Construct sample means and interpret finite sampling distributions

Three different distributions

A population distribution describes individual values X. One sample is a list of n observed values. A sampling distribution describes the statistic X̄ across repeated samples selected by the same rule. Do not plot all individual observations and label the result a distribution of sample means

A sample contains 0,2,4,6. Find its mean

One mean per repetition

Repeat the complete sampling procedure R times, keeping the within-sample size n fixed. Calculate one mean from each sample, then plot those R means. A dot represents a sample mean. R controls how many dots approximate the sampling distribution; n controls how much information goes into each mean

If 30 samples each contain 8 observations, how many dots belong in the sample-mean dot plot?

Which change increases the number of dots without changing the theoretical sampling distribution of the mean?

Construct an empirical dot plot, ungraded

Use the illustrative size-two samples (0,0),(0,2),(0,4),(2,2),(2,4),(4,4). Calculate every mean and draw a labelled dot plot on a uniform numerical scale. Stack repeated means and identify what one dot means. These supplied samples are an illustrative dataset rather than a random run

Model: calculate, locate and stack

The six means are 0,1,2,2,3,4. Put one dot at 0,1,3,4 and two dots at 2. The horizontal label is sample mean, in the original measurement units. There are six dots for six samples, and the observed range is 4−0=4. Repeated means represent separate samples and must not be collapsed into one dot

How many of these six sample means equal 2?

The six sample means are 0, 1, 2, 2, 3, 4. What fraction equal 2?

The six sample means are 0, 1, 2, 2, 3, 4. Find their average

The six sample means are 0, 1, 2, 2, 3, 4. Find their observed range

Compare on a common scale

For a second illustrative collection, six size-eight sample means are 1,1.5,2,2,2.5,3. Both sets are plotted on 0..4 in the same units, with the same population-mean reference 2. These selected teaching datasets illustrate a narrower spread; they do not prove a general rule by themselves

Find the observed range of the displayed size-eight sample means

Describe centre, spread and shape honestly

For an empirical sampling distribution, report the number of samples and sample size, then describe centre and spread in the original units. A range or empirical SD describes the realised means and is not automatically the population SD. Inspect asymmetry, gaps and outliers without assuming a normal shape. A small empirical collection can look irregular, and its average need not equal μ

An exact two-draw mean distribution

Suppose each independent draw is 0 or 4 with probability 1/2 each. The ordered pairs are (0,0),(0,4),(4,0),(4,4), each with probability 1/4. Their means are 0,2,2,4, so X̄ has probabilities 1/4,1/2,1/4 at 0,2,4. This exact distribution is obtained by counting every model outcome, not estimating from a finite run

Two independent draws each give 0 or 4 with probability 1/2 each. Find Pr(X̄=2)

Centre and spread under independent sampling

For n independent, identically distributed observations with mean μ and variance σ², the displayed mean, variance and standard deviation rules apply. The SD of the sample mean is often called its standard error. These are properties of repeated samples under the model, not a claim that an individual sample mean equals μ

If the population variance is 4, find the variance of the mean of 8 independent observations

If sample size increases from 2 to 8 under the same independent model, what is the larger-sample SD divided by the smaller-sample SD?

Derive the mean and variance rules, ungraded

Use the independent-sum and scaling rules from 6B to derive E(X̄), Var(X̄) and SD(X̄). State where independence and identical distributions are used, and why one realised mean need not equal μ

Model: sum, then divide

Write the sample mean as the sum of the observations divided by the sample size. Expectation adds, and identical marginal means suffice for that step. Independence makes the variance of the sum equal the sum of the variances. Dividing by the sample size squares the scale factor for variance; standard deviation is its nonnegative square root. Each equality describes a random variable’s distribution across samples, not each sample’s realised value

Design a comparison experiment, ungraded

Choose one finite population model and compare n=2 with n=8 using, for example, 200 repeated samples per size. State replacement and independence assumptions, implement fresh random draws, retain each sample mean, and draw both empirical distributions on the same horizontal scale. Record the number of repetitions separately from n.

Model: control everything except sample size

Use one fixed model, such as values 0,2,4 with probabilities 1/4,1/2,1/4. For each sample size, take 200 independent complete samples with replacement and store one mean per sample. Keep the model, axis units and repeat count fixed; change only n. Validate small cases using exact enumeration and retain random-source/seed details. Compare empirical centre and spread with the theoretical μ and σ/√n, while allowing finite variation. Reusing the same saved run is replay, not new evidence; no random results are asserted here

A larger sample happens to have a mean farther from μ than a smaller sample. Does this disprove the theoretical variance rule?

Check the sampling model before transferring formulas

Without replacement from a small finite population, draws are dependent and the independent σ²/n formula is not exact in general. A full census has zero sampling spread. Biased selection, changing populations, dependence or inconsistent measurement can also invalidate a simple comparison. Increasing sample size does not fix a wrong target or selection rule