Look up critical values from z, t, chi-square and F distributions for hypothesis testing.
A distribution table reference provides critical values and tail probabilities for the most-used probability distributions: standard normal (z), Student's t, chi-square (χ²), and F distribution. These tables are essential for hypothesis testing, confidence interval construction, and any quantitative statistical inference.
The standard normal distribution (mean 0, sd 1) underpins everything in classical statistics. Z = (X − μ)/σ converts any normal distribution to standard normal so a single table covers all cases. The t-distribution accommodates small samples; chi-square handles variance and goodness-of-fit; F handles ratios of variances.
Inputs: Common critical values for two-tailed α = 0.05
Use z (standard normal) when the population standard deviation σ is known, or when the sample size is large (n > 30) and you are using the sample standard deviation s as a stand-in. Use t when σ is unknown and the sample is small (n < 30). The t-distribution has heavier tails to account for the extra uncertainty from estimating σ; as n grows, t converges to z.
Degrees of freedom (df) measure the number of independent pieces of information in your data set after using some to estimate parameters. For a sample variance computed from n data points, df = n − 1 because one degree was used to estimate the mean. For chi-square goodness-of-fit on k categories with one parameter estimated, df = k − 2. Always check the specific df rule for the test you are using.
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