Chi Square Distribution: Definition & Examples

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  • 0:03 Probability Distributions
  • 2:27 Finding Chi Square
  • 4:18 Lesson Summary
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Lesson Transcript
Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

Chi square distributions are a way of mapping the probabilities of values. In this lesson, we'll look at distributions represented in graphs and tables. We'll also look at an example that uses chi square distribution to test the independence of variables.

Probability Distributions

Sally has a friend, Sarah, that believes certain college majors are more likely to be chosen by students of one gender over another. Sally does not agree with Sarah's claim, so she decided to survey students on her college campus. Once she collected the data, she used a chi square distribution to determine whether her prediction was correct.

Chi square distribution is a type of cumulative probability distribution. Probability distributions provide the probability of every possible value that may occur. Distributions that are cumulative give the probability of a random variable being less than or equal to a particular value. Since the sum of the probabilities of every possible value must equal one, the total area under the curve is equal to one.

To find the probability of a particular value, we find the area under the curve before the value. The area that's after the value is called the p-value, which is important for statistical tests that use chi square. In this figure, X^2 represents chi square and p represents the p-value:


Chi square distributions vary depending on the degrees of freedom. The degree of freedom is found by subtracting one from the number of categories in the data. For example, if you gather data about the gender of students enrolled in science, art, and education programs, you have three categories of students: one for each program. My degree of freedom would be 2 (or 3 - 1). In this graph, each curve represents the chi square distribution for a different degree of freedom:


No matter how many degrees of freedom there are, the shape of a chi square distribution is always skewed right. However, as the degree of freedom increases, the shape becomes closer to a normal distribution with a symmetrical bell shape.

Many statistical analyses involve using the p-value. However, calculating a portion of the area under the curve can be difficult. This graph can be used to find p-values for various degrees of freedom. For example, if the chi square value is 5 for a set of data that has a degree of freedom equal to 4, we can follow the curve to see that the p-value is approximately 0.3:


It's often more efficient to use a chi square table. In this table, each row represents a different degree of freedom along with several chi square values. The corresponding p-values are listed at the top of each column:


Finding Chi Square

Chi square is a calculation used to determine how closely the observed data fit the expected data. In the following chi square calculation formula, X represents chi, while o and e represent the observed and expected values, respectively:


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