Lesson 2: Understanding Measures of Center

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

The median is a number that divides an ordered data set into two parts with an equal number of values in each part.
To find the median, you must first put the values in order from lowest to highest.

• If there are an odd number of data points, the median is the number right in the middle.
• If there are an even number of data points, the median is the number halfway between the two middle values (their mean).

Examples

• For the data set {9, 2, 7, 5, 11}, we first order it: {2, 5, 7, 9, 11}. Since there are five values, the middle value is the 3rd one, so the median is 7.
• For the data set {14, 6, 8, 20}, we order it: {6, 8, 14, 20}. With an even number of values, the median is the mean of the two middle numbers: $\frac{8+14}{2} = 11$.
• The prices of five houses on a street are 200k, 210k, 225k, 240k, and 950k dollars. The median price is 225k dollars, which is a more typical value than the mean (365k dollars), which is skewed by the expensive house.