Unknown population standard deviation
If we know the value of the population standard deviation, σ, an interval estimate of the form
has a confidence level of 0.95 — i.e. it is a 95% confidence interval.
In practice however, the value of σ is rarely known.
It is tempting to simply replace σ in the formula by its sample equivalent, s.
However the confidence level for an interval of this form is lower than 95% — the extra variability caused by estimating σ means that it is less likely that the interval will include µ. When the sample size is large, the confidence level is close to 95% but the confidence level can be much lower if the sample size is small.
A confidence interval of this form would therefore give a misleading impression of accuracy — we would be claiming to be 95% confident that it included µ, but the true confidence level would be lower.
A different formula should be used if σ is unknown.
Simulation
The diagram below initially shows a random sample of n = 30 values from a normal population with µ = 12 and σ = 1. It can show interval estimates calculated using σ = 1 or using the sample standard deviation, s, depending on the setting of the checkbox Use sample sd.
Using known σ = 1 | Using sample sd, s |
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Click Accumulate and take about 100 samples. Observe that...
Reduce the sample size to 5 and take another 100 samples. Observe now that...
Repeat with a sample size of 3. When using the sample standard deviation, the confidence level is even lower — 81% instead of 95%.
The interval estimate clearly needs to be modified in order to have a confidence level of 95%, especially for small samples.