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Concept: Box-and-Whisker Plots

A box-and-whisker plot is a visual representation of how the data is spread out and how much variation there is. The main advantage of the box-and-whisker plot is that it is not cluttered by showing all the data values. It highlights only a few important features of the data. Therefore, the box-and-whisker plot makes it easier to focus attention on the median, extremes, and quartiles and comparisons among them. Another advantage of the box-and-whisker plot is that it does not become more complicated with more data values. A disadvantage of the box-and-whisker plot occurs when there are only a few data values.

Constructing a box-and-whisker plot:

Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
Seattle, WA 46 51 54 58 64 70 74 74 69 60 52 46
San Antonio TX 61 66 74 80 85 92 95 95 89 82 72 64
New York, NY 38 40 50 61 72 80 85 84 76 65 54 43

First, arrange the data in the table above in increasing order.

Seattle, WA : 46, 46, 51, 52, 54, 58, 60, 64 ,69, 70, 74, 74

San Antonio, TX: 61, 64, 66, 72, 74, 80, 82, 85, 89, 92, 95, 95

New York, NY: 38, 40, 43, 50, 54, 61, 65, 72, 76, 80, 84, 85

Second, find the extremes. The lower extreme is the lowest value and the upper extreme is the highest value.
The lowest value for Seattle is 46, and the highest value is 74.

Third, find the median.

There are 12 values, therefore the median is halfway between the 6th and the 7th values.
box 1
46, 46, 51, 52, 54, 58 | 60, 64, 69, 70, 74, 74
The median is 59. There are six numbers below 59 and six numbers above it.
Fourth, find the lower quartile (the median of the lower half of the data).
Consider only the data values to the left of the line.


46, 46, 51, 52, 54, 58
The median of these six numbers is halfway between the 3rd and the 4th values. This is the lower quartile. The lower quartile is 51.5.
Fifth, find upper quartile (the median of the upper half of the data).
Consider only the data values to the right of the line.
60, 64, 69, 70, 74, 74
Find their median. This is the upper quartile. The upper quartile is 69.5.
Sixth, plot the extremes (lower extreme 46, upper extreme 74), the quartiles (lower quartile 51.5 and upper quartile69.5), and the median (59) on a number line.

box 2

Seventh, draw a rectangular box extending from the lower quartile to the upper quartile. Indicate the median with a vertical line extending through the box.

Eighth, connect the lower extreme toto the lower quartile with a line (one "whisker") and the upper quartile to the upper extreme with another line (the other "whisker".)

box 4

Repeat steps one through eight to construct box-and-whisker plots using the data given for San Antonio and New York, respectively.

box 5

Note:

If there are an odd number of data values, the overall median is not included among the values used to calculate the medians of the lower or upper quartiles.

Example: 13 19 20 21 23 23 23 24 25 25 26 26 28

There are 13 data values here.
Overall median is 23.
Lower extreme is 13.
Lower quartile is 20.5 (the median of the lower quartile).
Upper extreme is 28.
Upper quartile is 25.5 (the median of the upper quartile).



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