Spread: Range, IQR, MAD and Standard Deviation

In this lesson you will: Compute the range, the interquartile range and the mean absolute deviation by hand; Interpret a standard deviation as a typical distance from the mean; Pair a measure of center with the measure of spread that belongs to it; Compare two data sets that have the same center but different spreads.

Two data sets can share a center and share nothing else

A courier company times two delivery routes, five runs each. Route A takes 68, 70, 72, 74 and 76 minutes. Route B takes 60, 66, 72, 78 and 84 minutes. Both average exactly 72 minutes.

A dispatcher who only knows the means would call these routes identical. A driver who has done both would laugh. Route A is never more than four minutes off its average; Route B has been twelve minutes either side of it.

Center answers \"around what value?\" Spread answers \"how tightly?\" Reporting one without the other is where most misleading statistics come from.

Three ways to measure how far the data wanders

The range is the maximum minus the minimum. It is one subtraction, and it is built entirely out of the two most unusual values in the data, which makes it fragile.

The interquartile range is Q_3-Q_1, the width of the middle half. It throws away the top and bottom quarters, so a single wild value cannot touch it. This is the spread that belongs with the median.

The mean absolute deviation asks a more direct question: on average, how far is a value from the mean? Subtract the mean from every value, take the absolute value of each difference, and average those. This is the spread that belongs with the mean.

Standard deviation, and what it is telling you

Standard deviation, written sigma, measures the same idea as the MAD but squares the distances before averaging them and takes a square root at the end. Squaring makes far-away values count for more, so sigma reacts harder to extremes than the MAD does.

You are not expected to grind out the formula by hand. You are expected to read one. For Route A, sigma ~ 2.8 minutes; for Route B, sigma ~ 8.5 minutes. In plain English: a Route A run is typically about 3 minutes off the 72-minute average, and a Route B run is typically about 8 or 9 minutes off.

Bigger sigma means less consistent. That comparison — which of these two is more predictable, and by roughly how much — is what the standard expects you to be able to say.

Common mistake — Computing the MAD without the absolute values

For 22, 25, 28, 31, 34 with a mean of 28, a student writes the deviations -6, -3, 0, 3, 6, adds them to get 0, divides by 5 and reports a MAD of 0.

Nothing about the arithmetic looks wrong. Every subtraction is correct, the addition is correct, the division is correct. The word \"absolute\" is one word in a definition that is otherwise all familiar operations, and it is easy to read past.

But a MAD of zero would mean every value equals the mean, and these five values clearly do not. The signed deviations from a mean always add to zero — that is exactly what makes the mean the balance point — so skipping the absolute values guarantees an answer of zero for every data set on earth.

Taken properly, the distances are 6, 3, 0, 3, 6, totaling 18, so the MAD is (18)/(5)=3.6 minutes.

How to catch it: Look at your column of deviations before you add. If it contains a negative number, you have not finished. Every entry in a MAD calculation is a distance, and a distance is never negative.