Sampling and the sample mean

The source compares independent samples from $X$ uniform on $\{1,2,\ldots,10\}$ and $Y\sim N(5,9)$ for $n=1,5,10,50,100$.

x	1	2	3	4	5	6	7	8	9	10
$P(X=x)$	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1

Here $E(X)=5.5$, $\operatorname{Var}(X)=8.25$, so the mean has variance $8.25/n$. The $n=1$ distribution is flat rather than normal; as $n$ grows, the exact discrete mean distribution becomes bell shaped and concentrates near 5.5. For $Y$, the mean is exactly $N(5,9/n)$ for every listed $n$. Both spreads shrink like $1/\sqrt n$.

n	$\operatorname{Var}(\bar X)$	$\operatorname{Var}(\bar Y)$
1	8.25	9
5	1.65	1.8
10	0.825	0.9
50	0.165	0.18
100	0.0825	0.09

The figure is an independent mathematical reconstruction: exact discrete convolutions are shown for $\bar X$ and exact normal densities for $\bar Y$, with a normal approximation overlaid for comparison.

It does not recreate the source’s random simulation run. Its ten original simulation chart images remain in the review evidence.

For a spreadsheet experiment, generate many independent size-$n$ samples, calculate each mean, and plot the means while keeping the population distribution fixed. Compare the observed centre and variance with the table.

The lecture links two external simulations: uniform discrete population and normal population. These are preserved source references; their external contents have not been reviewed. The Example 10 enrichment QR decodes to estimation theory; unbiased estimation is one form of estimation, and the destination is also outside this source review.

Before viewing each comparison, predict the centre, variance and shape of the distribution of means. In a spreadsheet, keep a separate plot of the individual observations. Explain why increasing the number of repeated samples makes the plot more stable, while increasing the size of each sample changes the spread of the means.