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Used for determining the confidence interval for means or for determining whether two means are significantly different. Developed by Gossett under the pseudonym “Student; hence, also referred to as Student’s t-distribution.

## What is a T Distribution?

The T distribution (also known as the Student’s t-distribution) is a type probability distribution similar to the normal distribution. However, it has larger tails and a bell shape. The T distributions are more likely to have extreme values than normal distributions. This is why they have fatter tails.

### KEY TAKEAWAYS

• The T distribution is a continuous probability distribution for the z-score if the estimated standard deviation is used as the numerator and not the true standard deviation.
• Like the normal distribution, the T distribution is bell-shaped, symmetrical, and has heavier tails. This means that it tends produce values far below its mean.
• Statistics use T-tests to determine significance.

## What does a T distribution tell you?

The parameter degreesof freedom determines the tail heaviness. Lower values give heavier tails. Higher values make the T distribution look like a normal distribution with a median of 0 and a standard deviation 1. The T distribution is also known by the “Student’s T Distribution.”

A sample of n observations taken from a normal distribution population with mean M and standard deviation d will yield the sample mean, M, and the sample standard deviation (d). This is due to the randomness of this sample.

## What is the difference between a T distribution and a normal distribution?

Normal distributions are used when the population distribution assumes to be normal. The T distribution is identical to the normal distribution but with fatter tails. Both assume normal distributions. Normal distributions are more kurtosis-friendly than T distributions. A T distribution has a higher chance of getting values that are very far off the mean than a normal distribution.

## Limitations to Using a T Distribution

The T distribution can cause deviations from the normal distribution. The T distribution is only useful when there is a need to maintain perfect normality. If the population standard deviation is unknown, the T-distribution should not be used. The normal distribution should be used if the population standard deviation can be determined and the sample size is sufficient to yield better results.

The t distribution is very similar to a normal distribution. It is mathematically defined. Let’s not get into complicated math. Instead, let us look at the t-distribution useful properties and why they are important for analyses.

• The t-distribution is similar to the normal distribution. It has a smooth form.
• The t-distribution, like the normal distribution is symmetric. Each side is identical if you fold it in half at the median.
• The t-distribution is similar to a normal distribution (or the z-distribution). It has a means of zero.
• Normal distribution assumes that there is a standard deviation in the population. This assumption is not made by the t-distribution.
• The degrees define the distribution. These are dependent on the sample size.
• The t-distribution works best for small sample sizes when the population standard deviation cannot be determined.
• The t-distribution becomes closer to a normal distribution as the sample size increases.