Library / Statistics / STA-03
Statistics
Confidence interval half-width
The mean is quoted as the sample value ± E.
- Ref
- STA-03
- Rev
- v1
- Field
- Statistics
- Result
- E
E is the half-width: the mean is quoted as x̄ ± E. z is the critical value for the confidence level (1.645 for 90 %, 1.96 for 95 %, 2.576 for 99 %), σ the population standard deviation (or s for a large sample) and n the sample size.
The mean is quoted as the sample value ± E. A larger z (more confidence) widens it; a larger sample narrows it.
Calculator
- Total width of the interval, 2E5.88
- Standard error σ/√n1.5
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 95 % interval means that if the sampling were repeated many times, about 95 % of such intervals would contain the true mean; it is not a 95 % chance for one fixed interval.
- At the defaults a mean of 100 is quoted as 100 ± 2.94, an interval about 5.9 wide.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Sample size | 2 to 10,000 | 0.294 | 20.79 | 20.49 | +1 | −0.496 % |
| σ Standard deviation | 0.1 to 100 | 0.0196 | 19.6 | 19.58 | +10% | +10 % |
| z Critical value (1.96 for 95 %) | 1 to 3.5 | 1.5 | 5.25 | 3.75 | +10% | +10 % |
Sweep of Critical value (1.96 for 95 %)
| Critical value (1.96 for 95 %) | Half-width of interval |
|---|---|
| 1 | 1.5 |
| 1.313 | 1.969 |
| 1.625 | 2.438 |
| 1.938 | 2.906 |
| 2.25 | 3.375 |
| 2.563 | 3.844 |
| 2.875 | 4.313 |
| 3.188 | 4.781 |
| 3.5 | 5.25 |
Sweep one variable
Variables and how each one contributes
Half-width of interval
What the equation gives you. Everything else feeds it.
Critical value (1.96 for 95 %)
The critical value for the confidence level wanted. Directly proportional: moving from 95 % (1.96) to 99 % (2.576) widens the interval by about 31 %.
Standard deviation
The standard deviation of the individual readings. Directly proportional: noisier data give a wider interval.
Sample size
The number of readings in the sample. Inverse square root: four times as many readings halve the interval.
Worked example with the values above
- Standard error: σ/√n = 15 / 10 = 1.5.
- Multiply by z = 1.96.
- E = 2.94, so the mean is quoted as x̄ ± 2.94.
Assumptions and limits
- Independent readings with approximately normal scatter, or a large n.
- σ known, or well estimated by s from a large sample.
Common mistakes
- Using z with a small sample and an estimated s; use the t value instead.
- Reading the interval as the range containing most of the individual readings; it covers the mean only.
Statistics and uncertainty
z = 1.96 gives a 95 % interval and 2.576 gives 99 %. The formula assumes σ is known or n is large; with a small sample and an estimated s use the t distribution instead.
Where it is used
- Stating how well a measured performance figure is known.
- Checking whether a result is consistent with a specification.
Maths you need
- Standard error: s/√n
- Normal distribution: z values for confidence levels