“Tell Me About Statistics!” vol.62 What is t distribution?

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In this section, we describe this distribution. This mountain graph is used when examining the results of various tests in terms of deviation values.

“t distribution” is a distribution used for statistical estimation and statistical tests, and the shape of the distribution is very similar to the z distribution.

The t-distribution was introduced by William Seeley Gossett in 1908. There seem to be various theories, but at the time, he was working at a beer brewing company. The company forbade the publication of employee papers; hence, he published his papers under the pseudonym “Student.” For this reason, the t-distribution came to be called the “Student’s t-distribution.”

The shape of the t-distribution graph depends on the parameter f (degree of freedom). When f is sufficiently large, it matches the z distribution (standard normal distribution). However, as f becomes smaller, the distribution becomes more gradual and wider than the z distribution (Figure 1).

[Figure1]

Note that the degree of freedom f depends on the sample size of the sample survey. The method for determining f differs depending on the estimation and test formula. The t distribution with degrees of freedom f = 1, f = 5, and f = 100 is shown in Figure 2.

[Figure2]

The number of data items that can take any value is called degrees of freedom. For example, if the sample mean calculated from data with a sample size of 3 is 5, the first and second values can be taken freely. Given 3 and 4, the third value must have a sample mean of 5, so it can only be “8.”

(3+4+x)÷3=5

x=8

In other words, the number of data items that can take any value is reduced by one, and the degree of freedom f is now two. The degree of freedom f for calculating the sample mean is the sample size n minus 1.

Degree of freedom f=n-1

There are two types of variance: sample variance and population variance.

[Formula]

Sample fraction: (observed data – sample mean) 2÷(n-1)

Sum the individual values found.

Population variance: (observed data – population mean) 2÷n

Sum the individual values found.

The sample mean varies because of survey errors, whereas the population mean does not. The denominator of the sample variance is degrees of freedom n-1, and the denominator of the population variance is n without considering the degrees of freedom.

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