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About the Median…
Take the 15 kids in Mrs. Jones’ local third grade class and measure their heights, and calculate the average. The average height might be, for example, four feet, six inches (54 inches). The average was obtained by measuring every single child, adding their heights together, and dividing by the number of children.
The median height, however, is calculated differently, and it is intended to give a different view than the average height. To get the median height, take those same 15 school kids, but first you must line them up carefully in order of their height, from shortest to tallest. The median height is then obtained simply by measuring the height of the 8th child in the line (the one who falls exactly in the middle). In this case, let’s say Joey, standing in the middle, is just 50 inches tall.
We would then report: “The height of the children in Mrs. Jones’ third grade class is an average of 54 inches, with the median being 50 inches.”
Averages and medians are intentionally two different calculations, with each giving a different angle of view concerning the mathematical “middle” or “center” of a series of measurements. Sometimes the average is close to the median, and therefore many people think of the average and the median as meaning pretty much the same thing.
Indeed, sometimes the average and median values are coincidentally the same. But this is certainly not always. The two numbers are more likely to coincide when you have a larger quantity of things to measure, and especially when you are measuring naturally occurring forms. For example, in measuring the height of all 500 third graders in the local community, the average and the median might both turn out to be 53 inches, or they could easily be two numbers very close together, such as an average of 53 inches and a median of 52.5 inches.
But more often than not, the median is not the same as the average; many times it isn’t even close to the average. This variance is proper and expected, because they are two different calculations for two different things. The average and median are most often different when measuring things that do not follow such a natural pattern of distribution. They include things like personal income, or business income, and personal opinions — as in movie ratings.
The difference in the average and the median height in the first example above occurs because the calculation of the average height takes into account the precise height of each single child by first adding them all together and then dividing by 15. The average is therefore influenced by every height, including the unusually tall children way over at the right side (the ones who are drinking too much milk), and by the very short kids at the left side (whose growth has been stunted by smoking).
The median height, however, is not impacted by every single height, except during the initial stage of arranging them in order of shortest to tallest. During this first step (essential to calculating the median), each height matters because it affects how you line them all up. But, once the children have been so arranged, there is now no need to precisely measure any of their heights except that of the child in the middle. The median or “middle” height is the height of that one child, but only because he or she happened to fall into the center as a direct result of the pre-arrangement process.
Here it is important to note that the median is not influenced by any unusually tall or short children that might appear at either end of the line. For example, if you replaced the child at the right end (60 inches) with a much taller child (seven feet), the median would not change one bit. The median isn’t impacted, really, by anyone but the kid in the middle, except to say that every other kid’s height during the pre-sorting process contributed to making one particular child fall exactly into the middle. The child so chosen receives the glorious honor of representing the median height for that particular class.
In most elementary schools, the child in the middle would be flanked immediately on either side by children of very similar height. As you move away from the center toward the children at the either end, the heights would typically tend to slope upwards or downwards rather gently, with maybe a few small jumps here and there, especially if the children are all similar in age. This assumes the class is not a wild conglomeration of kids of widely different ages (which would impact natural height distributions).
The median and the average are both useful measures of the “center” of a set of values. But a median, as noted above, is much more stable than an average. In the same way, the median rating of a film by a group of 15 movie critics is far more stable than an average rating. Some people prefer the average rating because it is a more familiar number, while some (like us) prefer the median for its greater stability.
By John Simmons