t-distribution

The t-distribution has several key properties that distinguish it from the normal distribution.

• Degrees of Freedom: The t-distribution is defined by its degrees of freedom (df), which is typically the sample size minus one (n-1). As the degrees of freedom increase, the t-distribution approaches the normal distribution.
• Shape: The t-distribution is symmetric and bell-shaped, but with heavier tails compared to the normal distribution. This means it is more likely to produce values that fall far from the mean.
• Mean and Variance: The mean of the t-distribution is zero, and its variance is greater than one, especially for small sample sizes. As the sample size increases, the variance approaches one.

## Applications of the t-distribution

The t-distribution is primarily used in hypothesis testing and constructing confidence intervals, especially when dealing with small sample sizes or unknown population variances.

• Hypothesis Testing: The t-distribution is widely used in hypothesis testing, particularly in the t-test, which assesses whether the means of two groups are statistically different from each other. It is especially useful when the sample sizes are small and the population variance is unknown.
• Confidence Intervals: It is also used in constructing confidence intervals for population means when the sample size is small. The t-distribution provides a more accurate interval estimate under these conditions.

### When to use the t-distribution

The t-distribution is most appropriate in the following scenarios:

• When dealing with small sample sizes (typically less than 30).
• When the population variance is unknown.
• When the data can be assumed to follow a normal distribution.

Understanding related concepts can enhance your comprehension of Student’s t-distribution and its applications. Consider exploring topics such as hypothesis testing, confidence intervals, degrees of freedom, and the normal distribution. These will provide additional context and deepen your understanding of the t-distribution’s role in statistics.

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