Question Video: Estimating Areas Under a Normal Distribution Curve Mathematics

For a normally distributed data set with mean 32.1 and standard deviation 2.8, between which two values would you expect 99.7% of the data set to lie?

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Video Transcript

For a normally distributed data set with mean 32.1 and standard deviation 2.8, between which two values would you expect 99.7 percent of the data set to lie?

Remember, a normal distribution with a mean 𝜇 and a standard deviation 𝜎 is represented by this bell curve. The curve is symmetrical about the mean. And the total area under the mean represents 100 percent. For a normally distributed data set, we say that approximately 68 percent of the data values occur within one standard deviation of the mean. That’s the area shown.

We say that proximally 95 percent of the data values occur within two standard deviations of the mean. And 99.7 percent lie within three standard deviations of the mean. We are told that this data set has a mean of 32.1 and a standard deviation of 2.8. And we expect that 99.7 percent of them will lie within three standard deviations of the mean, three standard deviations of 32.1. That’s 32.1 plus three multiplied by 2.8 and 32.1 minus three multiplied by 2.8.

32.1 plus three multiplied by 2.8 is 40.5. And 32.1 minus three multiplied by 2.8 is 23.7. We would therefore expect for a data set with a mean of 32.1 and a standard deviation of 2.8, 99.7 percent of the data set would lie between the values of 23.7 and 40.5.