Mean

After collecting, organizing, and illustrating data using diagrams, we need to calculate certain statistical measures to describe the data more clearly. These are called measures of central tendency, and the main ones are:

• The mean (arithmetic average)
• The median
• The mode

These measures help us to summarize and compare different sets of data.

The arithmetic mean is what most people think of as the "average." It is calculated by adding all the values and dividing the total by the number of values.

If we have n values: $x_1, x_2, x_3, ..., x_n$

Then the mean is:

$\text{mean} = \frac{x_1 + x_2 + x_3 + ... + x_n}{n}$

Example 1

The masses of some parcels are: 5 kg, 8 kg, 20 kg, and 15 kg.

Find the mean mass.

Solution:

Total mass = 5 + 8 + 20 + 15 = 48 kg

Number of parcels = 4

Mean mass = 48 ÷ 4 = 12 kg

Note:

The mean is useful, but sometimes it can be misleading, especially when some values are much higher or lower than the others.

Example 2

John and Mussa played for the local cricket team. In their last six batting innings, they scored:

John: 64, 0, 1, 2, 4, 1

Mussa: 15, 20, 13, 11, 10, 3

Find the mean score of each player. Which player would you rather have in your team?

Solution:

John's mean = (64 + 0 + 1 + 2 + 4 + 1) ÷ 6 = 72 ÷ 6 = 12

Mussa's mean = (15 + 20 + 13 + 11 + 10 + 3) ÷ 6 = 72 ÷ 6 = 12

Even though both have the same mean score (12), John's scores are very inconsistent, while Mussa's are more steady.

If you want a reliable player, you would likely choose Mussa.

• If you add or subtract a number (say $a$) to all the values, the mean will also increase or decrease by the same number.
• If you multiply all values by a number $a$, the mean will also be multiplied by $a$.

When data is presented in a frequency table, we can still find the mean using a slightly different method.

Steps:

1. Multiply each value (x) by its frequency (f) to get fx.
2. Add all the fx values to get the total.
3. Add all the frequencies to get the total number of items.
4. Divide the total fx by the total frequency to get the mean.

Example:

Mean = Total fx ÷ Total f = 38 ÷ 10 = 3.8