The Mean

The mean sometimes called the arithmetic mean, is the most common average.

The mean of a set of numbers or values is found by simply adding all the values together and then dividing by the number of values. i.e.

Mean = \( \frac{sum\:of\:the\:given\:values}{number\:of\:values} \)

To find the sum of the values we rearrange the formula i.e.

Sum of the given values = Mean \(\scriptsize \times \) number of values.

Example 1:

Find the mean of the numbers 3, 6, 5, 6, 4, 9. (Round your answer to 1 decimal place). 

Solution: 

Sum of all numbers = 3 + 6 + 5 + 6 + 4 + 9 = 33

There are 6 numbers, so divide by 6

Mean = \( \frac{sum\:of\:the\:given\:values}{number\:of\:values} \\ = \frac{3 \:+ \:6 \:+ \:5\: +\: 6 \:+\: 4 \:+ 9}{6} \\ = \frac{33}{6} \\ = \scriptsize 5.5 \)

Example 2:

The mean of 5 numbers is 6 if four of the numbers are 5, 6, 7, and 8. What is the fifth number? 

Let the fifth number be x

Mean = \( \frac{sum\:of\:the\:given\:values}{number\:of\:values} \)

From the question,

Mean = 6

Number of values = 5

Substitute the given values in the question into the formula

∴ 6 = \( \frac{5\: + \: 6 \: + \: 7 \: + \: 8 \: + \: x}{5} \)

∴ \( \frac{6}{1} = \frac{26 \: + \: x}{5} \)

To find x let us cross multiply

\( \scriptsize 6 \: \times \: 5 = 26 \: + \: x \)

Move x to the Left-hand side

\( \scriptsize 26 \: + \: x = 6 \: \times \: 5 \)

∴ \( \scriptsize 26 \: + \: x = 30 \)

Collect the like terms

\( \scriptsize x = 30 \: – \: 26\)

\( \scriptsize x = 4\)

∴ The fifth number is 4.