Lesson 3: Measures of Center and Variability

Property

The most common measure of center is the mean.
The mean is the arithmetic average, often referred to simply as “average.”
The procedure of computing the mean is to add up all the data values and then divide by the number of data values.
The significance of the mean is that it represents a fair share of the total.
For a data set of $N$ values, $a_1, a_2, …, a_N$, the formula is:

$$

= \frac{a1 + a2 + a3 + \cdots + aN}{N} $$
Another way to view the mean is as a balance point: the sum of the distances of the data points from the mean for those points below the mean is equal to the same sum for all the points above the mean.

Examples

• A student's scores on five math tests are 85, 90, 75, 88, and 82. The mean score is calculated as $\frac{85+90+75+88+82}{5} = \frac{420}{5} = 84$.
• Four friends collected stamps: 30, 42, 25, and 35. To share them equally, they find the mean: $\frac{30+42+25+35}{4} = \frac{132}{4} = 33$. Each friend gets 33 stamps.
• The mean height of three plants is 15 cm. Two plants measure 12 cm and 18 cm. The height of the third plant, $h$, is found by solving $\frac{12+18+h}{3} = 15$, which gives $h = 15$ cm.

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.