Lesson 5: Summarize Data Using Measures of Variability

Property

The interquartile range (IQR) measures the spread of the middle 50% of a data set. To find the IQR:

1. Find Q1 (the median of the lower half of the data)
2. Find Q3 (the median of the upper half of the data)
3. Calculate: IQR = Q3 - Q1

Examples

Property

Once we have calculated the mean for a set of data, we want to have some sense of how the data are arranged around the mean.
The mean absolute deviation (MAD) is calculated this way: for each data point, calculate its distance from the mean.
Now the MAD is the mean of this new set of numbers.
When we use the mean and the MAD to summarize a data set, the mean tells us what is typical or representative for the data and the MAD tells us how spread out the data are.
The MAD tells us how much each score, on average, deviates from the mean, so the greater the MAD, the more spread out the data are.

Examples

• For the data set {3, 5, 6, 10}, the mean is $\frac{3+5+6+10}{4} = 6$. The distances from the mean are $|3-6|=3$, $|5-6|=1$, $|6-6|=0$, and $|10-6|=4$. The MAD is $\frac{3+1+0+4}{4} = 2$.
• Two friends track their nightly sleep. Alex's hours are {7, 8, 9} and Ben's are {5, 8, 11}. Both have a mean of 8 hours. Alex's MAD is $\frac{1+0+1}{3} \approx 0.67$. Ben's MAD is $\frac{3+0+3}{3} = 2$. Ben's sleep is more spread out.
• A bowler's scores are {150, 155, 160}. The mean is 155. The deviations are $|150-155|=5$, $|155-155|=0$, and $|160-155|=5$. The MAD is $\frac{5+0+5}{3} \approx 3.33$, showing high consistency.

Explanation

The Mean Absolute Deviation (MAD) measures how spread out your data is. A small MAD means all the numbers are bunched up close to the average. A large MAD means the numbers are widely scattered far from the average.