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Measures of central tendency – The mean

📘 Notes

This topic covers how to calculate the mean for raw data, frequency tables and grouped data, including formulas and worked examples. All math is rendered with MathJax.

1. Definition

The mean is a measure of central tendency — it gives the average value of a set of data.

For a set of data values \( x_1, x_2, x_3, \dots, x_n \), the mean is \[ \bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} \] where \(n\) is the number of data values.

Example (Raw data):
The masses (in kg) of 5 students are: 50, 55, 52, 53, 60.
\[ \bar{x} = \frac{50 + 55 + 52 + 53 + 60}{5} = \frac{270}{5} = 54 \] Mean mass = \(54\) kg.

2. Mean from a Frequency Table

If data are presented in a frequency table with values \(x\) and frequencies \(f\):

xf
13
25
32

The mean is given by: \[ \bar{x} = \frac{\sum fx}{\sum f} \]

Using the table above:
\(\sum fx = 1(3)+2(5)+3(2)=3+10+6=19\), \(\sum f=3+5+2=10\).
\(\bar{x}=\frac{19}{10}=1.9\)

3. Mean from Grouped Data

For grouped data (e.g. class intervals), the mean is estimated using midpoints.

Class IntervalMidpoint \(x\)Frequency \(f\)
0–1053
10–20154
20–30253

Use the same formula: \[ \bar{x} = \frac{\sum fx}{\sum f} \]

\(\sum fx = 5(3)+15(4)+25(3)=15+60+75=150\), \(\sum f=10\).
\(\bar{x}=\frac{150}{10}=15\).

4. Using an Assumed Mean (Coding Method)

When the numbers are large, it's often easier to use coding. Choose an assumed mean \(A\) and define \[ y = \frac{x - A}{c} \] where \(c\) is a class width or convenient scaling factor. Then \[ \bar{x} = A + c\bar{y} \] where \(\bar{y} = \frac{\sum fy}{\sum f}\).

Suppose midpoints are 10, 20, 30 with frequencies 2, 3, 5. Choose \(A = 20\), \(c = 10\). Then \(y = (x - 20)/10\):
  • x = 10 → y = -1
  • x = 20 → y = 0
  • x = 30 → y = 1
\(\sum fy = 2(-1)+3(0)+5(1)=3\), \(\sum f=10\). \(\bar{y} = \frac{3}{10} = 0.3\). \(\bar{x} = 20 + 10(0.3) = 23\).

5. Key Points

  • The mean uses all data values, making it sensitive to extreme values (outliers).
  • For grouped data, the mean is only an estimate because midpoints are used.
  • The coding method simplifies calculation without changing the result.
  • In exam answers, always show working: \(\sum fx\), \(\sum f\), and the formula used.

6. Summary Table

Data TypeFormula for Mean
Raw Data \(\displaystyle \bar{x} = \frac{\sum x}{n}\)
Frequency Table \(\displaystyle \bar{x} = \frac{\sum fx}{\sum f}\)
Grouped Data \(\displaystyle \bar{x} \approx \frac{\sum f(\text{midpoint})}{\sum f}\)
Coding Method \(\displaystyle \bar{x} = A + c\bar{y}\)