11.1.7 Measuring the Spread of a Data Set

As well as finding a typical value or average for a set of data, it is also important to see how the data values are spread about the typical value.

Last year, Sam grew two varieties of sunflowers and measured the heights of the flowers. Here are Sam’s results:

Heights of Sunflowers, in Inches

Variety A 60 59 66 55 55 62 58 65
Variety B 52 69 61 60 54 67 50 67

Sam wants compare how each sunflower variety performed so wants to look at the data in more detail.

Solution symbolActivity: How Tall Are the Sunflowers on Average?

(a) Find the mean height of the sunflowers of variety A, using the data from the table above.

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Answer

(a) The sum of the heights of the sunflowers of variety A is equation left hand side sum with, 8 , summands 60 plus 59 plus 66 plus 55 plus 55 plus 62 plus 58 plus 65 equals right hand side 480 times.

So the mean height of the sunflowers of variety A is 480 inches divided by eight equals 60 inches.

Remember to include the units, in this case, inches; otherwise your answer won’t make much sense.

(b) Find the mean height of the sunflowers of variety B, using the data from the table above.

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Answer

(b) For variety B, the sum of the heights is sum with, 8 , summands 52 plus 69 plus 61 plus 60 plus 54 plus 67 plus 50 plus 67 equals 480.

So the mean height is 480 inches divided by eight equals 60 inches.

On average, the sunflowers of the two varieties are the same height, 60 inches.

When Sam looked at the sunflowers, they did look about the same height on average, but there was still a difference. To investigate this, you can plot the heights on number lines, like this:

The blue dots represent the heights in inches of the sunflowers of variety A, and the red dots represent the heights in inches of the sunflowers of variety B. Can you see that the red dots are more spread out than the blue dots? The heights of sunflowers of variety A are more consistent, closer together, while the heights of the sunflowers of variety B are more variable, more spread out.

One way of measuring the spread of a data set is called the range. It is the largest value minus the smallest value.

Finding the Range

range of a data set equals largest value minus smallest value

For the sunflowers of variety A, the smallest value is 55 inches and the largest value is 66 inches, so the range in inches is 66 minus 55 equals 11.

Remember to include the units when you state the range: The range in the heights of variety A sunflowers is 11 inches.

Solution symbolActivity: The Range for Sunflowers of Variety B

Find the range in the heights of the sunflowers of variety B.

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Answer

The smallest value in the data set for variety B is 50 inches, and the largest value is 69 inches. So the range in inches is 69 minus 50 equals 19.

Therefore, the range in the heights of variety B sunflowers is 19 inches.

So even though both sunflower varieties had the same mean height there was much more variation, that is a bigger range, of heights seen in variety B compared to variety A. If you grew sunflowers commercially can you think of any reasons why knowing this information about the range may be significant?  You might like to discuss this with friends and see what they think.

The range gives an idea of the spread of the data, although it can be affected by either very high or very low values. There are other ways of measuring the spread of a data set, and you’ll meet these if you continue studying mathematics or statistics.

11.1.6 Using Different Averages

11.1.8 More About Data