Three indispensable sampling distributions. Understand why small samples demand heavier tails, why categorical tests use chi-square, and how ANOVA relies on the F-distribution. Adjust degrees of freedom and watch the shapes transform in real time.
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t-Distribution: Symmetric, heavier tails than normal. Used with small samples and unknown σ.
Low df → very heavy tails. t-critical values are much larger than z-critical values.
Each panel shows t (solid teal) vs standard normal (dashed gray). The badge shows the density overlap: the share of probability area the two curves have in common.
Three key sampling distributions, each with a distinct use case.
The t-distribution is your go-to whenever the population standard deviation is unknown and you're working with small samples.
Example
Sample of 15 students (mean GPA 3.2, s = 0.4). Test if GPA differs from 3.0. Use t with df = 14.
Chi-square tests work on counts and frequencies. The distribution itself is always positive and right-skewed.
Example
Roll a die 300 times. Test if it is fair. Use chi-square goodness-of-fit with df = 5.
The F-distribution is defined as the ratio of two scaled chi-square variables. It arises whenever you compare variances.
Example
Three production lines, 30 samples each. Test if they have equal output variances using Levene's test, an F-test with df1 = 2, df2 = 87.