Chi Square Distribution & Hypothesis Test

The Chi-Square (χ2) distribution is the best method to test a population variance against a known or assumed value of the population. A Chi-Square distribution is a continuous distribution with degrees of freedom.

The best part of a Chi-Square distribution is that it describes the distribution of a sum of squared random variables. It is also used to test the goodness of fit of data distribution, whether a data series is independent, and for judging the confidences surrounding variance and standard deviation for a random variable from a normal distribution.

History of Chi-Square

Karl Pearson (1857 – 1936), the father of modern statistics (founded the first statistics department in the world at University College London), came up with the Chi-Square distribution. Pearson’s work in statistics began when he developed a mathematical method for studying the process of heredity and evolution. Later, the Chi-Square distribution came about as Pearson tried to find a measure of the goodness of fit of other distributions to random variables in his heredity model.

Chi-Square Statistic

A Chi-Square distribution may skew to the right or with a long tail toward the significant values of the distribution. The overall shape of the distribution will depend on the number of degrees of freedom in a given problem. The degree of freedom is one less than the sample size.

Chi-Square Properties

• The mean of the distribution is equal to the number of degrees of freedom: μ=ϑ.
• The variance equals two times the number of degrees of freedom: σ2 = 2*ϑ.
• When the degrees of freedom are greater than or equal to 2, the maximum value for Y occurs when χ2=ϑ-2.
• As the degrees of freedom increase, the Chi-Square curve nears a normal distribution.
• As the degrees of freedom increase, the symmetry of the graph also increases.
• Finally, It may be skewed to the right, and since the random variable it is based on is squared, it has no negative values. As the degrees of freedom increase, the probability density function begins to appear symmetrical.

The formula for the probability density function of the Chi-Square distribution is

Where ϑ is the shape parameter, and Γ is the gamma function.

The formula for the gamma function is

Chi-Square (χ²) Hypothesis Test

Usually, the goal of the Six Sigma team is to find the level of variation of the output, not just the mean of the population. Above all, the team would like to know how much variation the production process shows about the target to see what changes are needed to reach a process free of defects.

For a comparison between several sample variances or a comparison between frequency proportions, the standard test statistic called the Chi-Square χ2 test will be used. So, the distribution of the Chi-Square statistic is called the Chi-Square distribution.

Types of Chi-Square Hypothesis Tests

There are two types of Chi-Square tests:

• Chi-Square Test of Independence: Determines whether there is any association between two categorical variables by comparing the observed and expected frequencies of test outcomes when there is no defined population variance.
• Chi-Square Test of Variance: Compare the variances when the variance of the population is known.

Chi-Square Test of Independence

The Chi-Square Test of Independence determines whether there is an association between two categorical variables (like gender and course choice). For example, the Chi-Square Test of Independence examines the association between one category, like gender (male and female), and another category, like the percentage of absenteeism in a school. The Chi-Square Test of Independence is a non-parametric test. In other words, you do not need to assume a normal distribution to perform the test.

A Chi-Square test uses a contingency table to analyze the data. Each row shows the categories of one variable. Similarly, each column shows the categories of another variable. Each variable must have two or more categories. Each cell reflects the total number of cases for a specific pair of categories.

Assumptions of Chi-Square Test of Independence

• Variables must be nominal or categorical
• Categories of variables are mutually exclusive
• The sampling method is a simple random sampling
• The data in the contingency table are frequencies or count

Steps to Perform Chi-Square Test of Independence

Step1: Define the Null Hypothesis and Alternative Hypothesis

• Null Hypothesis (H₀): There is no association between two categorical variables
• Alternative Hypothesis (H₁): There  is a significant association between two categorical variables

Step2: Specify the level of significance

Step 3: Compute χ2 statistic

• O is the observed frequency
• E is the expected frequency

The expected frequency is calculated for each cell = (frequency of columns * frequency of rows)/ n

Step 4: Calculate the degree of freedom = (number of rows -) * (number of columns -1)= (r-1) * (c-1)

Step 5: Find the critical value based on degrees of freedom

Step 6: Finally, draw the statistical conclusion: If the test statistic value is greater than the critical value, reject the null hypothesis, and hence, we can conclude that there is a significant association between two categorical variables.

Chi-Square Test of Independence Example

For example, 1000 middle school students are asked which their favorite superhero is: Superman, Ironman, or Spiderman. At a 95% confidence level, would you conclude that there is a relationship between gender and superhero characters?

• Null Hypothesis (H₀): There is no association between gender and favorite superhero characters.
• Alternative Hypothesis (H₁): There is a significant association between gender and favorite superhero characters.

Level of significance: α=0.05:

First, calculate the expected frequency:

For the cell (Boys, Superman) = (200 * 600)/ 1000 = 120

Similarly, determine the expected frequency of all cells:

Degrees of freedom = (r – 1) * (c – 1) = (2 – 1) * (3 – 1) =2

Chi-Square critical value for 2 degrees of freedom =5.991

The test statistic value is greater than the critical value; hence, we can reject the null hypothesis.

So, we can deduce a significant association between gender and favorite superhero characters.

Download Chi-Square Test of Independence Excel Exemplar

A: 57.7. and 4 degrees of freedom.

First, we will calculate the degrees of freedom. This is an easy way to eliminate half the answers on the page.

There are five rows and two columns in the chart.

So, Degrees of freedom = (rows -1) * (columns – 1) = (5-1) * ( 2 – 1) = 4* 1 = 4.

Now we’ll run the equation Chi Squared = X^2 = Σ (((o-E)^2 )/ E) = 100 / 20 + 25 / 20 + 900 / 20 + 25/20 + 100 / 20 = 1150 / 20 = 57.5

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