What Is And How To Use Chebyshev's Theorem And The Empirical Rule Formula In Statistics Explained
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In this video we discuss
In this video we discuss what is, and how to use Chebyshev's theorem and the empirical rule for distributions in statistics. We define both of these topics and go through an example of each.
Transcript/notes
The variance and standard deviation help us understand the spread or dispersion of a variable. In comparing 2 variables, the variable with the smaller standard deviation will be less spread out. So, we could have 2 variables with the same mean, but variable 1 has a smaller standard deviation than variable 2, so the data for variable 2 will be more spread out as you see here in these graphs.
Now to Chebyshev’s theorem, which says the portion of values in a data set within k standard deviations, with k being greater than one, of the mean is at least 1 minus 1 over k squared, where k is a number greater than one.
In simple terms, what this means is that you can plug in any number for k, greater than 1, and this formula will give you a percentage. That percentage is at the least, the percentage of values in the data set that will lie within whatever number you plugged in for k deviations.
For example, if we plug in 2, for k, so 2 deviations from the mean, the formula gives us 75%, so at least 75% of the values in the data set lie within 2 deviations of the mean. And 3 for k, 3 deviations from the mean gives us 88.89%. We could look at 1.5 deviations from the mean, which gives us 55.6%. And this theorem can be applied to any distribution regardless of its shape.
For example here is a right skewed distribution with a mean of 638, a median of 550, a mode of 500, and a standard deviation of 208. Here is a rough sketch of a right skewed distribution with the mean, x bar here of 638. 1 deviation to the right of the mean is 846, 638 + 208, which is x bar + 1s, 2 deviations to the right is 1054, which is x bar + 2s, and 3 deviations to the right is 1262. 1 deviation to the left is 430, 638 minus 208, which is x bar minus 1s, 2 deviations to the left is 222, and 3 deviations left is 14.
With what we computed earlier, we know that at least 75% of all values in the data set lie within 2 deviations of the mean, so in this example, 75% of all values in the data set lie between 222 and 1054. And at least 88.89% of all values in the data set lie between 14 and 1262, which is 3 deviations from the mean.
And this is Chebyshev’s theorem and it applies to any distribution regardless of its shape.
Now for the empirical rule, which applies only to a distribution that is bell shaped or normal, like the example on the screen.
It states that approximately 68% of the data values will lie within 1 standard deviation of the mean, approximately 95% of the data values will lie within 2 standard deviations of the mean, and approximately 99.7% of the data values will lie within 3 standard deviations of the mean.
So, if we have a data set with a mean of 88 and a standard deviation of 11, approximately 68% of the data values will lie between 77 and 99, or within 1 deviation of the mean. And approximately 95% of the data values will lie between 66 and 110, or within 2 deviations of the mean, and approximately 99.7% of the data values will lie between 55 and 121, which is within 3 standard deviations of the mean.
Timestamps
0:00 Standard Deviation Explained
0:20 Chebyshev's Theorem Explained
1:06 Example Problem For Chebyshev's Theorem
2:11 Empirical Rule Explained
Transcript/notes
The variance and standard deviation help us understand the spread or dispersion of a variable. In comparing 2 variables, the variable with the smaller standard deviation will be less spread out. So, we could have 2 variables with the same mean, but variable 1 has a smaller standard deviation than variable 2, so the data for variable 2 will be more spread out as you see here in these graphs.
Now to Chebyshev’s theorem, which says the portion of values in a data set within k standard deviations, with k being greater than one, of the mean is at least 1 minus 1 over k squared, where k is a number greater than one.
In simple terms, what this means is that you can plug in any number for k, greater than 1, and this formula will give you a percentage. That percentage is at the least, the percentage of values in the data set that will lie within whatever number you plugged in for k deviations.
For example, if we plug in 2, for k, so 2 deviations from the mean, the formula gives us 75%, so at least 75% of the values in the data set lie within 2 deviations of the mean. And 3 for k, 3 deviations from the mean gives us 88.89%. We could look at 1.5 deviations from the mean, which gives us 55.6%. And this theorem can be applied to any distribution regardless of its shape.
For example here is a right skewed distribution with a mean of 638, a median of 550, a mode of 500, and a standard deviation of 208. Here is a rough sketch of a right skewed distribution with the mean, x bar here of 638. 1 deviation to the right of the mean is 846, 638 + 208, which is x bar + 1s, 2 deviations to the right is 1054, which is x bar + 2s, and 3 deviations to the right is 1262. 1 deviation to the left is 430, 638 minus 208, which is x bar minus 1s, 2 deviations to the left is 222, and 3 deviations left is 14.
With what we computed earlier, we know that at least 75% of all values in the data set lie within 2 deviations of the mean, so in this example, 75% of all values in the data set lie between 222 and 1054. And at least 88.89% of all values in the data set lie between 14 and 1262, which is 3 deviations from the mean.
And this is Chebyshev’s theorem and it applies to any distribution regardless of its shape.
Now for the empirical rule, which applies only to a distribution that is bell shaped or normal, like the example on the screen.
It states that approximately 68% of the data values will lie within 1 standard deviation of the mean, approximately 95% of the data values will lie within 2 standard deviations of the mean, and approximately 99.7% of the data values will lie within 3 standard deviations of the mean.
So, if we have a data set with a mean of 88 and a standard deviation of 11, approximately 68% of the data values will lie between 77 and 99, or within 1 deviation of the mean. And approximately 95% of the data values will lie between 66 and 110, or within 2 deviations of the mean, and approximately 99.7% of the data values will lie between 55 and 121, which is within 3 standard deviations of the mean.
Timestamps
0:00 Standard Deviation Explained
0:20 Chebyshev's Theorem Explained
1:06 Example Problem For Chebyshev's Theorem
2:11 Empirical Rule Explained
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