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

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