Assume that a normal distribution has a mean of 21 and a standard deviation of 2. What is the percentage of values that lie above 23?

Use the empirical rule.

1 Answer
Jun 27, 2018

16% of values are expected to be above 23

Explanation:

The question asks us to apply the empirical rule for normal distributions which states that 68%,95%, and 99.7% of values lie within 1,2, and 3 standard deviations of the mean, respectively. First, we need to know which of these ranges we are in, i.e. how many standard deviations from the mean. (68-95-99.7 rule)

We are looking for the percentage of the population above 23 where the mean is μ=21 and the standard deviation is σ=2 which means that the point we were given was the mean plus one standard deviation, i.e. μ+1σ.

The empirical rule tells us that 68% of our population lies within ±1σ from the mean. That tells us that 32% lies outside that range, but on both sides - both above and below. Since the normal distribution is symmetric (the same on both sides) we know that 16% is below μ1σ and that 16% is above μ+1σ.

Therefore, 16% of values are expected to be above 23.

Check:

We can use the following diagram of a normal curve to check our answer:

https://en.wikipedia.org/wiki/File:Standard_deviation_diagramsvghttps://en.wikipedia.org/wiki/File:Standard_deviation_diagramsvg

All we need to do is add up the percentages of the population above the 1σ point which are:

13.6%+2.1%+0.1%=15.8%16%

Therefore our answer makes sense.