Mean, median, mode and range
In a nutshell
Mean, median and mode are all calculations to determine the average of a set of data. The range is a calculation to determine the spread of a set of data. The "best" estimate for the average of a data set depends on the data provided and so there are situations where mean, median or mode are preferred over any of the other forms.
Average
Definition
The average of a set of data is the measure of the "centre" of the set. It is the value such that other data is oriented best around it, and is estimated with the mean, median and mode.
Average estimate | Description
| Calculation
|
| Mean
| This is the arithmetic average of all the data | Mean=sum of data valuesnumber of data values\text{Mean}=\dfrac{\text{sum \ of \ data \ values}}{\text{number \ of \ data \ values}}Mean=number of data valuessum of data values
|
| Median | This is the value such that half of the data lies below, and half of the data lies above it | Median=n+12value\text{Median} = \dfrac{n+1}{2} ^{\text{}} \text{value}Median=2n+1value, where nnn = number of data values
|
| Mode | This is the most frequent value in the data | Most common value |
Example 1
Calculate the mean, median and mode of the following set of data:
| 151515
| 121212
| 191919
| 202020
| 181818
| 161616
| 202020 | 171717
| 202020
| 151515 |
To find the mean use the formula.
Mean=sum of data valuesnumber of data values=15+12+19+20+18+16+2+17+20+1510=17.2‾\text{Mean}=\dfrac{\text{sum \ of \ data \ values}}{\text{number \ of \ data \ values}} = \dfrac{15+12+19+20+18+16+2+17+20+15}{10}=\underline{17.2}Mean=number of data valuessum of data values=1015+12+19+20+18+16+2+17+20+15=17.2
To find the median first sort the data as follows:
| 121212
| 151515
| 151515
| 161616
| 171717
| 181818
| 191919
| 202020
| 202020
| 202020
|
As there are 101010 values find the 10+12=5.5th\dfrac{10+1}{2}=5.5^{th}210+1=5.5th value. This occurs between 171717 and 181818 and so the median is 17+182=17.5‾\dfrac{17+18}{2}= \underline{17.5}217+18=17.5
To find the mode find the most frequent value, which is 20‾\underline{20}20.
Spread
Definition
The spread of a set of data is a measure of how close the data is to each other. It allows you to describe how consistent a set of data is, and is estimated with the range.
| Range
| This is the difference between the greatest and smallest values in the data set | Range= greatest value - smallest value\text{Range= greatest \ value - smallest \ value}Range = highest value − lowest value |
Example 2
Calculate the range of the following set of data:
To find the range subtract the lowest value from the greatest value as follows: 20−12=8‾20 - 12 = \underline{8}20−12=8.
Outliers
Definition
An outlier is an extreme value of data that doesn't match the rest of the data in the set. They may occur during experiments if a piece of equipment is faulty, or a result is recorded wrong.
Impact on averages
Outliers are values in the data set and so will affect the mean and the median. The median remains relatively unaffected by outliers as they will not change value greatly, whereas the mean will be heavily affected by outliers and so is less reliable in a set containing outliers.
