The Normal Density and Percentiles

The distance from the mean doesn't grow linearly

Figure 1: An animation showing how the distance from the mean to a percentile grows.

Figure 1 has a lot going on, so let's go on to Figure 2 and then come back to the animation. The 84th, 89th, 94th, and 99th percentiles are denoted with a vertical line in Figure 2, which depicts a standard normal density with a mean value of zero and a standard deviation of one. The values found along the x-axis represent the distances between the various percentiles. In other words, the distance along the x-axis required to go from the 84th to the 89th percentile is 0.23, whereas the distance required to move from the 89th to the 94th percentile is 0.33. Taking it a step further, the gap between the 94th and 99th percentiles is 0.77. The main point being made here is that progressing from one percentile to the next does not take place in a linear fashion. If this were the case, then the spaces would stay the same.

Figure 2: The 84th, 89th, 94th, and 99th percentiles in a standard normal density with the distance in standard deviations between them.

To better understand this, let's put it into context. In the previous piece that I wrote about the differences in normal density tails, I used the example of the difference in height between men and women.

The Centers for Disease Control and Prevention (CDC) reports that the average height of a woman in the United States who is 20 years old or older is 62.5 inches (5' 3.5''), with a standard deviation of 5.2 inches. The average height of males was found to be 69.0 inches (5' 9''), with a standard deviation of 5.0 inches.

Let's say that a couple has two sons. The physician is currently keeping tabs on the percentile range of heights for each child. The one who is older falls inside the 89th percentile, whereas the one who is younger falls within the 94th percentile. The younger brother is beyond ecstatic as a result of the fact that they are going to be taller. There is bound to be some friction between the brothers as a result of this. The parents believe that an increase of just 5 percentile points won't make much of a difference in either of their heights and tell them that they will be nearly the same height. However, the younger brother would be 1.6 inches taller (75.13 to 76.77), making him noticeably taller. If instead we were talking about the 55th and 60th percentiles, then the difference between them would be only 0.64 inches.

Going back to Figure 1, this is again a standard normal density curve. The red line that is moving represents a percentile, and the text in the upper right corner contains its value. On the x-axis of the inset graph is the percentile, and the distance that percentile is from the mean is shown along the y-axis. There is a correspondence between each dot in the inset graph and the percentile line in the main graph. It is important to note that the growth depicted in the inset graph is not linear; rather, it bends upward or is concave up. 

It is important to keep in mind that attaining elite status becomes progressively more difficult. For instance, in major league baseball, we tend to underestimate the degree to which a pro player's skill set differs from that of a player in AAA. This is true of prominent scholars, musicians, and other professionals in their respective disciplines.

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