Mean absolute deviation, often called MAD, measures how far data values typically are from the mean. It is a measure of spread, so it helps you describe whether a data set is tightly clustered or widely scattered. Unlike the mean alone, MAD tells you how consistent the values are.
It is useful in statistics, science labs, test scores, sports data, and any situation where variation matters.
To find MAD, first compute the mean of the data set. Then find each absolute deviation by subtracting the mean from each value and taking the absolute value. Finally, average those absolute deviations.
MAD is similar in purpose to standard deviation, but it uses ordinary distances from the mean instead of squared distances, which often makes it easier to interpret.
Key Facts
- Mean = sum of all data values / number of data values
- Absolute deviation = |x - mean|
- Mean absolute deviation = (sum of |x - mean|) / n
- MAD measures the typical distance of data values from the mean.
- A larger MAD means the data values are more spread out.
- Standard deviation squares deviations, while MAD uses absolute values: SD = sqrt(sum of (x - mean)^2 / n) for a population.
Vocabulary
- Mean
- The mean is the arithmetic average found by adding all data values and dividing by the number of values.
- Deviation
- A deviation is the difference between a data value and the mean.
- Absolute Value
- Absolute value is the distance of a number from zero, so it is always nonnegative.
- Mean Absolute Deviation
- Mean absolute deviation is the average of the absolute distances between each data value and the mean.
- Spread
- Spread describes how far apart or how varied the values in a data set are.
Common Mistakes to Avoid
- Forgetting to use absolute values, which is wrong because positive and negative deviations can cancel out and make the spread look smaller than it really is.
- Finding deviations from the median instead of the mean, which is wrong when the question specifically asks for mean absolute deviation because MAD is centered on the mean.
- Dividing by the wrong number of values, which is wrong because the final step averages all absolute deviations using n, the total count of data values.
- Confusing MAD with standard deviation, which is wrong because standard deviation squares deviations before averaging while MAD averages absolute distances directly.
Practice Questions
- 1 Find the mean absolute deviation of the data set 4, 6, 8, 10, 12.
- 2 A student recorded these daily study times in minutes: 20, 30, 30, 40, 80. Find the mean and the mean absolute deviation.
- 3 Two classes have the same mean quiz score of 80. Class A has a MAD of 3 and Class B has a MAD of 12. Explain which class has more consistent scores and why.