Library / Foundations: Maths / FMA-33
Foundations: Maths
Standard deviation from the mean of squares
Spread of data from the mean of squares and the mean.
- Ref
- FMA-33
- Rev
- v1
- Field
- Foundations: Maths
- Result
- σ
The population variance for a frequency table. Take the square root for the standard deviation. Both x and the mean must use consistent units. The calculator takes Σfx²/Σf and the mean as inputs.
Variance is the average of the squares minus the square of the average; it measures how spread out the data are.
Calculator
- Variance σ²6
- Coefficient of variation σ/x̄20.41 %
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.
- A small spread relative to the mean signals consistent data.
- Subtracting two large, similar numbers loses accuracy; keep unrounded values.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Σfx²/Σf Mean of the squares | 20 to 500 | 2.449 | 18.87 | 16.42 | +10% | +87.1 % |
| x̄ Mean | 1 to 15 | 1.077 | 12.21 | 11.13 | +10% | — |
Sweep of Mean of the squares
| Mean of the squares | Standard deviation |
|---|---|
| 20 | — |
| 29.91 | — |
| 44.72 | — |
| 66.87 | — |
| 100 | — |
| 149.5 | 2.353 |
| 223.6 | 8.922 |
| 334.4 | 13.8 |
| 500 | 18.87 |
Sweep one variable
Variables and how each one contributes
Standard deviation
What the equation gives you. Everything else feeds it.
Mean of the squares
Average of x² over all the data. Raises the spread.
Mean
The mean of the data. Raises x̄² and so lowers the spread.
Worked example with the values above
- Square the mean: 12² = 144.
- Variance: 150 − 144 = 6.
- σ = √6 = 2.449.
Assumptions and limits
- Population variance (divide by Σf).
- Consistent units for x and the mean.
Common mistakes
- Forgetting to square the mean.
- Taking the square root before subtracting.
Statistics and uncertainty
This formula subtracts nearly equal numbers when the spread is small, so rounding errors are magnified; keep full precision until the end. For a sample of n readings use n − 1 in the denominator.
Where it is used
- Spread of repeated measurements.
- Precision of firing groups and instruments.
Maths you need
- Mean: Σfx/Σf
- Square roots: standard deviation