Table of Critical Value Calculation N and Population

Critical values are essential in statistical hypothesis testing. They help determine whether sample data is statistically significant or if it could have occurred by random chance. This guide explains how to calculate critical values for different sample sizes and population distributions.

What is a Critical Value?

A critical value is a threshold value from a statistical table that is used to determine whether results are statistically significant. When comparing a test statistic to a critical value, you can decide whether to reject or fail to reject the null hypothesis.

Critical values are typically found in statistical tables for common distributions like the t-distribution, normal distribution, chi-square, and F-distribution. The specific critical value you need depends on:

The type of test (one-tailed or two-tailed)
The significance level (α) you're using
The degrees of freedom (n-1 for sample size n)
The distribution being used

For example, in a t-test with a sample size of 15 and a significance level of 0.05, you would look up the critical value in the t-distribution table with 14 degrees of freedom.

How to Calculate Critical Values

The exact method for calculating critical values depends on the statistical test you're performing. However, the general approach is:

Determine the degrees of freedom (df = n - 1 for sample size n)
Choose the appropriate distribution (t, normal, chi-square, etc.)
Select the significance level (α) you want to use
Look up the critical value in the appropriate table or use statistical software

Formula for Degrees of Freedom

df = n - 1

Where n is the sample size

For most common tests, you can use the calculator on this page to find critical values. The calculator uses standard statistical tables and provides results for common distributions.

Critical Value Table

The following table shows critical values for common distributions at different significance levels:

Distribution	Degrees of Freedom	α = 0.05	α = 0.01
t-Distribution	10	2.228	3.169
t-Distribution	20	2.086	2.528
t-Distribution	30	2.042	2.457
Normal (Z)	N/A	1.960	2.576
Chi-Square	10	18.307	23.209
Chi-Square	20	30.144	36.191

Note: These values are approximate. For precise calculations, use statistical software or more comprehensive tables.

Example Calculation

Let's calculate the critical value for a one-sample t-test with n=15 and α=0.05:

Calculate degrees of freedom: df = n - 1 = 15 - 1 = 14
Look up the t-distribution table for df=14 and α=0.05 (two-tailed)
The critical value is approximately 2.145

Interpretation

This means that if your calculated t-statistic is greater than 2.145 or less than -2.145, you can reject the null hypothesis at the 0.05 significance level.

FAQ

What's the difference between a critical value and a p-value?

A critical value is a fixed threshold from a statistical table, while a p-value is a calculated probability that your results occurred by random chance. Both are used to determine statistical significance, but they're based on different approaches.

How do I know which distribution to use?

The distribution depends on your specific statistical test. Common choices include t-distribution for small samples, normal distribution for large samples, chi-square for variance tests, and F-distribution for ratio tests.

What if my degrees of freedom aren't listed in the table?

For degrees of freedom not in standard tables, you can use statistical software or linear interpolation between nearby values to estimate the critical value.