Increase n
If you quadruple n, what happens to the standard error?
Show explanation
SE=σ/√n: quadrupling n halves SE. Try n=10 and n=100 to see the direction of change.
Probability · interactive path
From population to intervals: a guided laboratory
Six connected stages: from a theoretical distribution to data, sample means, intervals and their coverage.
01 / 06
Explore a distribution
Choose a model and adjust its parameters. The curve describes the theoretical population, not a sample.
Formula:
- Population mean
- Population variance
Explain the theory
A random variable maps uncertain outcomes to numbers. Discrete variables have point probabilities; continuous variables have densities, with probabilities given by areas. Mean and variance belong to the model.
02 / 06
Generate a sample
Draw fresh data from the same model: parameters remain fixed while sample mean and variance change.
- n
- x̄
- s²
- s
Show sample values
Explain the theory
Sample mean x̄ and variance s²=Σ(xᵢ−x̄)²/(n−1) are statistics: they depend on the sampled data. A new sample changes statistics, not population parameters.
03 / 06
Repeat sampling
Draw many samples of the same size and keep one mean from each. The chart now shows means, not individual observations.
- Samples
- Mean of means
- Theoretical standard error
- Spread of means
Explain the theory
For independent observations with mean μ and finite variance σ², E(x̄)=μ and SE(x̄)=σ/√n. Under suitable conditions, the central limit theorem describes how the distribution of means approaches a Normal as n grows.
04 / 06
Build an interval
Calculate an interval for the current sample mean. Use z only when the population σ is known in the model; otherwise use t.
Explain the theory
Known σ: x̄ ± z·σ/√n. Unknown σ: x̄ ± t·s/√n with n−1 degrees of freedom. For small samples the t interval is exact only with a Normal population. The interval concerns μ, not individual data.
05 / 06
Check coverage
Repeat the experiment: how many intervals contain the true population mean? The chart displays at most the first 80 intervals.
- Contain μ
- Observed coverage
- Nominal level
Green: contains μ; orange: misses μ. The vertical line marks μ. At most 80 intervals are drawn.
Explain the theory
A 95% level describes the procedure: across many repetitions, about 95 intervals out of 100 contain μ if assumptions hold. It does not assign a 95% probability to μ after a particular interval has been calculated.
06 / 06
Guided experiments
First make a prediction; then press “Try” and compare it with the explanation.
90%, 95%, 99%
With the same data, which interval will be widest?
Show explanation
99% needs a larger critical value than 95% or 90%: nominal coverage increases, but so does interval width.
Change σ
Does a more dispersed population produce wider intervals?
Show explanation
Yes: for a z interval at fixed n, the margin is proportional to σ. For a t interval, sample deviation s varies from sample to sample.
z versus t
Why does t generally give a wider interval at small n?
Show explanation
Unknown σ is estimated by s: t has heavier tails and a larger critical value. The exact comparison assumes a Normal population.
t tails
What happens to t as ν increases?
Show explanation
Its tails approach those of the standard Normal. Its mean exists only for ν>1 and its variance only for ν>2.
Central limit theorem
Do the means of Exponential samples become more symmetric as n grows?
Show explanation
Yes, under the theorem’s conditions. The population remains skewed; it is the distribution of means that changes.
Simulations run in your browser. Samples are not sent to the server.