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 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.
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
|
89
|
6
|
03569
|
7
|
01123
|
Next we will make a box-and-whisker plot.
Box-and-Whisker Plot
To create a , we need to identify five things.
- Minimum value
- Lower
- value
- Upper quartile
- Maximum value
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: 58Maximum 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 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
6th and
7th value.
Median: 266+69=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:260+63=61.5Upper quartile:271+71=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.
Outliers
In order to identify any we will first need to find the . The interquartile range
IQR is the range between the
lower quartile Q1 and the
upper quartile Q3.
IQR=Q3−Q1
Let's find the
IQR.
IQR=Q3−Q1
IQR=71−61.5
IQR=9.5
Thus, the interquartile range is
9.5. Now we will write two inequalities that determine the outliers,
O.
Inequality I O<Q1−1.5(IQR)⇒O<47.25Inequality II O>Q3+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.