Library / Statistics / STA-01
Statistics
Sample mean
The average: total of the readings divided by how many there are.
- Ref
- STA-01
- Rev
- v1
- Field
- Statistics
- Result
- x̄
Σx is the total of all the readings and n the number of readings. The mean is the balance point of the data. It is a sample estimate of the true average, with uncertainty given by the standard error.
The average: total of the readings divided by how many there are.
Calculator
- New mean if one extra reading of 2 × the mean is added27.27
- Share of the mean carried by each reading10 %
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.
- Each of the ten default readings carries 10 % of the mean, so an outlier pulls the mean far more in a small sample than in a large one.
- The mean says nothing about spread; two data sets can share a mean and differ widely in scatter.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Σx Sum of readings | 1 to 10,000 | 0.1 | 1,000 | 999.9 | +10% | +10 % |
| n Number of readings | 1 to 1,000 | 0.25 | 250 | 249.8 | +1 | −9.09 % |
Sweep of Number of readings
| Number of readings | Mean |
|---|---|
| 1 | 250 |
| 2.371 | 105.4 |
| 5.623 | 44.46 |
| 13.34 | 18.75 |
| 31.62 | 7.906 |
| 74.99 | 3.334 |
| 177.8 | 1.406 |
| 421.7 | 0.5928 |
| 1,000 | 0.25 |
Sweep one variable
Variables and how each one contributes
Mean
What the equation gives you. Everything else feeds it.
Sum of readings
The sum of all the readings. Directly proportional: one large reading pulls the mean up by its share of the extra.
Number of readings
How many readings were taken. Inversely proportional for a fixed total; more readings of the same total spread the value more thinly.
Worked example with the values above
- Add up the readings: Σx = 250.
- Count them: n = 10.
- x̄ = 250 / 10 = 25.
Assumptions and limits
- The readings come from one process and are equally trustworthy.
- No reading has been left out or recorded twice.
Common mistakes
- Dividing by n − 1; that is for the sample variance, not for the mean.
- Averaging averages from groups of different size without weighting.
Statistics and uncertainty
The sample mean is itself an estimate. Its uncertainty is the standard error s/√n, so the mean of a small sample can sit well away from the true value.
Where it is used
- Typical value of repeated range-test readings.
- Average of repeated measurements to reduce random error.
Maths you need
- Sums: Σ notation
- Division: total divided by count