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?

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 #mu=21# and the standard deviation is #sigma=2# which means that the point we were given was the mean plus one standard deviation, i.e. #mu + 1 sigma#.

The empirical rule tells us that #68%# of our population lies within #+-1sigma# 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 #mu - 1 sigma# and that #16%# is above #mu + 1 sigma#.

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:

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

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

Therefore our answer makes sense.