Let’s start with a definition of the measures of variation. When analyzing data, if you want to find out the spread of the data, the way to describe this spread is by the measures of variation.
Go back to the test scores on a statistics exam and say there were 31 students in the fourth period class. These were their scores:
Organize them in stem-and-leaf plot would look like this:
One way to measure the spread of the data is from end to end.
The minimum number is 51 in this data.
The maximum number is 98.
So, you could say that the data is between, and including, 51 through 98.
Subtract 51 from 98 to find the range (the distance from the minimum to the maximum). The range is 47 because there are 47 units, from the lowest number to the highest.
Another way to measure the spread of the data is to divide it into quarters or fourths, called quartiles.
There are 31 numbers, so the 16 is the middle or median number.
That makes 79 the median.
There are 15 numbers on each side of the median, which is 79.
So the eighth number would divide the first half into two parts.
The eighth number from the other end (which is the 24th) would divide the second half into two parts.
That makes 71 the first quartile marker and 82 the third quartile marker.
The median is the second quartile, so we just call it the median.
To put it all together, here is the data with the markers that explain where the fourths of the data occur:
Minimum 51
First Quartile 71
Median 79
Third Quartile 82
Maximum 98
Measures of Variation
range = maximum – minimum
minimum = least number of the data
first quartile (Q1) = the middle of the lower half of the data
median = the middle of the data
third quartile (Q3) = the middle of the upper half of the data
maximum = greatest number of the data
Example 1: Finding the Measures of Variation
Junior Varsity Basketball Team
Height (cm)
Frequency
175
1
178
1
180
2
181
1
184
3
185
2
188
1
192
1
Total
12
Find the median, first quartile, third quartile, and range.
You know that there are 12 numbers, so the median is between the sixth and seventh number. Looking at the frequency, the sixth number is 184 and 184 is also the seventh number. The median is 184.
Now looking at the first six numbers in the data, the halfway mark is between the third and fourth numbers. They are both 180. Q1 is 180.
In the last six numbers, the halfway mark is between the ninth and tenth numbers. They are both 185. Q3 is 185
The range is maximum – minimum = 192 – 175 = 17.
Quick Practice
Find the minimum, first quartile, median, second quartile, maximum, and range for each of the following.
Number of minutes spent online in the social studies course by Mr. Park’s students:
Measures of Variation
