# Mean (Average)

Mean is the most commonly used measure of central tendency. It's often called an average (and sometimes called an arithmetic mean).

**To find the mean of a dataset, you add up all of the data points and divide by the number of datapoints.**

When dealing with a mean in a word problem, it's often helpful to think of this equation:

$\dfrac{\text{Sum of numbers}}{\text{Number of numbers}}=\text{Mean or Average}$

**Let's look at the process of finding a mean in detail:**

Look at your dataset. Presume you have earned the following grades on your last 5 spelling tests.

$9$, $8$, $3$, $6$, $9$

Now, you will add all of the numbers up and divide by the number of numbers:

$\dfrac{\text{Sum of numbers}}{\text{Number of numbers}}=\dfrac{9+8+3+6+9}{5}=\dfrac{35}{5}=7$

The mean of your spelling test scores is $7$.

Sometimes means take a while to calculate (there's a lot of math, and when there are a lot of numbers, be careful of careless mistakes!). The only time means get tricky is when you have zeros in your dataset. **Remember: zero counts as a number!** So, even though, when you add in a zero, it does not change the sum, it does change the "number of numbers" that you are dividing by.

Let's say you were absent for the sixth spelling test. Here are your scores:

$9$, $8$, $3$, $6$, $9$, $0$

Now, you will add all of the numbers up and divide by the number of numbers:

$\dfrac{\text{Sum of numbers}}{\text{Number of numbers}}=\dfrac{9+8+3+6+9+0}{6}=\dfrac{35}{6}=5.833$

The mean of your spelling test scores is now $5.833$. So, even though you got a zero, and your sum did not change, your average went down because the number of numbers changed (and this makes sense, if you get a 0 on a test, your average will go down!).

Sometimes, you will be asked to find a datapoint (or the sum of datapoints) based on an average. This can feel complicated, but if you use the formula above, it's just a matter of plugging in numbers and doing algebra.

*Example:*

*Your teacher tells you that your average test score for the semester is 82. You know that you took 4 tests and that you got scores of 67, 99, and 87, but you can't find the last test. What score did you get on that test?*

Just as before, use the formula, plug in what you know, use variables for numbers you don't know, then use algebra to solve:

$$\eqalign{\dfrac{\text{Sum of scores}}{\text{Number of scores}}&=\text{Average of scores}\qquad&&\text{Start with the basic Mean equation}\\\dfrac{67+99+87+x}{4}&=82\qquad&&\text{Plug in what you know; use a variable for the missing test}\\\dfrac{253+x}{4}&=82\qquad&&\text{Combine like terms}\\\times 4 &=\times4 \qquad&&\text{Multiply each side by 4 to get rid of the 4 in the denominator}\\253+x&=328\\-253&=-253\qquad&&\text{Subtract 253 from each side}\\x&=75\qquad&&\text{Your missing score was 75}}$$

Problems that give you the average and ask you to work backwards, seem harder, but use the formula and they become simple algebra problems.

Remember: to find an average or mean, you just need the SUM of datapoints. You don't actually need to know each data point. Average and mean problems often try to trick students into thinking that they need to know each datapoint (there's often no way to do that, so that the students give up). You just need the sum, and you can find that by adding up the datapoints OR by multiplying the mean by the number of datapoints.

#### Practice Problems:

## Mean (Average)

Find the mean of the data.

1. $-4,10,-3,7,-5$

2. $-28,6,-3,8,-11,4$

3. $-7,6,2,-28,12,-9$

4. $21,-7,-3,13$

5. $2,-1,23,-11,7$

6. $-16,4,9,-5$

7. $8,3,-12,-1,7$

8. $18,5,-6,-3,11$

9. $-3,9,-20,-7,2,-5$

10. $14,-1,-3,-9,19,7,-6$

#### Answer Key: