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Chi-Square Statistic

Chi-Square Statistic

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This lesson will explain the chi-square statistic.

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Tutorial

What's Covered

This tutorial will cover the chi-square statistic and how it's calculated. You’ll learn about:

  1. The Chi-Square Statistic

1. The Chi-Square Statistic

In this tutorial, you will not run any significance tests because the chi-square tests have many different versions, and each of them will have their own tutorial. This tutorial is going to focus on how the statistic is calculated, as it's calculated the same regardless of the test you're running.

What is the Chi-Square Statistic?

Term to Know

  • Chi-Square Statistic
  • The sum of the ratios of the squared differences between the expected and observed counts to the expected counts.

Example  Suppose you have a tin of colored beads. And you claim that the tin contains the colored beads in these proportions: 35% blue, 35% green, 15% yellow, and 15% red.

You draw 10 beads from the tin: 4 red, 3 blue, 1 green, and 2 yellow. This is called the observed counts.

  • Is what you drew consistent with the percentages you claimed or not? Why or why not?

The two yellow seems fairly consistent with the 15% claim. But the four red don't seem all that consistent with the 15% claim for red.

  • How can you measure that discrepancy?

If the claim were true, you would have expected that out of 10 beads, 3 1/2 of them would be blue, 3 1/2 green, 1 1/2 yellow, and 1 1/2 red. This is called the expected counts.

Terms to Know

  • Observed Counts
  • The frequencies within each of the categories in a qualitative distribution.
  • Expected Counts
  • The frequencies we would have expected within each of the categories in a qualitative distribution if the null hypothesis were true.

You can't actually pull 3 1/2 blue beads, because you can't have half of a bead. So this is sort of an idealized scenario, representative of what you might expect in the long-term in samples of 10.

In your one sample of 10 beads, what you actually got was: 3 blue, 1 green, 2 yellow, and 4 red.

Think About It

How can you measure the discrepancy between what you observed and what you expected?

Blue and yellow we're pretty close to what we expected, whereas, green and red were pretty far off.

The statistic that we use to measure discrepancy from what we expect is called chi-square, which is calculated this way:

  1. take the observed values
  2. subtract the expected values
  3. square that difference
  4. divide by the expected
  5. add up all of those fractions

The 3 1/2, 3 1/2, 1 1/2, and 1 1/2 was expected. The observed were the 3, 1, 2, and 4. So the chi-square statistic value is 6.1905.

You can use a table to calculate the chi-square statistic or you can use technology.

Now, it's worth noting that in this case, the conditions for inference with a chi-square test are not met. This is only meant to illustrate how a chi-square statistic would be calculated, although you can't do any real chi-square inference on this because the sample size isn't large enough.


Summary

The chi-square statistic is a measure of discrepancy across categories from what you would have expected in categorical data. You can only use it for data that appear in categories or qualitative data. The expected values may not be whole numbers since the expected values are long-term average values.

Thank you and good luck!

Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS

TERMS TO KNOW
  • Expected Counts

    The frequencies we would have expected within each of the categories in a qualitative distribution if the null hypothesis were true.

  • Observed Counts

    The frequencies within each of the categories in a qualitative distribution.

  • Chi-Square Statistic

    The sum of the ratios of the squared differences between the expected and observed counts to the expected counts.