Library / Foundations: Maths / FMA-05
Foundations: Maths
Mean from a frequency table
Average of grouped or tallied data.
- Ref
- FMA-05
- Rev
- v1
- Field
- Foundations: Maths
- Result
- x̄
Multiply each value x by its frequency f, add the products (Σfx) and divide by the total frequency (Σf). For grouped data use class midpoints for an estimate.
Add up all the values, counting each as often as it occurs, then divide by how many values there are.
Calculator
- Sum of values372
- Number of values30
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The mean is the balance point of the data; one extreme value can pull it far.
- With grouped data the mean is only an estimate because the exact values inside each class are unknown.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Σfx Sum of frequency × value | 50 to 5,000 | 1.667 | 166.7 | 165 | +10% | +10 % |
| Σf Total frequency | 5 to 500 | 0.744 | 74.4 | 73.66 | +10% | −9.09 % |
Sweep of Sum of frequency × value
| Sum of frequency × value | Mean |
|---|---|
| 50 | 1.667 |
| 88.91 | 2.964 |
| 158.1 | 5.27 |
| 281.2 | 9.372 |
| 500 | 16.67 |
| 889.1 | 29.64 |
| 1,581 | 52.7 |
| 2,812 | 93.72 |
| 5,000 | 166.7 |
Sweep one variable
Variables and how each one contributes
Mean
What the equation gives you. Everything else feeds it.
Sum of frequency × value
The total of every value counted by its frequency. Directly proportional.
Total frequency
The number of data values. Inversely proportional.
Worked example with the values above
- Total of all values: Σfx = 372.
- Number of values: Σf = 30.
- Mean = 372 / 30 = 12.4.
Assumptions and limits
- Frequencies are counts of equally weighted data.
- For grouped data, values are spread evenly around the midpoint.
Common mistakes
- Dividing Σfx by the number of classes instead of Σf.
- Using class boundaries instead of midpoints.
Statistics and uncertainty
The mean of n measurements has a standard error s/√n, so quadrupling the number of readings halves the uncertainty in the mean.
Where it is used
- Average measurement from repeated trials.
- Summarising tallied test results.
Maths you need
- Tables: read frequencies
- Division: total divided by count