A measure of spread is a way of quantifying how spread out, or different, the points in a data set are. A small spread means data points are similar, while a large spread means they are different. This is illustrated by the two data sets below. Both have a mean, median and mode of 3, but, we can assume the second data set has a larger spread because of how different its data points are.
One way to measure the spread of a data set is to find its range. This is done by calculating the difference between the maximum and minimum value of the data set.
range=maximum value−minimum value
Standard deviation is a commonly used measure of spread. It is a measure of how much a randomly selected value from a data set is expected to differ from the mean. To denote the standard deviation, the Greek letter σ is used, which is read as "sigma."
The standard deviation, σ, of a data set is calculated using the rule σ=n(x1−xˉ)2+(x2−xˉ)2+…+(xn−xˉ)2, where n is the number of values in the data set and xˉ is the mean of the set. Performing this calculation in one step makes for a convoluted expression. Therefore, it is best divided into a few, smaller steps. Consider the following data set as an example. 1,5,3,8,3,12
For each data value, x−xˉ can now be calculated and added to a table. This shows how much each data point varies from the mean.
x | x−xˉ |
---|---|
1 | 1−4=-3 |
5 | 5−4=1 |
3 | 3−4=-1 |
8 | 8−4=4 |
3 | 3−4=-1 |
12 | 12−4=8 |
Square the deviations, and add them to a new column in the table.
x | x−xˉ | (x−xˉ)2 |
---|---|---|
1 | -3 | (-3)2=9 |
5 | 1 | 12=1 |
3 | -1 | (-1)2=1 |
8 | 4 | 42=16 |
3 | -1 | (-1)2=1 |
12 | 8 | 82=64 |
Finally, take the square root of the just found quotient to get the standard deviation. Here, the fraction is used instead of the quotient, to avoid rounding errors. σ=692≈3.92 Thus, a randomly chosen value from this data set is expected to deviate roughly 4 units from the mean.
One way to measure the spread of a data set is with the interquartile range, which is the difference between the third and the first quartile. The quartiles are found by dividing the data set into four equal-sized groups. A set of 12 data values would be divided into groups with three data values in each.
The quartiles are the three values that divide the data set into four groups. They are denoted with Q1,Q2 and Q3. Notice that Q2 is the median.
The interquartile range, or IQR, is calculated by subtracting the first quartile, Q1, from the third, Q3.
Interquartile range=Q3−Q1
First, identify the median of the data set. Since the number of values is even, the median is the mean of the two middle values. 1,3,4,4,5,6,6,8,8,10,10,11
The interquartile range is calculated by subtracting the first quartile, Q1, from the third, Q3. For the example, this gives IQR=Q3−Q1=9−4=5.
Each of these three measures of spread, range, standard deviation, and interquartile range, have advantages and drawbacks.
Calculating the range is a relatively simple process. However, since it only takes into account two data points, the variability of the entire data set is not known.
Because the standard deviation is found using all data points, it is representative of the entire data set. A drawback, though, is that for a large data set it is a very time-consuming calculation to do by hand.