Math Core

When someone asks "What's a typical value?", you need one number that sums up a whole data set. That number is called a measure of center. The most common one is the mean, which is what most people mean when they say "average."

The mean as a fair share

Four friends go fishing. They catch 3, 5, 2 and 6 fish. If they put all the fish in one bucket and share them equally, how many does each friend get?

Total fish: $3 + 5 + 2 + 6 = 16$ .
Share equally among 4 friends: $16 \div 4 = 4$ .

Each friend gets 4 fish. That fair share is the mean. It's the value every data point would have if the total were spread out evenly.

Definition

Mean

The mean of a data set is the sum of the values divided by the number of values:

\text{mean} = \frac{\text{sum of the values}}{\text{number of values}}

The mean as a balance point

You can also picture the mean on a dot plot. Here are the fish again:

Measure how far each value is from the mean, 4:

Below 4: the 2 is 2 away and the 3 is 1 away. Total: $2 + 1 = 3$ .
Above 4: the 5 is 1 away and the 6 is 2 away. Total: $1 + 2 = 3$ .

The distances on each side are equal. The mean is the point where the data would balance, like a seesaw. This is a good way to check your answer: the mean should always land somewhere in the middle of the data, never below the smallest value or above the largest.

Worked example: Test scores

Jada's scores on five quizzes were $84, \ 90, \ 76, \ 95, \ 80$ . Find her mean score.

Step 1: add. $84 + 90 + 76 + 95 + 80 = 425$ .

Step 2: divide by the number of values. There are 5 scores. $425 \div 5 = 85$ .

Jada's mean score is 85.

Check: 85 is between the lowest score, 76, and the highest, 95. ✓

The mean doesn't have to be one of the data values, and it doesn't have to be a whole number.

Worked example: Mean from a dot plot

The dot plot shows how many siblings each of 10 students has.

Number of siblings

Find the mean number of siblings.

Step 1: find the sum. Each value counts once for every mark above it.

0 \times 2 + 1 \times 4 + 2 \times 2 + 3 \times 1 + 4 \times 1 = 0 + 4 + 4 + 3 + 4 = 15

Step 2: divide. There are $2 + 4 + 2 + 1 + 1 = 10$ students. $15 \div 10 = 1.5$ .

The mean is 1.5 siblings. No student has 1.5 siblings, but that's fine. It describes the center of the group.

Common mistake

When the data is in a dot plot or a frequency table, don't just add the numbers on the number line. In the siblings plot, adding $0 + 1 + 2 + 3 + 4$ ignores that 4 students have 1 sibling. Multiply each value by how many times it appears. Also divide by the number of data values (10 students), not the number of different values (5).

Working backward from the mean

If you know the mean and the number of values, you can find the total: multiply.

\text{sum} = \text{mean} \times \text{number of values}

That lets you find a missing value.

Worked example: What score do I need?

Carlos has taken three tests, with scores of 78, 85 and 88. What score does he need on the fourth test to have a mean of 85?

Step 1: find the total he needs. A mean of 85 over 4 tests means a total of $85 \times 4 = 340$ points.

Step 2: find what he has so far. $78 + 85 + 88 = 251$ .

Step 3: subtract. $340 - 251 = 89$ .

He needs an 89 on the fourth test.

Check: $\dfrac{78 + 85 + 88 + 89}{4} = \dfrac{340}{4} = 85$ . ✓

Tip

Before you calculate, estimate. If the values are 78, 85, 88 and something, and the mean should be 85, the missing score must make up for the 78, so it has to be a bit above 85.