Math 1 · Unit 12 · Lesson 3 · part 2 of 4
Same center, different spread: the range and the IQR
Shape, center, and spread, every time.
Same center, different spread: the range and the IQR
Shape, center, and spread, every time. This lesson is spread, also called variability: how far apart the values are. Two data sets can have the same center and look completely different. Both of these have a mean of 6.
The simplest measure of spread is the range, maximum minus minimum: 4 for the tight set and 10 for the wide set. But the range uses only the two end values, so one extreme value can make it huge. Two better measures follow.
The interquartile range, IQR
The quartiles Q1 and Q3 are the values one quarter and three quarters of the way through the ordered data: the median of the lower half and the median of the upper half (leaving out the overall median when the count is odd, the way a graphing calculator does it).
IQR = Q3 − Q1: the width of the box in a box plot, the spread of the middle half of the data. Like the median, it resists extreme values.
Tight set: 4, 5, 6, 7, 8. Lower half 4, 5, so Q1 = 4.5. Upper half 7, 8, so Q3 = 7.5. IQR = 3. Wide set: Q1 = 2, Q3 = 10, IQR = 8.
The mean absolute deviation, from 7th grade
Find how far each value is from the mean (its deviation), ignore the signs, and average those distances.
For 2, 4, 6, 8, 10 with mean 6: distances 4, 2, 0, 2, 4. Average: 125 = 2.4. A typical value is about 2.4 from the mean.
The standard deviation does almost the same thing, with one change.