Mean - Calculating the Average
The mean is what most people mean when they say "average". It is calculated by adding up all the values in a dataset and dividing by how many values there are. The mean takes every value into account, which makes it very useful but also sensitive to extremely large or small values.
The Formula
Mean = Sum of all values ÷ Number of values
In symbols: Mean = Σx / n (where Σx means "the sum of all x values" and n is the count).
Worked Examples
Sum = 4 + 7 + 9 + 3 + 12 = 35. Count = 5. Mean = 35 ÷ 5 = 7.
Sum = 72 + 85 + 60 + 91 + 77 = 385. Mean = 385 ÷ 5 = 77.
Total = 9 × 4 = 36. Known sum = 6 + 11 + 8 = 25. Fourth = 36 − 25 = 11.
First: (3+5+4+6+4)/5 = 22/5 = 4.4. Second: (3+5+4+6+100)/5 = 118/5 = 23.6. An outlier (100) pulls the mean up dramatically.
Mean from a Frequency Table
Multiply each value by its frequency, sum those products, then divide by the total frequency.
| Goals (x) | Frequency (f) | f × x |
|---|---|---|
| 0 | 3 | 0 |
| 1 | 5 | 5 |
| 2 | 4 | 8 |
| 3 | 2 | 6 |
| Total | 14 | 19 |
Mean = 19 ÷ 14 ≈ 1.36 goals per match.
Key Takeaways
- Mean = Sum of values ÷ Number of values.
- The mean uses every value, so it is affected by outliers.
- If the mean is known, the total sum = mean × count (useful for finding a missing value).
- For a frequency table: mean = Σ(f × x) ÷ Σf.
Practice Questions
- Find the mean of: 15, 22, 8, 30, 10.
- The mean of five numbers is 14. Four of them are 10, 16, 12, 18. Find the fifth.
- A shop sells 30, 45, 28, 52, 35, and 40 items on six days. Find the mean daily sales.
- Shoe sizes in a sample: size 6 (4 people), size 7 (9 people), size 8 (7 people), size 9 (4 people), size 10 (1 person). Find the mean shoe size.
- A class mean is 65. After one more test (score 85) is added, there are now 21 results. Find the new mean.