[JK10 page 363] Definition: A Sampling Distribution of a Sample Statistic is the distribution of values for
a sample statistic obtained from repeated samples, all of the same size and all drawn from the same
population.
Thinking about the meaning of that: See [JK10 Page 366 Figure 7.5]
One sample of size n.
Then another sample of size n.
And another and another and another and another.
For each sample, calculate some statistic (the mean, or the range, or the variance, or the standard
deviation, or whatever).
You get a value from each sample.
The collection of all of those values – what does the distribution look like.
[JK10 page 363 Example 7.1] The set { 0, 2, 4, 6, 8 }
They list all the possible samples of size 2, they calculate the mean of each sample. They make a
histogram of the sampling distribution of the mean. Then they calculate the range of each sample and
examine that sampling distribution.
[JK10 page 364 Example 7.2] The set { 1, 2, 3, 4, 5}; 30 samples of size 5. Histogram shows the
distribution of the sample means.
My side notes on Example 7.2:


Population has 55 = 3125 elements
an Excel exercise with this would be interesting to do. You could easily observe the distribution
of samples’ range, variance, standard deviation, too. RANDBETWEEN, COUNTIF, etc. are used.
[JK10 page 365] Definition: A random sample is a sample obtained in such a way that each possible
sample of fixed size n has an equal probability of being selected.
[JK10 page 369] Definition: SDSM, Sampling Distribution of Sample Means – If all possible random
samples, each of size n, are taken from any population with mean 𝜇 and standard deviation 𝜎, the the
sampling distribution of sample means will have a mean, 𝜇𝑥̅ = 𝜇 and a standard deviation 𝜎𝑥̅ =
𝜎
.
√𝑛
Furthermore, if the sampled population has a normal distribution, then the sampling distribution of 𝑥̅
will also be normal for samples of all sizes.
My comments:


Note “any” population. Even a population that doesn’t have a normal distribution.
Note the denominator in the standard deviation fraction.

Want to do an Excel worksheet with a big experiment of this.
[JK10 page 370] Definition: Standard Error of the Mean isanother name for the standard deviation of
the SDSM, 𝜎𝑥̅ =
𝜎
.
√𝑛
[JK10 Page 370] The Central Limit Theorem says that “The sampling distribution of sample means will
more closely resemble the normal distribution as the sample size increases.” JK10 notes:


If the sampled population is not normal, the CLT tells us that the sampling distribution will still
be approximately normally distributed under the right conditions. If the sample population
distribution is nearly normal, the 𝑥̅ distribution is approximately normal for fairly small 𝑛,
perhaps as small as 15. But when the sampled population distribution lacks symmetry, 𝑛 may
have to be quite large, 50 or more, maybe, before the normal distribution provides a
satisfactory approximation.
The number of repeated samples used in an empirical situation has no effect on the standard
error.
[JK10 pages 373-374] Uniform Distribution and U-Shaped Distribution and J-Shaped Distribution: one
diagram shows the population distribution; subsequent diagrams show the sampling distribution of 𝑥̅ for
values of = 2, 5, 30 . Visually, it’s easy to see how the sampling distribution of 𝑥̅ appears more like a
normal distribution as 𝑛 increases. [JK10 page 378 Figure 7.12] similar, just the overall concept, without
specific values of n.
[JK10 page 377] Section 7.4 – Application of the sampling distribution of sample means:


[JK10 page 377 Example 7.5] – given a normal population with its 𝜇 and 𝜎, and given a range of
𝑥 scores, and a sample size of 𝑛, what is the probability that the sample mean will be in that
range? [JK page 379 Example 7.6 is similar.]
[JK10 page 377 Example 7.7] – similar, but they say “the middle 90%”. Instead of converting the
endpoints of an 𝑥 range into a 𝑧 range, you instead find the 𝑧 range from the clue “the middle
90%” (of a normal distribution). Then work backwards to find the endpoints of the x range.