Video: Estimating Areas Under a Normal Distribution Curve

For the normal distribution shown, approximately what percent of the data points lie in the shaded region?

02:17

Video Transcript

For the normal distribution shown, approximately what percent of the data points lie in the shaded region?

So we have this diagram, which shows the bell curve shape of a normal distribution. The curve is symmetrical about the mean of the distribution 𝜇. The reason why this bell curve is useful is because the probability that an outcome from this normal distribution that lies in a certain interval is just the area under the curve above that interval.

So suppose we wanted to find the probability that 𝑥 lies between 𝑎 and 𝑏, we just mark 𝑎 and 𝑏 on the 𝑥-axis. And then the probability that we’re looking for is just the area under the curve between these two points, 𝑎 and 𝑏. So the percent that we’re looking for is the probability that 𝑥 lies between 𝜇 minus 𝜎 and 𝜇.

So in other words, that’s a probability that an outcome from the normal distribution is below the mean, but below it by less than one standard deviation. It turns out that if 𝑥 is normally distributed then this probability that 𝑥 is between 𝜇 minus 𝜎 and 𝜇 is about 34 percent. This is true for any normal distribution, no matter what the value of the mean and the standard deviation actually are.

Just to be completely clear, remember that 𝜇 stands for the mean and 𝜎 for the standard deviation. This percent and other related ones are very useful to remember. This is because they tell you roughly how spread out you should expect normally distributed data to be. A related fact is that about 68 percent of any normally distributed data will be within one standard deviation of the mean.

And if you remember this percent, then you can easily find the percent that we’re looking for in this question by considering the symmetry of the normal distribution and hence just dividing by two. Whether you remember the value of 34 percent or instead remember 68 percent and then divide it by two, you will get the same answer that about 34 percent of the data points lie in the shaded region.

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