Prices, location and spread
Prices, location and spread

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Prices, location and spread

A measure of spread

Now we can define a new measure of spread based entirely on the lower and upper quartiles.

The interquartile range

The interquartile range (sometimes abbreviated to IQR) is the distance between the lower and upper quartiles:

IQR = uppercase Q sub 3 minus uppercase Q sub 1 .

Note that this value is independent of the sizes of uppercase E subscript uppercase U end and uppercase E subscript uppercase L end.

Example 16 The prices of small televisions, yet again!

For the batch of 20 television prices in Example 14 (Subsection 3.2),

IQR = uppercase Q sub 3 minus uppercase Q sub 1 = 180 minus 130 = 50.

So the interquartile range is £50.

Activity 11 Coffee prices again

Calculate both the range and the interquartile range of the batch of 15 coffee prices, last seen in Figure 17 (Subsection 3.2).

Discussion

The range is the distance between the extremes:

range = uppercase E subscript uppercase U end minus uppercase E subscript uppercase L end = 369 p minus 268 p = 101 p .

The interquartile range is the distance between the quartiles:

IQR = uppercase Q sub 3 minus uppercase Q sub 1 = 299 p minus 268 p = 31 p .

Activity 12 Interquartile range of gas prices

In Activity 10(b) (Subsection 3.2) you found the quartiles of the 14 gas prices from Activity 2 (Subsection 1.2). Find the interquartile range.

Discussion

The quartiles, before rounding, are uppercase Q sub 1 =3.75575 and uppercase Q sub 3 =3.80175. So

IQR = uppercase Q sub 3 minus uppercase Q sub 1 =3.80175 minus 3.75575 =0.046 comma

and the interquartile range is 0.046p per kWh.

You may be wondering why you are being asked to learn a new measure of spread when you already know the range. As a measure of spread, the range open bracket uppercase E subscript uppercase U end minus uppercase E subscript uppercase L end close bracket is not very satisfactory because it is not resistant to the effects of unrepresentative extreme values. (Resistant measures were explained in Subsection 1.4.) The interquartile range, by contrast, is a highly resistant measure of spread (because it is not sensitive to the effects of values lying outside the middle 50% of the batch) and it is generally the preferred choice.

Example 17 Comparing the resistance of the range and the IQR

Suppose the price of the most expensive jar of coffee is reduced from 369p to 325p. How does this affect the range and the interquartile range of the batch of coffee prices in Figure 17 (Subsection 3.2)?

The new range is

uppercase E subscript uppercase U end minus uppercase E subscript uppercase L end = 325 p minus 268 p = 57 p comma

a lot less than the original value of 101p (found in Activity 11). The interquartile range is unchanged.

M140_1

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