Chi Square Distribution

Chi-Square distribution is used to test whether or not two factors are independent or dependent. Chi square is a test of dependence or independence.

Chi-square distribution is not bimodal or level. It seems to be a skewed bell shape. Going with skewed.

As the degrees of freedom increase the symmetry of the graph increases.

It is skewed to the right, and since the random variable on which it is based is squared, it has no negative values. As the degrees of freedom increases, the probability density function begins to appear symmetrical in shape.


chi square equation


Chi Square and Hypothesis testing

See additional notes on Hypothesis testing.

Don’t need knowledge of population variation

Evaluates sample variances (Chi squared distribution) (Chi squared test for the variance)


Chi Square Examples


Example 1:

The Barnes Company manufactures a DVD player and claims that the mean number of hours of use before repairs are needed is 400, with a standard deviation of 10 hours. The specified variance, therefore, is σo2 = 102 = 100 hours2. A new company marketing representative suspects that the “before repair” variance is actually less than 100 hours2. To verify this, she tests nine machines and finds a sample mean of 410 hours and a standard deviation of 5.5. Is the sample variance statistically significantly less than the currently claimed variance? Use α = 0.05.

chi square ex1 chart

chi square ex1


Six Sigma Black Belt Certification Chi Square Questions:

Question: The time for a fail-safe device to trip is thought to be a discrete uniform distribution from 1 to 5 seconds. To test this hypothesis, 100 tests are conducted with results as shown below.
chi square question
On the basis of these data, what are the chi square (c2) value and the number of degrees of freedom (df)?

(A) (c2) value = 57.5, degrees of freedom = 4
(B) (c2) value = 57.5, degrees of freedom = 5
(C) (c2) value = 1,150.0, degrees of freedom = 4
(D) (c2) value = 1,150.0, degrees of freedom = 5

Answer:  57.7. and 4 degrees of freedom.

First we will figure out the degrees of freedom. It’s an easy way to eliminate half the answers on the page.

There are 5 rows and 2 columns in the chart.

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





3 comments… add one
  • Praful bhusari

    What is difference in between chi square test and chi square distribution.

    • The distribution refers to what probability an arrangement of values of a variable showing their observed or theoretical frequency of occurrence. The Chi Square distribution looks like a skewed bell curve.

      The Ch Square test is a mathematical procedure used to test whether or not two factors are independent or dependent. Chi square is a test of dependence or independence. In other words, you use this test (which makes use of the chi square distribution) to see if there is a statistically valid dependence of one thing on another. Check out the examples above and you’ll see.


    In the example 1 above (The Barnes Company), X² statistic is < X² (table) and the decision was "we reject the H0".
    However, for the F-test (see example :, F statistic is < F (table) and the decision was "we fail to reject the H0". same decision taken for T-test (see example :
    Could you please clarify ?
    Best regards.

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