Chi-Square Test Calculator Formula

Understand the math behind the chi-square test calculator. Each variable explained with a worked example.

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Formulas Used

Chi-Square Statistic

chi_sq = c1 + c2 + c3 + c4

Degrees of Freedom (k-1)

df = 3

Total Observed

total_observed = o1 + o2 + o3 + o4

Total Expected

total_expected = e1 + e2 + e3 + e4

Variables

VariableDescriptionDefault
o1Observed 145
e1Expected 140
o2Observed 235
e2Expected 240
o3Observed 325
e3Expected 330
o4Observed 445
e4Expected 440
c1Derived value= pow(o1 - e1, 2) / e1calculated
c2Derived value= pow(o2 - e2, 2) / e2calculated
c3Derived value= pow(o3 - e3, 2) / e3calculated
c4Derived value= pow(o4 - e4, 2) / e4calculated

How It Works

How to Perform a Chi-Square Test

Formula

chi-square = Sum of [(Oi - Ei)^2 / Ei]

Compare the computed chi-square statistic to the chi-square distribution with k-1 degrees of freedom (where k is the number of categories). If the statistic exceeds the critical value, reject the null hypothesis that the observed distribution matches the expected distribution.

Worked Example

Four categories: observed 45, 35, 25, 45; expected 40, 40, 30, 40.

o1 = 45e1 = 40o2 = 35e2 = 40o3 = 25e3 = 30o4 = 45e4 = 40
  1. 01c1 = (45-40)^2/40 = 25/40 = 0.625
  2. 02c2 = (35-40)^2/40 = 25/40 = 0.625
  3. 03c3 = (25-30)^2/30 = 25/30 = 0.8333
  4. 04c4 = (45-40)^2/40 = 25/40 = 0.625
  5. 05Chi-square = 0.625 + 0.625 + 0.8333 + 0.625 = 2.7083
  6. 06df = 4 - 1 = 3
  7. 07Critical value at alpha=0.05, df=3 is 7.815
  8. 08Since 2.7083 < 7.815, do not reject H0

Frequently Asked Questions

What is the null hypothesis for a chi-square test?

For a goodness-of-fit test, H0 is that the observed frequencies match the expected distribution. For a test of independence, H0 is that the two categorical variables are independent.

What if expected frequencies are less than 5?

The chi-square approximation is unreliable when expected frequencies are below 5. Consider combining categories or using Fisher's exact test instead.

Can chi-square detect the direction of the difference?

No. Chi-square only detects that a difference exists, not the direction. To understand which categories differ, examine the individual components (residuals).

Learn More

Guide

Hypothesis Testing Guide

Learn how hypothesis testing works step by step. Covers null and alternative hypotheses, test statistics, p-values, significance levels, and common pitfalls to avoid.

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