Measures of Spread (Range, IQR, MAD)

Two classes can have the same average test score and still be completely different — one where everyone scored near the average, and one with a mix of very high and very low scores. Measures of center (mean, median, mode) can't tell those classes apart. Measures of spread can.

There are three you need: range, IQR, and MAD. Each one answers the same question — how spread out is this data? — but measures it a different way. Smaller spread means more consistent data.

Range: the full stretch

Range $=$ maximum $-$ minimum. It measures the full stretch of the data using only its two most extreme values.

That makes range fast but fragile — one unusual value changes it completely. A data set of $5, 6, 6, 7, 50$ has a range of $45$ , even though four of the five values sit within $2$ of each other. The dot plot below makes it visible: four values cluster near $6$ while a lone value at $50$ stretches the range wide.

IQR: the middle half

The interquartile range is $IQR = Q_3 - Q_1$ , where $Q_1$ is the median of the lower half of the data and $Q_3$ is the median of the upper half. It measures the span of the middle half of the data.

Because it ignores the lowest quarter and the highest quarter, the IQR doesn't flinch at outliers. It's the spread measure that pairs with the median, and it's exactly the width of the box in a box plot.

MAD: average distance from the mean

The mean absolute deviation is the mean of each value's distance from the mean. Find the mean, find how far each value is from it (distances are always positive), then average those distances.

MAD answers: on average, how far is a typical value from the mean? A MAD of $2$ says values typically land about $2$ away from the mean — some closer, some farther.

Worked examples

Example 1: range

Find the range of $7,\; 2,\; 11,\; 5$ .

Find the maximum

11

Find the minimum

2

Subtract

11 - 2 = 9

Answer: The range is $9$

Example 2: MAD

Find the MAD of $1,\; 3,\; 5,\; 7$ .

Find the mean

\dfrac{1 + 3 + 5 + 7}{4} = 4

Find each value's distance from the mean

3,\; 1,\; 1,\; 3

Average the distances

\dfrac{3 + 1 + 1 + 3}{4} = 2

Answer: The MAD is $2$

Example 3: IQR

Find the IQR of $4,\; 6,\; 8,\; 10,\; 12,\; 14,\; 16$ .

The data is already in order; the median is

10

Find

Q_1

, the median of the lower half

4, 6, 8

Q_1 = 6

Find

Q_3

, the median of the upper half

12, 14, 16

Q_3 = 14

Subtract

14 - 6 = 8

Answer: The IQR is $8$

Try one yourself

Common questions

Which measure of spread should I use?

Range for a fast rough answer. IQR when the data has outliers or you're working with a median or box plot. MAD when you're working with the mean and want the typical distance from it. Test questions usually name the one they want.

Why are the distances in MAD always positive?

Because distance measures how far, not which direction. A value $3$ below the mean and a value $3$ above it are both a distance of $3$ away. If you kept the signs, they would cancel and the average would come out $0$ every time.

Do I include the median when splitting the data into halves for IQR?

With an odd number of values, no — set the middle value aside and take the halves on either side of it. With an even number of values, the data splits cleanly into two halves and nothing is set aside.

Can two data sets have the same mean but different spreads?

Yes, and that's the whole point of spread. $9, 10, 11$ and $0, 10, 20$ both have a mean of $10$ , but the first set is far more consistent. Center tells you where the data sits; spread tells you how tightly it sits there.

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