Lesson 2: Mean

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

Outliers are values that are significantly different from the rest of the data.
These are data points that stand far apart from the main group of values.
We can identify potential outliers by looking for values that seem unusually high or low compared to the other data points in the set.

Examples

Property

An outlier is an extreme value that is much higher or lower than the other values in a data set.
Outliers have a strong effect on the mean because the mean uses all values in its calculation.
When an outlier is present, it pulls the mean toward the extreme value, making the mean less representative of the typical values in the data set.

To calculate the mean: $\text{mean} = \frac{\text{sum of all values}}{\text{number of values}}$