Mean
Mean Revision
Mean
The mean, often called the average, is a measure of central tendency that represents the typical value in a set of numbers. It is calculated by adding all the values together and dividing by the total number of values. The mean is widely used in statistics, finance, and everyday decision-making to analyse data and identify trends, though it can be affected by extreme values, making it important to consider alongside other measures like the median and mode.
Learning Objectives:
After this topic students will be able to:
- Find the mean of a set of data.
Finding the Mean
Whenever people talk about an average value, they are usually referring to the mean. To calculate the mean, we need to add up all the values and divide by the number of values:
\text{Mean} = \dfrac{\text{Total of values}}{\text{Number of values}}
Note: the mean is affected by outliers – values that lie far outside the rest of the data.
Finding the Mean
Example 1:
Ryan measures his height and the height of six of his friends. Their heights in cm are:
155, 160, 153, 173, 164, 162, 160
Find the mean of their heights.
First to find the mean, we need to add up the values and divide by the number of values, which is 7:
155+160+153+173+164+162+160=1127
\text{Mean}=\dfrac{1127}{7}=161 cm
Example 2:
A teacher records the scores of five students on a maths test: 78, 85, 92, 74, and 81.
What is the mean score of the students?
First to find the mean, we need to add up the values and divide by the number of values, which is 5:
78+85+92+74+81=410
\text{Mean}=\dfrac{410}{5}=82
Mean Example Questions
Question 1: Calculate the mean of the following numbers:
5, 8, 7, 4, 8, 3, 1, 12
To calculate the mean, add up all the numbers and divide by the number of values:
\text{mean} = \dfrac{5+8+7+4+8+3+1+12}{8}=6
Question 2: The mean length of 7 planks of wood is 1.35 m. When an extra plank of wood is added, the mean length of a plank of wood increased to 1.4 m.
What is the length of the extra plank of wood that was added?
In most questions involving the mean, we are given the total and need to work out the mean from the total. In this question, we have been given the mean, so we are going to have to calculate the total from the mean.
If the mean length of 7 planks of wood is 1.35 m, then the total length of all these planks of wood combined can be calculated as follows:
7 \times 1.35 m = 9.45 m
When the extra plank of wood is added, the mean length of a plank of wood increases to 1.4 m. This means there are now 8 planks of wood, with a combined length of:
8 \times 1.40 m = 11.2 m
Therefore, by adding this additional plank of wood, the combined length has increased from 9.45 m to 11.2 m, so the length of this extra plank of wood is therefore:
11.2 m - \, 9.45 m = 1.75 m
Question 3: In a rowing team, the weight of 8 women, in kilograms, is:
63, \, 60, \, 57, \, 66, \, 62, \, 65, \, 69, \, 58
In order to be a more competitive team, the coach has said that each team member should try to increase their overall muscle mass which will result in a 2\% gain in overall body weight.
What will be the mean weight of the team if all 8 are successful in precisely meeting this 2\% weight gain?
In this question, we do not need to work out a 2\% increase in weight for each individual team member (it would not be wrong to do so, just unnecessarily time-consuming).
The combined weight of all 8 members is:
63 + 60+57+66+62+65+69+58 = 500kg
If each team member increases their weight by 2\%, then this is the same as the team increasing their combined weight by 2\%. Therefore, if the team is successful in achieving this 2\% weight gain, then the combined weight of the team can be calculated as follows:
1.02\times500 = 510kg
Since there are 8 team members in total, then mean weight following this weight gain is:
510kg \div \, 8 =63.75kg
