Mathematics » Introducing Decimals » Averages and Probability

# Calculating the Mean of a Set of Numbers

## Calculating the Mean of a Set of Numbers

The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size.

Suppose Ethan’s first three test scores were $$85,88,\text{and}\phantom{\rule{0.2em}{0ex}}94.$$ To find the mean score, he would add them and divide by $$3.$$

$$\begin{array}{c}\cfrac{85+88+94}{3}\\ \cfrac{267}{3}\\ 89\end{array}$$

His mean test score is $$89$$ points.

### Definition: The Mean

The mean of a set of $$n$$ numbers is the arithmetic average of the numbers.

$$\text{mean}=\cfrac{\text{sum of values in data set}}{n}$$

### How to Calculate the mean of a set of numbers.

1. Write the formula for the mean
$$\text{mean}=\cfrac{\text{sum of values in data set}}{n}$$

2. Find the sum of all the values in the set. Write the sum in the numerator.
3. Count the number, $$n,$$ of values in the set. Write this number in the denominator.
4. Simplify the fraction.
5. Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.

## Example

Find the mean of the numbers $$8,12,15,9,\text{and}\phantom{\rule{0.2em}{0ex}}6.$$

### Solution

 Write the formula for the mean: $$\text{mean}=\frac{\text{sum of all the numbers}}{n}$$ Write the sum of the numbers in the numerator. $$\text{mean}=\frac{8+12+15+9+6}{n}$$ Count how many numbers are in the set. There are 5 numbers in the set, so $$n=5$$. $$\text{mean}=\frac{8+12+15+9+6}{5}$$ Add the numbers in the numerator. $$\text{mean}=\frac{50}{5}$$ Then divide. $$\text{mean}=10$$ Check to see that the mean is ‘typical’: 10 is neither less than 6 nor greater than 15. The mean is 10.

## Example

The ages of the members of a family who got together for a birthday celebration were $$16,26,53,56,65,70,93,\text{and}\phantom{\rule{0.2em}{0ex}}97$$ years. Find the mean age.

### Solution

 Write the formula for the mean: $$\text{mean}=\frac{\text{sum of all the numbers}}{n}$$ Write the sum of the numbers in the numerator. $$\text{mean}=\frac{16+26+53+56+65+70+93+97}{n}$$ Count how many numbers are in the set. Call this $$n$$ and write it in the denominator. $$\text{mean}=\frac{16+26+53+56+65+70+93+97}{8}$$ Simplify the fraction. $$\text{mean}=\frac{476}{8}$$ $$\text{mean}=59.5$$

Is $$59.5$$ ‘typical’? Yes, it is neither less than $$16$$ nor greater than $$97.$$ The mean age is $$59.5$$ years.

Did you notice that in the last example, while all the numbers were whole numbers, the mean was $$59.5,$$ a number with one decimal place? It is customary to report the mean to one more decimal place than the original numbers. In the next example, all the numbers represent money, and it will make sense to report the mean in dollars and cents.

## Example

For the past four months, Daisy’s cell phone bills were $$\text{\42.75},\text{\50.12},\text{\41.54},\text{\48.15}.$$ Find the mean cost of Daisy’s cell phone bills.

### Solution

 Write the formula for the mean. $$\text{mean}=\frac{\text{sum of all the numbers}}{n}$$ Count how many numbers are in the set. Call this $$n$$ and write it in the denominator. $$\text{mean}=\frac{\text{sum of all the numbers}}{4}$$ Write the sum of all the numbers in the numerator. $$\text{mean}=\frac{42.75+50.12+41.54+48.15}{4}$$ Simplify the fraction. $$\text{mean}=\frac{182.56}{4}$$ $$\text{mean}=45.64$$

Does $$\text{\45.64}$$ seem ‘typical’ of this set of numbers? Yes, it is neither less than $$\text{\41.54}$$ nor greater than $$\text{\50.12}.$$

The mean cost of her cell phone bill was $$\text{\45.64}$$