Chi-Square Test

chi square groups
Single: 47   Married: 71   Divorced: 35
chi square groups
Single: 44   Married: 85   Divorced: 40


Groups and Numbers

You research two groups and put them in categories of single, married or divorced:


The numbers are definitely different, but ...


The Chi-Square Test gives a "p" value to help you decide!

Example: "Which holiday do you prefer?"

  Beach Cruise
Men 209 280
Women 225 248

Does Gender affect Preferred Holiday?

If Gender (Man or Woman) does affect Preferred Holiday we say they are dependent.

By doing some special calculations (explained later), we come up with a "p" value:

p value is 0.132

Now, p < 0.05 is the usual test for dependence.

In this case p is greater than 0.05, so we believe the variables are independent (ie not linked together).

In other words Men and Women probably do not have a different preference for Beach Holidays or Cruises.

It was just random differences which we expect when collecting data.

Understanding "p" Value

"p" is the probability the variables are independent.

Imagine that the previous example was in fact two random samples of Men each time:

chi square group 1 chi square group 2
Beach 209, Cruise 280
Beach 225, Cruise 248

Is it likely you would get such different results surveying Men each time?

Well the "p" value of 0.132 says that it really could happen every so often.

Surveys are random after all. We expect slightly different results each time, right?

So most people want to see a p value less than 0.05 before they are happy to say the results show the groups have a different response. 

Let's see another example:

Example: "Which pet do you prefer?"

  Cat Dog
Men 207 282
Women 231 242

By doing the calculations (shown later), we come up with:

P value is 0.043

In this case p < 0.05, so this result is thought of as being "significant" meaning we think the variables are not independent.

In other words, because 0.043 < 0.05 we think that Gender is linked to Pet Preference (Men and Women have different preferences for Cats and Dogs).

Just out of interest, notice that the numbers in our two examples are similar, but the resulting p-values are very different: 0.132 and 0.043. This shows how sensitive the test is!

Why p<0.05 ?

It is just a choice! Using p<0.05 is common, but we could have chosen p<0.01 to be even more sure that the groups behave differently, or any value really.

Calculating P-Value

So how do we calculate this p-value? We use the Chi-Square Test!

Chi-Square Test

Note: Chi Sounds like "Hi" but with a K, so it sounds like "Ki square"

And Chi is the greek letter Χ, so we can also write it Χ2

Important points before we get started:

Our first step is to state our hypotheses:

Hypothesis: A statement that might be true, which can then be tested.

The two hypotheses are.

Lay the data out in a table:

  Cat Dog
Men 207 282
Women 231 242

Add up rows and columns:

  Cat Dog  
Men 207 282 489
Women 231 242 473
  438 524 962

Calculate "Expected Value" for each entry:

Multiply each row total by each column total and divide by the overall total:

  Cat Dog  
Men 489×438962 489×524962 489
Women 473×438962 473×524962 473
  438 524 962

Which gives us:

  Cat Dog  
Men 222.64 266.36 489
Women 215.36 257.64 473
  438 524 962

Subtract expected from observed, square it, then divide by expected:

In other words, use formula (O−E)2E where

  Cat Dog  
Men (207−222.64)2 222.64 (282−266.36)2 266.36 489
Women (231−215.36)2 215.36 (242−257.64)2 257.64 473
  438 524 962

Which gets us:

  Cat Dog  
Men 1.099 0.918 489
Women 1.136 0.949 473
  438 524 962

Now add up those calculated values:

1.099 + 0.918 + 1.136 + 0.949 = 4.102

Chi-Square is 4.102

From Chi-Square to p

Degrees of Freedom

First we need a "Degree of Freedom"

Degree of Freedom = (rows − 1) × (columns − 1)

For our example we have 2 rows and 2 columns:

DF = (2 − 1)(2 − 1) = 1×1 = 1


The rest of the calculation is difficult, so either look it up in a table or use the Chi-Square Calculator.

The result is:

p = 0.04283


Chi-Square Formula

This is the formula for Chi-Square:

Χ2 = Σ(O − E)2E

So we calculate (O−E)2E for each pair of observed and expected values then sum them all up.