## Comparing Growth Charts

Suppose both an 8-year old boy and a 10-year old boy are 54 inches tall (four feet six inches). Physically they are of the same heights. But a better way to compare is to find out where each boy stands in the distribution for his age group. One convenient way to do this is to compare the growth charts for these two ages. A clinical growth chart from the Center for Disease Control and Prevention contains the percentiles for heights for boys from age 2 to age 20. The following two tables list out the ranking in heights for age 8 and age 10. A copy of the chart is found here.

$\displaystyle \text{in centimeter} \ \ \ \begin{bmatrix} \text{Percentile}&\text{ }&\text{Age 8}&\text{ }&\text{Age 10} \\\text{ }&\text{ }&\text{ } \\\text{95th}&\text{ }&\text{138 cm}&\text{ }&\text{150 cm} \\\text{90th}&\text{ }&\text{135.5 cm}&\text{ }&\text{147 cm} \\\text{75th}&\text{ }&\text{132 cm}&\text{ }&\text{143 cm} \\\text{50th}&\text{ }&\text{128}&\text{ }&\text{138.5 cm} \\\text{25th}&\text{ }&\text{124 cm}&\text{ }&\text{134 cm} \\\text{10th}&\text{ }&\text{121 cm}&\text{ }&\text{130.5 cm} \\\text{5th}&\text{ }&\text{118.5 cm}&\text{ }&\text{128 cm} \end{bmatrix}$

$\displaystyle \text{in inch} \ \ \ \ \ \ \ \ \ \ \ \begin{bmatrix} \text{Percentile}&\text{ }&\text{Age 8}&\text{ }&\text{Age 10} \\\text{ }&\text{ }&\text{ } \\\text{95th}&\text{ }&\text{53.82 inches}&\text{ }&\text{58.5 inches} \\\text{90th}&\text{ }&\text{52.845 inches}&\text{ }&\text{57.33 inches} \\\text{75th}&\text{ }&\text{51.48 inches}&\text{ }&\text{55.77 inches} \\\text{50th}&\text{ }&\text{49.92 inches}&\text{ }&\text{54.015 inches} \\\text{25th}&\text{ }&\text{48.36 inches}&\text{ }&\text{52.26 inches} \\\text{10th}&\text{ }&\text{47.19 inches}&\text{ }&\text{50.895 inches} \\\text{5th}&\text{ }&\text{46.215 inches}&\text{ }&\text{49.92 inches} \end{bmatrix}$

With charts as in the above table, a doctor or a parent knows right away that about 90% of all 8-year old boys are between 46 inches (2 inches under four feet) and 54 inches (four feet 6 inches). Any 8-year old boy outside of this range is unusually short or unusually tall. If an 8-year old boy is taller than 54 inches, he is extremely tall (only 5% of this age group belongs to this height range).

If a 10-year old boy is 56 inches tall, his parent should be satisfied that the boy is developing normally. Not only the height of 56 inches for a 10-year old is above average, this height is at the 75th percentile (taller than 75% of the boys in this age group).

Back to the two 54-inch tall boys mentioned at the beginning. According to the above chart, the 8-year old is at the 95th percentile while the 10-year old is at the 50th percentile (the median). The 8-year old is exceedingly tall for his age while the 10-year old is just average. Though they have the same height measurement, relative to their age groups, one is very tall and the other is average.

The 25th percentile, 50th percentile and the 75th percentile are called the first quartile, the second quartile, and the third quartile, respectively because these three percentiles divide the data (or the distribution) into quarters. The first quartile and the third quartile bracket the middle 50% of the data, which indicates the middle range of the heights. The upper percentiles (like 90th and 95th) indicate the upper end of the growth range. The lower percentiles (like 10th and 5th) indicate the lower end of the growth range.

The take way is that measures of positions such as median, quartiles and other upper and lower percentiles are easy to use. These several percentiles are included in the growth charts. Their inclusion makes the growth chart so easy to use. A doctor or nurse takes one look at the growth chart and knows immediately where the child stands in his age group, whether the child is average or above average. If the child is above average, the doctor will know how much above average. Using the percentiles requires no additional calculation.

The alternative to using measures of position is to report the growth in heights with a mean and a standard deviation. Doing so is also a valid approach, but requires additional calculation. For example, the growth chart for the 10-year old can be reported with a mean of 54 inches and a standard deviation of 2.6 inches. Since height measurements are described by a normal curve, the doctor can know that 68% of all 10-year old boys are within one standard deviation (2.6 inches) from the mean (54 inches). So about 68% of 10-year old boys are between 51.4 inches (54 – 2.6) and 56.6 inches (54 + 2.6).

To find out what a tall height is and what a short height is, the doctor will have to look at two standard deviations away from the mean. About 95% of the 10-year old boys are between 48.8 inches (54 – 2 x 2.6) and 59.2 inches (54 + 2 x 2.6). So any 10-year old boy outside of this range is unusually tall or unusually short. So a growth chart made up of mean and standard deviation is not as easy to use. You have to do some calculation before knowing how tall the boy is in relation to other boys (the mean by itself does not tell you much). A busy doctor is probably not going to reach out for a calculator for this kind of analysis.

In contrast, the median and other percentiles are so easy to use. You take one look and immediately know where the child stands in relation to other boys in the same age group.

If the data are skewed (e.g. income data), we obviously want to use median and quartiles since measures of positions are resistant to extreme data values. On the other hand, if the data are symmetric and have no outliers, we can describe the data using mean (as center) and standard deviation (as spread). This kind of discussion is in many introductory statistics texts. I make sure that this is presented to my students. However, the growth chart example here tells us that there is a good reason to use median and quartiles and other percentiles even if the data are normal (described by a bell curve such as the height measurements in this example).

The median and the quartiles are very useful for describing a data set. This is the case even in the situations where we are taught to use mean and standard deviation.

Interestingly, if we decide to use median and other percentile to describe the data we work with, the median is the center and the other percentiles (such as quartiles) will play the role of a spread. For example, the first quartile and third quartile will tell us how spread out the middle 50% of the data are. If we also include the 90th percentile and the 10th percentile, we know the spread for the middle 80% of the data. If we include the 95th percentile and the 5th percentile, we know the spread of the middle 90% of the data. This is essentially the information provided by the growth chart we examine.