1. A box-and-whisker plot uses a number line to represent the data visually. I.E. It shows visually the
GRE - Mean, Median, Mode, and Range2. It displays a data set along a number line using medians. Quartiles divide the data set into four equal parts. The median (second quartile) divides the data set into two halves. The median of the lower half is the first quartile. The median of the upper half is the third quartile.
3. The box plot does not show
individual data but it analyzes the
spread of data. A measure of spread is a single number that describes how to spread out or clustered together the data are.
4. Another measure of spread is the interquartile range (IQR). The “Interquartile Range” is the difference between the smallest value and the largest value of the middle 50% of a set of data.
The "Interquartile Range" is from Q1 to Q3:
To find the interquartile range of a set of data:
- First put the list of numbers in order
- Then cut the list into four equal parts
- The quartiles are the “cuts”
- The interquartile range is the distance between the two middle sets of data
Example: Data Set 24, 30, 30, 22, 25, 22, 18, 25, 28, 30, 25, 27
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a) Order the data. Find the median and the quartiles.
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b) Draw a number line that includes the least and greatest values. Graph points above the number line for the least value, greatest value, median, first quartile, and third quartile.
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c) Draw a box using the quartiles. Draw a line through the median. Draw whiskers from the box to the least and greatest values.
Tips:
- A box-and-whisker plot shows the variability of a data set.
- A long whisker or box indicates data are more spread out.
More ExamplesThe first step in constructing a box-and-whisker plot is to first find the median (Q2), the lower quartile (Q1) and the upper quartile (Q3) of a given set of data.
Box plot with odd numbers in that data setThe first step in constructing a box-and-whisker plot is to first find the median (Q2), the lower quartile (Q1) and the upper quartile (Q3) of a given set of data.
Step 1: Find the median. The median is the value exactly in the middle of an ordered set of numbers.
Step 2: Consider the values to the left of the median: 18 27 34 52 54 59 61 Find the median of this set of numbers. The median is 52 and is called the lower quartile.
Step 3: Consider values to the right of the median: 78 82 85 87 91 93 100 Find the median of this set of numbers. The median 87 is therefore called the upper quartile.
Note that when there is an odd number of values, as in this example. we don't include the median in the set of numbers used to calculate the upper and lower quartiles.
We are able now to find the interquartile range (IQR). The interquartile range is the difference between the upper quartile and the lower quartile. In example 1, the IQR = Q3 — Q1 = 87 - 52 = 35.
The IQR is a very useful measurement. It is useful because it is less influenced by extreme values as it limits the range to the middle 50% of the values.
Box plot with even numbers in that data setStep 1: Find the median. The median is the value exactly in the middle of an ordered set of numbers.
The median in this example is in-between 70 and 73, so the median is calculated by taking the mean of 70 and 73:
Median \(= \frac{70+73}{2} =71.5\)
Step 2: Consider the values to the left of the median: 42 63 64 64 70 Find the median of this set of numbers. The median is 64.
Step 3: Consider the values to the right of the median: 72 76 77 81 81
Note that when the number of values is even the median lies between the two middle values. As in this example, we include the data value just below the median in the set of numbers used to calculate the lower quartile. and the number just above the median in the set of numbers used to calculate the upper quartile.
The median is 77 and is called the upper quartile. We are able now to find the interquartile range (IQR). The interquartile range is the difference between the upper quartile and the lower quartile. The IQR = Q3 — Q1 = 77 - 64 = 13. The IQR is a very useful measurement. It is useful because it is less influenced by extreme values as it limits the range to the middle 50% of the values.
The Box Plots can be either verticalOr horizontalhttps://students.flinders.edu.au/support/slssAttachment:
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