Spread: the range and the deviations from the mean

Science and society, grade 11 (3 units) Lesson 8 of 16Learning

Spread: the range and the deviations from the mean

The same mean, a different spread

Two sets of data can have the same mean and look completely different. What tells them apart is the spread: how far the data stray from the mean.

The points of two basketball players in their last five games:
Noa: 14, 16, 12, 18, 15.
Tamar: 4, 26, 10, 20, 15.

Noa’s mean: $\displaystyle {\frac{14+16+12+18+15}{5}}={\frac{75}{5}}={15}$ .
Tamar’s mean: $\displaystyle {\frac{4+26+10+20+15}{5}}={\frac{75}{5}}={15}$ .
Both have a mean of 15, but Noa scores about 15 points in every game, and Tamar sometimes very few and sometimes many.

A measure that describes how spread out the data are is called a measure of spread.

The range

The range is the simplest measure of spread: the greatest datum minus the least.

\displaystyle x_{\max}-x_{\min}

Noa’s range: $\displaystyle {18-12}={6}$ points.
Tamar’s range: $\displaystyle {26-4}={22}$ points.

Tamar’s range is larger: her points are more spread out.

The weakness of the range

The range depends on two data only – the greatest and the least – and says nothing about the rest.

The temperature at noon, in degrees, over a week in two towns:
Town A: 18, 22, 22, 22, 22, 22, 26.
Town B: 19, 20, 21, 22, 23, 24, 25.

In both towns the mean is 22.
The range in town A is $\displaystyle {26-18}={8}$ , and in town B $\displaystyle {25-19}={6}$ .
Yet on five of the seven days town A had exactly 22 degrees. Two unusual days set its range.

To describe the spread of all the data, we need a measure that every datum affects.

The deviation from the mean

The deviation of a datum from the mean is the difference between the datum and the mean:

\displaystyle x_i-\bar{x}

A datum greater than the mean has a positive deviation; a datum less than the mean has a negative one.
The further a datum is from the mean, the larger its deviation (ignoring the sign).

The deviations of the points from the mean of 15:

Noa’s points	14	16	12	18	15
Deviation	$\displaystyle -1$	$\displaystyle 1$	$\displaystyle -3$	$\displaystyle 3$	$\displaystyle 0$

Tamar’s points	4	26	10	20	15
Deviation	$\displaystyle -11$	$\displaystyle 11$	$\displaystyle -5$	$\displaystyle 5$	$\displaystyle 0$

Tamar’s deviations are much larger – her points are far from the mean.

The sum of the deviations

The deviations from the mean always add up to 0: the positive and the negative deviations cancel each other out.

Noa: $\displaystyle {-1+1-3+3+0}={0}$ .
Tamar: $\displaystyle {-11+11-5+5+0}={0}$ .

This happens in every set of data, because the mean is exactly the point where the sums on its two sides balance. So the mean of the deviations is always 0, and says nothing about the spread.

So that the deviations do not cancel out, each deviation is squared – the square of a negative number is positive. That is how the standard deviation is built, in the next lesson.

Exercise 1

Two pupils’ marks in five tests:
Dana: 70, 75, 80, 85, 90.
Yossi: 50, 100, 80, 95, 75.

Both have a mean of 80.

Work out the range of Dana’s marks.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power