Distinguish the data from the statistic

Try

Draw two independent values from a population that is0 or2 with equal probability. List ordered samples: (0,0), (0,2), (2,0), (2,2). Each is equally likely. Compute each sample mean before looking at a formula. The sample data vary, and so does the summary calculated from them.

Predict

Predict whether the mean of all possible sample means matches the population mean. Also predict whether the spread of sample means is smaller than the spread of individual observations. These are different distributions of different random quantities, even though their measurement units match.

Experiment

The four means are0,1,1,2. Their distribution has probabilities1/4,1/2,1/4, giving expectation1 and variance1/2. The population itself has mean1 and variance1. Averaging two independent observations halves the variance.

Use the widget below for a different exact model: independent binary draws with replacement. Start with the representative population model and increase n. Inspect the probability mass on sample proportions and the standard error. Then select the biased subgroup model. Increasing n now concentrates estimates around the subgroup proportion, not around the target population proportion.

Live experiment: precision is not the same as lack of bias

In this synthetic population, 40% have a specified characteristic. Half belong to a subgroup with 70%, and the other half to a subgroup with 10%. Predict what happens if every sample is drawn only from the first subgroup, then increase the sample size.

Sampling method Uniform draws from the entire populationDraw only from the 70% subgroup Independent draws per sample 125102040 Reset experiment

Every model draw is with replacement, independently and uniformly within the chosen sampling group. Filled bars give exact-model probabilities. You can separately generate seeded trials below and compare their empirical frequencies with these probabilities.

Expected sample proportion: 0.4. Bias relative to the population: 0.0000.

Standard deviation of the sample proportion: √(0.4 × (1 − 0.4) / 5) ≈ 0.2191.

Probability of a result from 0.3 through 0.5, inclusive: ≈ 34.56%.

The expected sample proportion is0.4 for every sample size. Individual samples still vary. The probability inside a fixed interval need not increase at every small change of n because only the fractions k/n are possible.

Inspect exact-model probabilities

Successes k	Proportion k/n	Probability
0	0.0000	7.78%
1	0.2000	25.92%
2	0.4000	34.56%
3	0.6000	23.04%
4	0.8000	7.68%
5	1.0000	1.02%

Run a reproducible sampling investigation

Define a success as drawing an item with the characteristic. Each trial draws n items and records the proportion. Predict how often that proportion will fall from 0.3 through 0.5, then run 100 trials. Compare the simulated fraction with the exact probability above.

Simulation seed Run 100 simulated samples

The simulation uses a deterministic pseudorandom generator to approximate independent model draws. It is not a new real-world sample and cannot repair an unrealistic sampling model. Changing settings hides mismatched simulated results. Reset or navigation clears the temporary run.

Probabilities are calculated from the binomial model and displayed with rounding; a displayed0.00% may be small but nonzero. Sampling without replacement needs a different model when the sampling fraction is material. Settings are temporary and do not count as assessment evidence.

The population success proportion is0.4. The selected subgroup, half of the population, has proportion0.7; the remaining half has0.1, so their equally weighted average is0.4. The widget first enumerates exact binomial probabilities. Predict the fraction of sample proportions between 0.3 and 0.5, then use a chosen seed to generate 100 simulated samples. Inspect the trace and compare observed frequencies with the exact model. Repeating a seed repeats the same trials; change it to examine another reproducible run. These are synthetic pseudorandom trials, not newly collected real data or saved assessment evidence.

Understand

The population distribution describes individual values in the target population. A sample distribution describes the values in one collected sample. A sampling distribution describes a statistic across hypothetical repetitions of the sampling procedure with the same sample size and design. It is not a histogram of all individual observations pooled together.

For independent identically distributed observations with mean μ and finite variance σ², sample mean x̄ has expectation μ and variance σ²/n. Its standard deviation is σ/√n, commonly called the standard error of the mean when discussing the sampling distribution. When σ is unknown, estimating it requires further work; a computed estimate is not the same as a known population quantity.

The expectation equality makes x̄ unbiased under this model. Unbiasedness means its repeated-sample average equals the target, not that each realized estimate is correct. Variability describes how estimates differ across samples. An estimator that always adds5 to x̄ has bias5 even as larger n reduces its variability. Precision around the wrong target is not accuracy.

Sampling without replacement creates dependence. For a simple random sample from a finite population, a finite-population correction reduces the variance relative to independent sampling. The common10% sampling-fraction check supports an approximate independence calculation; it is not a theorem making the draws literally independent.

To halve σ/√n under the same model, multiply n by four. More data can be costly, and this rule addresses sampling variation only. Measurement error, nonresponse and selection bias can dominate even when a model-based standard error is small.

Apply

Record the widget’s target, selected-distribution mean and standard error for two sample sizes in each mode. Explain why a narrowing biased distribution is a warning rather than success. Then describe the three distributions in the height-study practice using the phrase “one observation in this distribution represents…” for each.

Continue

Next distinguish the law of large numbers from the central limit theorem. One concerns concentration around a mean; the other concerns the shape of a standardized sampling distribution.