Using Averages (Mean, Median, Mode)

Types of Averages

Mean

The mean is the arithmetic average of a set of numbers. It is calculated by adding all the values together and then dividing by the number of values.

Mean formula:

$$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$$

For example, if rainfall (in $\mathrm{mm}$) over 5 days is 10, 12, 8, 15, and 5, the mean rainfall is:

$$\frac{10 + 12 + 8 + 15 + 5}{5} = \frac{50}{5} = 10\,\mathrm{mm}$$

Median

The median is the middle value when the data are arranged in order from smallest to largest. If there is an even number of values, the median is the average of the two middle numbers.

For example, for the set of temperatures (in $^\circ\mathrm{C}$) 6, 8, 12, 15, 20, the median is 12 because it is the middle value.

If the data are 6, 8, 12, 15, 20, 25, the median is the average of 12 and 15:

$$\frac{12 + 15}{2} = 13.5^\circ\mathrm{C}$$

Mode

The mode is the value that appears most frequently in a data set. There can be more than one mode or none if all values are unique.

For example, in the data set 3, 5, 7, 5, 8, 5, 10, the mode is 5 because it appears three times, more than any other number.

Mode is useful for categorical data or to find the most common value. It can also be used with numerical data to identify the most frequent value.

Using Averages in Geography

Averages help summarise large sets of geographical data so we can understand general trends and patterns without looking at every individual value.

Summarising data sets: Averages provide a single value that represents the whole data set, making it easier to communicate information like average rainfall, temperature, or population density.

Comparing geographical data: Averages allow comparison between different places or times. For example, comparing the mean annual rainfall of two regions helps identify which is wetter.

Identifying trends and patterns: By calculating averages over time, such as mean temperature each year, we can spot climate trends like warming or cooling.

For instance, if the average temperature in London over five years is calculated, it helps show if the city is getting warmer or cooler overall, despite yearly fluctuations.

Advantages and Limitations

When to Use Each Average

• Mean: Best for data without extreme values. It uses all data points, so it reflects the overall dataset well.
• Median: Useful when data have outliers or are skewed, as it is not affected by very high or low values.
• Mode: Ideal for categorical data or to find the most common value, such as the most common land use type in an area.

Impact of Outliers

Outliers are values much higher or lower than the rest of the data. They can distort the mean but have little effect on the median.

For example, if average incomes in a town are mostly around \(\text{\£}20,000\) but one person earns \(\text{\£}1,000,000\), the mean income will be much higher than most people’s actual income, but the median will better represent the typical income.

Representativeness of Data

No average perfectly represents all data points. The mean can be misleading if data are skewed, the median ignores some data values, and the mode might not exist or be unique.

Always consider the data distribution and whether outliers or multiple modes exist before choosing which average to use.