Mean, Median and Mode

Mean

The mean is the average value of a set of data. It is found by adding all the data values together and then dividing by the number of values.

The formula for the mean is:

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

For example, if five students scored the following marks in a test: 12, 15, 14, 10, and 9, the mean mark is calculated by adding these scores and dividing by 5.

Sum = $12 + 15 + 14 + 10 + 9 = 60$

Mean = $\frac{60}{5} = 12$

Effect of outliers: The mean is sensitive to outliers, which are values much higher or lower than the rest of the data. For example, if one student scored 50 instead of 9, the mean would increase significantly, even though most scores are low.

Median

The median is the middle value when the data is arranged in order from smallest to largest.

- If there is an odd number of values, the median is the middle one.

- If there is an even number of values, the median is the mean of the two middle values.

The median is less affected by outliers than the mean, making it a more robust measure of central tendency when data contains extreme values.

For example, consider the data set: 3, 7, 9, 15, 20.

The median is the middle value, which is 9.

If the data set is 3, 7, 9, 15, 20, 25 (even number of values), the median is the average of the two middle numbers (9 and 15):

$$\text{Median} = \frac{9 + 15}{2} = 12$$

Mode

The mode is the most frequently occurring value in a data set.

- There can be more than one mode if multiple values appear with the same highest frequency.

- If no value repeats, the data set is said to have no mode.

The mode is especially useful for categorical data where calculating a mean or median is not possible.

For example, in a survey of favourite colours, the mode is the colour chosen by the most people.

Example: The data set 2, 3, 3, 5, 7, 7, 7, 8 has mode 7 because it appears most often.