Explain why t distributions tend to be flatter and more spread out than the normal distribution.

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Harold Beltran · Accepted Answer

We can solve this exercise by analyzing the formulas of the z-scores, which form a normal distribution, and of the tt statistics, which form the tt distribution.The formula for the z-score is:  z  =            M      −      μ              s              M              =            M      −      μ                                σ                      2                          n            Analyzing the formula above, we can notice that the only variable for the z distribution is the value of the sample mean M, which varies from one sample to another. All other values (  μ ,       σ          2      , and n) are constants from the population, meaning their values do not change.On the other hand, the formula for the tt statistic is:  t  =            M      −      μ              σ              M              =            M      −      μ                                s                      2                          n            Analyzing it, we can notice that we have variables on the numerator and on the denominator of the formula. Like in the formula for the z-score, the value of M is variable. But a side from M, we also have the sample variance       s          2      , which also varies from one sample to another. This makes the t statistic more variable than the z-score, which makes the t distribution flatter and more spread out than the normal z distribution.

Explain why t distributions tend to be flatter and more spread out than the normal distribution.

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