13. Representing Data
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To create a box-and-whisker plot you must identify five things — the minimum value, lower quartile, median value, upper quartile, and the maximum value.
Stem-and-Leaf Plot:
| Stem | Leaf |
|---|---|
| 5 | 8 9 |
| 6 | 0 3 5 6 9 |
| 7 | 0 1 1 2 3 |
Box-and-Whisker Plot:
Outliers: Since there are no outliers, they do not affect the quartiles.
The following set of data is given. {65,63,69,71,73,59,60,70,72,66,71,58} We will make a stem-and-leaf plot and a box-and-whisker plot. To do that it will help to order the data values from least to greatest. {58,59,60,63,65,66,69,70,71,71,72,73} Now, we can start with stem-and-leaf plot.
In a stem-and-leaf plot, the digits of the least place value usually form the leaves, and the rest of the digits form the stems. Thus, we can make the stem-and-leaf plot as shown below.
| Stem | Leaf |
|---|---|
| 5 | 8 9 |
| 6 | 0 3 5 6 9 |
| 7 | 0 1 1 2 3 |
Next we will make a box-and-whisker plot.
To create a box-and-whisker plot, we need to identify five things.
To find these values we need to have the data arranged from least to greatest. 58,59,60,63,65,66,69,70,71,71,72, 73 Now we can immediately identify the minimum and maximum value. &Minimum Value: 58 &Maximum Value: 73 The median is the value that is in the middle of the data set. Since the number of data values is an even number, 12, there is no unique median. Instead, the median will be average of the data values that are closest to the middle 58,59,60,63,65, 66, 69,70,71,71,72,73 The median is the mean of the 6^\text{th} and 7^\text{th} value. Median:& 66+ 69/2=67.5 The lower and upper quartile is the value that is in the middle of the lower and upper half of the data set. Again we have to use the values that are closest to the middle. Let's mark these for the lower and upper quartile. 58,59, 60, 63,65,66, | 69,70, 71, 71,72,73 Now we can calculate the lower and upper quartile for the data set. &Lower quartile: 60+ 63/2=61.5 &Upper quartile: 71+ 71/2=71 Let's summarize the information in a table.
| Minimum value | 58 |
| Lower quartile | 61.5 |
| Median | 67.5 |
| Upper quartile | 71 |
| Maximum value | 73 |
Now we can make our box-and-whisker plot.
In order to identify any outliers we will first need to find the interquartile range. The interquartile range IQR is the range between the lower quartile Q_1 and the upper quartile Q_3. IQR=Q_3-Q_1 Let's find the IQR.
Thus, the interquartile range is 9.5. Now we will write two inequalities that determine the outliers, O. Inequality I O< Q_1-1.5(IQR) ⇒ O< 47.25 [0.7em] Inequality II O> Q_3+1.5(IQR) ⇒ O> 85.25 Numbers less than 47.25 and greater than 85.25 are outliers. Since there are no outliers in the set of data, they do not affect the quartiles.