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6. Descriptive Statistics
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Integrated Math 3

6.03 Measures of spread

Lesson
Practice
Lesson

Concept summary

Measures of spread describe how similar or varied a data set is. Measures of spread include the range, the interquartile range, variance, and deviation.

Summarizing a data set can help us understand the data points, especially when the data set is large. The measures of spread summarize the data set in a way that shows how scattered the values are.

Range

The distance between the smallest and largest data point

\displaystyle \text{Range}=\text{Maximum}-\text{Minimum}
\bm{\text{Maximum}}
The largest value in the data set
\bm{\text{Minimum}}
The smallest value in the data set

We can divide a data set in half with the median, and then can also divide each half of the data set into quartiles using the upper and lower quartiles, Q_1 and Q_3. The boundaries for the quartiles are called the 5-number summary.

A box plot is a type of data display that shows the 5-number summary and any extreme data points.

Interquartile range (IQR)

The distance between the upper and lower quartiles

\displaystyle IQR=Q_3-Q_1
\bm{Q_1}
Lower quartile
\bm{Q_3}
Upper quartile

Worked examples

Example 1

The number of fatal accidents from 2000 to 2014 for different airlines are displayed in the box plot:

Number of Fatal Accidents
-2
0
2
4
6
8
10
12
14
16
18
20
22
24
26
a

Identify and interpret the range of the data set.

Approach

The range of the data set is the distance between the minimum value and the maximum value.

Solution

The maximum of the data set is at 24 and the minimum is at 0 so 24-0=24 is the range.

From 2000-2014 the number of fatal accidents with different airlines varied by 24 accidents.

Reflection

All statements about the range that describe variance should be sentences phrased in terms of the variable of interest, and include the correct units of measurement.

b

Identify and interpret the interquartile range of the data set.

Approach

The interquartile range of the data set is the distance between the upper and lower quartiles.

Solution

The upper quartile is at 5.5 and the lower quartile is at 1 so:

\displaystyle IQR\displaystyle =\displaystyle Q_3-Q_1
\displaystyle =\displaystyle 5.5-1
\displaystyle =\displaystyle 4.5

From 2000-2014 the number of fatal accidents for the middle half of all airlines varied by 4.5 accidents.

Reflection

Since each quartile of a box plot represents \approx 25\% of the data set, the IQR represents the middle 50\% of the data set.

c

Explain what will happen to the range and interquartile range if the extreme value at 24 is removed.

Approach

From parts (a) and (b) we know that the range and IQR are 24 and 4.5, respectively. We need to recalculate, or estimate, the new range and IQR without the point at 24.

Solution

Without the point at 24, the maximum of the data set is 17 and the minimum is still 0. The new range is 17-0=17 which means the range lowers by 7 accidents when the point at 24 is removed.

Without seeing the data set itself, we can't calculate the exact IQR once the point at 24 is removed. However, the lower quartile can't be less than 0 and can't be more than 1 so it won't change by more than 1 accident. The upper quartile can't be less than 3 and can't be more than 5.5 so the IQR may be less, but won't lower by more than 2.5 accidents.

Reflection

Similar to mean and median, the range is greatly affected when extreme data points are added or removed, but the interquartile range should change very little.

Outcomes

M3.N.Q.A.1

Use units as a way to understand real-world problems.*

M3.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.

M3.S.ID.A.2

Use statistics appropriate to the shape of the data distribution to compare center (mean, median, and/or mode) and spread (range, standard deviation) of two or more different data sets.*

M3.S.ID.A.3

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points.*

M3.MP2

Reason abstractly and quantitatively.

M3.MP3

Construct viable arguments and critique the reasoning of others.

M3.MP4

Model with mathematics.

M3.MP5

Use appropriate tools strategically.

M3.MP6

Attend to precision.

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