Averages from a list

When analysing data from a list, we often calculate the averages (mean, median, and mode) and the range to summarise the data and understand its spread. Each measure provides different insights into the data.

1. Mean:

The mean (or arithmetic average) is calculated by dividing the sum of all the numbers in the list by the total number of values. The formula is:

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

Example: For the list $[4,8,6,10]$: $$\text{Sum of values}=4+8+6+10=28$$ $$\text{Number of values}=4$$ $$\text{Mean}=\frac{28}{4}=7$$

2. Median:

The median is the middle value when the numbers are arranged in ascending order. If there is an even number of values, the median is the mean of the two middle numbers.

Example: For the list $[4,8,6,10]$:

• Arrange in order: $[4,6,8,10]$
• There are 4 values, so the median is the mean of the two middle values: $$\text{Median}=\frac{6+8}{2}=7$$

For an odd-numbered list, such as $[4,6,8]$, the median is the middle value: $\text{Median}=6$.

3. Mode:

The mode is the value (or values) that occur most frequently in the list. A list may have one mode, more than one mode, or no mode at all.

Example:

• For $[4,6,6,8,10]$, the mode is $6$ (it appears twice).
• For $[4,6,8,10]$, there is no mode (no value repeats).
• For $[4,6,6,8,8]$, the modes are $6$ and $8$ (bimodal).

4. Range:

The range measures the spread of the data and is calculated as the difference between the largest and smallest values. The formula is:

$$\text{Range}=\text{Largest value}-\text{Smallest value}$$

Example: For the list $[4,8,6,10]$: $$\text{Range}=10-4=6$$

5. Summary of Steps:

1. To find the mean, sum all values and divide by the number of values.
2. To find the median, arrange the numbers in order and find the middle value (or the mean of the two middle values).
3. To find the mode, identify the most frequently occurring value(s).
4. To find the range, subtract the smallest value from the largest value.