Chi Square Formula

The Chi-square formula is used in the Chi-square test to compare two statistical data sets. Chi-Square is one of the most useful non-parametric statistics. The Chi-Square test is used in data consist of people distributed across categories, and to know whether that distribution is different from what would expect by chance.

  • A very small Chi-Square test statistic means that your observed data fits your expected data extremely well.
  • A very large Chi-Square test statistic means that the data does not fit very well. If the chi-square value is large, you can reject the null hypothesis.

Chi-Square is one way to show a relationship between two categorical variables. There are two types of variables in statistics: numerical variables and non-numerical variables. The value can be calculated by using the given observed frequency and expected frequency.

Formula for Chi-Square Test

The Chi-Square is denoted by χ2 and the formula is:

χ2 = ∑ (O − E)2 / E


  • O = Observed frequency
  • E = Expected frequency
  • ∑ = Summation
  • χ2 = Chi-Square value

Solved Example

Question: Calculate the chi-square value for the following data:

Male Female
Full Stop 6(observed)

6.24 (expected)

6 (observed)

5.76 (expected)

Rolling Stop 16 (observed)

16.12 (expected)

15 (observed)

14.88 (expected)

No Stop 4 (observed)

3.64 (expected)

3 (observed)

3.36 (expected)


Now calculate Chi Square using the following formula:

χ2 = ∑ (O − E)2 / E

Calculate this formula for each cell, one at a time. For example, cell #1 (Male/Full Stop):

Observed number is: 6
Expected number is: 6.24

Therefore, (6 – 6.24)2 /6.24 = 0.0092

Continue doing this for the rest of the cells, and add the final numbers for each cell together to get the final Chi-Square number. There are 6 total cells, so at the end, you should be adding six numbers together for your final Chi-Square number.


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