1. Meaning of Mean
The mean is the value obtained by sharing the total of all observations equally. It is often called the 'average' because it represents a central value of the data.
If several numbers are added together and the total is divided equally among all values, the amount received by each is the mean.
For example, if the numbers are 4, 6, and 10, their total is 20. Sharing 20 equally among 3 values gives \( \dfrac{20}{3} \), which is the mean.
2. Formula for Mean
The basic formula for mean when data values are known individually is:
\( \text{Mean} = \dfrac{\sum x}{n} \)
Here:
- \( x \) = each data value
- \( \sum x \) = sum of all data values
- \( n \) = number of values
2.1. Example
Find the mean of 3, 7, 5, and 5.
Total = \( 3 + 7 + 5 + 5 = 20 \)
Number of values = \( 4 \)
\( \text{Mean} = \dfrac{20}{4} = 5 \)
3. Mean for Frequency Data
When values repeat many times, it is better to use a frequency table. Instead of writing each value again, we multiply each value by its frequency.
The formula becomes:
\( \text{Mean} = \dfrac{\sum (f \cdot x)}{\sum f} \)
Here:
- \( x \) = data value
- \( f \) = frequency of that value
- \( \sum (f \cdot x) \) = sum of 'value × frequency'
- \( \sum f \) = total frequency
3.1. Example
Consider the frequency table:
| Value (x) | Frequency (f) |
|---|---|
| 2 | 3 |
| 4 | 2 |
| 6 | 1 |
Now calculate:
- \( 2 \times 3 = 6 \)
- \( 4 \times 2 = 8 \)
- \( 6 \times 1 = 6 \)
So,
\( \sum (f \cdot x) = 6 + 8 + 6 = 20 \)
\( \sum f = 3 + 2 + 1 = 6 \)
\( \text{Mean} = \dfrac{20}{6} \approx 3.33 \)