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Introduction to Statistics: Help and Review9 chapters | 137 lessons

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Lesson Transcript

Instructor:
*Mia Primas*

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

In this lesson, you will learn the formula and method for calculating chi square. You can then take a brief quiz to see what you learned. You will learn how to interpret the value of chi square in a separate lesson.

**Chi square** is a method used in statistics that calculates the difference between observed and expected data values. It is used to determine how closely actual data fit expected data. The value of chi square helps us to answer the question, 'is the difference in expected and observed data statistically significant?' A small chi square value tells us that any differences in actual and expected data are due to chance, so the data is not statistically significant. A large value tells us the data is statistically significant and there is something causing the differences in data. From there, a statistician may explore factors that may be responsible for the differences.

Chi is a Greek symbol that looks like the letter *x* as you can see in the 'chi square formula' image on screen now. To calculate chi square, we take the square of the difference between the observed (*o*) and expected (*e*) values and divide it by the expected value. Depending on the number of categories of data, we may end up with two or more values. Chi square is the sum of those values. It's a common mistake for students to take the square root, as you would if you were solving for chi. However, the value that we are looking for is chi square, so we do not need to take the square root.

Once we have the value of chi square, we would use a chi square table to determine whether the data is statistically significant.

A poll was taken of 100 college freshmen to determine whether chocolate or vanilla was the preferred ice cream flavor. It was expected that 25% would prefer chocolate and 75% would prefer vanilla. Let's look at the results of the survey. Instead of 25% of the students polled preferring chocolate, it turns out that 39% preferred it, and instead of 75% of the students polled preferring vanilla, it turned out that 61% preferred it.

Chocolate | Vanilla | |
---|---|---|

Expected |
25 | 75 |

Observed |
39 | 61 |

Because we have two categories, chocolate and vanilla, we will need to calculate two values and take their sum. This will give us the value of chi square. The expected value gets substituted for *e* in the equation, and the observed value gets substituted for *o*.

Notice that in the numerator of each fraction, the expected value is subtracted from the observed value. It is then squared to make sure that the result is a positive value, as chi square must be positive.

After completing the calculation, we see that our chi square value is 10.45. To understand the significance of this value, we would need to use a chi square table. You will learn how to do this in a separate lesson.

In the next example, we will calculate chi square for a sample with four categories. When planning the schedule for incoming college freshmen, the art department expected that 40% of art students would want to take graphic design, while other students would want to take photography, sculpture or abstract painting. Let's take a look at the results and compare them to the expected numbers.

The image below shows the number of students who actually enrolled in each course.

Graphic Design | Photography | Sculpture | Abstract Painting | |
---|---|---|---|---|

Expected |
384 | 192 | 192 | 192 |

Observed |
367 | 201 | 180 | 212 |

The process for calculating chi square will be the same as our first example. The only difference is that we have four categories, so we will take the sum of four values instead of two.

**Chi square** is used in statistics to determine how close observed data is to the data that was expected. When we have values for the observed and expected data, we can use the chi square formula to calculate its value. To calculate chi square, take the square of the difference between the observed (*o*) and expected (*e*) values and divide it by the expected value. Depending on the number of categories of data, we may end up with two or more values. Remember that chi looks like the letter *x*, so that's the letter we use in the formula.

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Introduction to Statistics: Help and Review9 chapters | 137 lessons

- Descriptive & Inferential Statistics: Definition, Differences & Examples 5:11
- Difference between Populations & Samples in Statistics 3:24
- Defining the Difference between Parameters & Statistics 5:18
- Estimating a Parameter from Sample Data: Process & Examples 7:46
- What is Quantitative Data? - Definition & Examples 4:11
- What is Categorical Data? - Definition & Examples 5:25
- Discrete & Continuous Data: Definition & Examples 3:32
- Nominal, Ordinal, Interval & Ratio Measurements: Definition & Examples 8:29
- The Purpose of Statistical Models 10:20
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- How Randomized Experiments Are Designed 8:21
- Analyzing & Interpreting the Results of Randomized Experiments 4:46
- Confounding & Bias in Statistics: Definition & Examples 3:59
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- Misleading Uses of Statistics 8:14
- Causation in Statistics: Definition & Examples 3:28
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- Skewness in Statistics: Definition, Formula & Example 6:49
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- Chi Square Distribution: Definition & Examples 4:55
- Chi Square Practice Problems 6:53
- Chi Square: Definition & Analysis 4:04
- How to Calculate a Chi Square: Formula & Example 4:13
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