Range, interquartile range and mean absolute deviation
Explain why the centre alone cannot describe a distribution.
Before you begin
- Mean and median
- Absolute value
Measure the span and middle spread
Two data sets can share a mean while differing greatly in variability. Range is maximum minus minimum and uses only the extremes. The interquartile range, IQR, is the upper quartile Q3 minus the lower quartile Q1 and describes the middle half. Here we find each quartile as the median of its half of the ordered data, excluding the overall median when the count is odd; other conventions can differ.
Measure distances from the mean
Mean absolute deviation, MAD, is the average distance of the observations from their mean. Find the mean, take each absolute difference from it, add those distances and divide by the count. Absolute values prevent positive and negative deviations from cancelling. MAD is measured in the original units. A larger MAD means more average spread about the mean, but its interpretation still depends on the context.
Worked example
Find the range, quartiles and IQR of 2, 4, 4, 6, 8, 10, 12 and 14.
Show the worked solution
- The range is 14 − 2 = 12. The median is (6 + 8)/2 = 7.
- The lower half is 2,4,4,6, whose median is Q1 = 4. The upper half is 8,10,12,14, whose median is Q3 = 11.
- The middle-half spread is IQR = 11 − 4 = 7.
Range 12; Q1 = 4; median 7; Q3 = 11; IQR 7.
Try it yourself
Which data set has greater spread: A = {4, 5, 6} or B = {1, 5, 9}?
Common mistakes
- State a quartile convention when comparing answers from different tools.
- Average absolute deviations, not signed deviations.
What can you explain now?
Find the MAD of 4, 4, 6, 6 and 10 minutes.
Compare with the explanation
1.6 minutes.
The mean is 30/5 = 6. Absolute deviations are 2, 2, 0, 0 and 4, totalling 8. MAD = 8/5 = 1.6 minutes.
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Common Core mathematics standards ↗