Library / Statistics / STA-02
Statistics
Standard error of the mean
How much a sample mean is expected to wobble from sample to sample.
- Ref
- STA-02
- Rev
- v1
- Field
- Statistics
- Result
- SE
s is the sample standard deviation of the readings and n the sample size. The standard error measures how far a sample mean is likely to be from the true mean; it is not the same as s, which measures the scatter of individual readings.
How much a sample mean is expected to wobble from sample to sample. It shrinks with the square root of the sample size, so four times the data halves it.
Calculator
- Approximate 95 % half-width, 1.96 × SE3.92
- Standard error with four times as many readings1
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.
- With s = 12 and 36 readings the mean is known to about ±2, six times better than a single reading.
- Precision costs a lot: halving the standard error needs four times the work.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Sample standard deviation | 0.1 to 100 | 0.01667 | 16.67 | 16.65 | +10% | +10 % |
| n Sample size | 2 to 10,000 | 0.12 | 8.485 | 8.365 | +1 | −1.36 % |
Sweep of Sample standard deviation
| Sample standard deviation | Standard error |
|---|---|
| 0.1 | 0.01667 |
| 0.2371 | 0.03952 |
| 0.5623 | 0.09372 |
| 1.334 | 0.2223 |
| 3.162 | 0.527 |
| 7.499 | 1.25 |
| 17.78 | 2.964 |
| 42.17 | 7.028 |
| 100 | 16.67 |
Sweep one variable
Variables and how each one contributes
Standard error
What the equation gives you. Everything else feeds it.
Sample standard deviation
The scatter of the individual readings. Directly proportional: noisier readings give a less certain mean.
Sample size
The number of readings in the sample. Inverse square root: four times as many readings halve the standard error, nine times as many cut it to a third.
Worked example with the values above
- Square root of the sample size: √36 = 6.
- Divide the standard deviation: 12 / 6.
- SE = 2.
Assumptions and limits
- The readings are independent and come from one population.
- The sample is large enough, or the data roughly normal, for s to be a good estimate.
Common mistakes
- Quoting s instead of s/√n as the uncertainty of the mean.
- Using the formula for readings that are correlated, such as repeated looks at the same sample.
Statistics and uncertainty
This is the core statistical result for averages: the standard error falls as 1/√n. Quadrupling the sample halves it, which is why precision gets expensive.
Where it is used
- Deciding how many trial repeats are needed.
- Stating the uncertainty on an average result.
Maths you need
- Standard deviation: spread of readings
- Square roots: √n