Objectives
By physically generating the data and by
calculating the same statistic themselves for each of several samples,
students will understand intuitively that a statistic’s value varies from
sample to sample, and that the distribution of the statistic’s values is
different from the distribution of the original observations. By comparing
the shape, center, and spread of the (non-normal) distribution of the
original observations to the corresponding features of the sampling
distribution from samples of two different sizes, students will discover
the Central Limit Theorem’s description of the sampling distributions of
the sample mean and sample proportion.
Materials and equipment
 |
One fair
six-sided die. (Several dice will speed
up the process and reduce the tedium
somewhat.) |
|
|
One penny.
(Pennies minted in the early 1960’s have
the most severely beveled edges and are
ideal.) |
|
|
Data on a
quantitative variable and a categorical
variable for all members of a small
population. (The SAT math scores and
the home state for incoming new students
at Wittenberg University in 1995 are
provided. Instructors are encouraged to
replace these data with data of more
local interest.) |
Time involved
|
|
30 minutes
out of class for each of the two
activities |
|
|
20 minutes
in class for each of the two activities |
Activity description -
Classroom analysis of sample means
Students should start with a histogram
of the individual observations (i.e., individual dice rolls), noting that
the shape is non-normal, that the mean is about 3.5, and that the standard
deviation is about 1.7. (Students who have learned how to calculate the
mean and standard deviation of a probability distribution can be asked to
verify these values.) Students should then be directed to consider the
distribution of sample mean values from samples of 4 and 10 dice. It’s
important to display these distributions next to one another and on the
same scale, to lead the students to an understanding of the effects of
sample size on the variability in the sampling distribution. The following
results are from 180 samples, generated by 36 students.
Students quickly recognize that these
sampling distributions are roughly normal, despite the fact that the
distribution of individual rolls was highly non-normal (namely, uniform).
Some simple summary statistics will reinforce what their eyes tell them
about center and spread as well: the sampling distributions are centered
at 3.5, just like the distribution of individual rolls, but the
variability of sampling distributions decreases with the sample size.
VARIABLE N MEAN SD MEDIAN
MEAN_4 180 3.5583 0.8059 3.5000
MEAN_10 180 3.5133 0.5214 3.5000
At this point, students can be shown the
simple formula for calculating the standard deviation of the sample mean’s
distribution, and can verify that its prediction roughly agrees with the
standard deviation from their generated sample means.
To put this information on the sampling distribution of the sample mean
into a more meaningful context, students are also asked to repeatedly
sample from a tangible population. Given on the
student’s version
of the activity are the SAT Math scores for all 398 new students entering
Wittenberg University in 1995 who reported such scores (as opposed to ACT
scores alone). Now that they’ve encountered the Central Limit Theorem,
students should try to anticipate the distribution of the sample means
that they have collected – specifying the shape, center, and spread. To
make these predictions, students will need to know that the population
mean and standard deviation for individual scores are 554.1 and 100.2,
respectively.
At that point they can pull up the data file compiled by the instructor to
check their predictions. Most effective is a visual comparison, on the
same scale, of the distribution of individual scores and the distribution
of their sample means:
Students can easily verify – from the
histograms and from descriptive statistics – that the sampling
distribution of their sample means is centered at roughly the same place
as the distribution of individual scores, but that the standard deviation
is indeed much smaller.
Activity description -
Classroom analysis of sample
proportions
Students should again start with the distribution of individual
observations (i.e., penny spins) and note that it makes no sense even to
consider shape, center, and spread of a categorical variable’s
distribution. Students should then be directed to consider the
distribution of sample proportion values from samples of 10 and 20 spins.
Once again, it’s important to display these distributions on the same
scale, to lead the students to an understanding of the effects of sample
size on the variability in the sampling distribution. The following
results are from 205 samples, generated by 41 students.
Again, students quickly recognize that
these sampling distributions are roughly normal, despite the fact that the
original variable is not even quantitative, let alone normally
distributed. Here, too, some simple summary statistics will reinforce what
their eyes tell them about center and spread as well: the sampling
distributions are centered at about 0.4, which is roughly the proportion
that most spun pennies will land heads up, and the variability of sampling
distributions decreases with the sample size.
VARIABLE N MEAN SD MEDIAN
PROP_H_10 205 0.3576 0.1683 0.4000
PROP_H_20 205 0.3712 0.1482 0.3500
At this point, students can be shown the
simple formula for calculating the standard deviation of the sample
proportion’s distribution, and can check whether its prediction roughly
agrees with the standard deviation from their generated sample means.
In this case, the observed standard deviations are larger than predicted,
which is almost certainly due to the fact that the pennies used by these
classes were minted in different years and hence have different
probabilities of landing heads-up. For this very reason, Scheaffer et al.
(Activity-Based Statistics, 1996, p.129) recommend that all pennies used
be minted in the same year. If students use pennies from different years,
as was the case with the above results, have the students report the year
each penny was minted, so that they can then construct a scatterplot of
their sample proportions against minting year:
Although this has nothing to do with
sampling distributions or the Central Limit Theorem, it may be of interest
to see that the probability of landing heads-up has indeed risen
considerably since the early 1960’s, due to changes in the subtle angle at
which the edges are beveled to help the pennies fall easily out of the
minting trays.
If this differing probability of landing heads-up is a concern, and if
it’s not feasible to have all students use pennies from the same minting
year, instructors can consider alternative experiments with dichotomous
outcomes. One alternative is to flip or drop thumbtacks and note what
proportion land point-up – though students would need to be given
identical tacks, so that the probability of landing point-up would be the
same for all flips. Another alternative, described by
Richardson, Curtiss,
and Gabrosek (2002), is to toss Hershey’s Kisses, presumed to be of
uniform size and shape, and note what proportion land on the base.
To put this information on the sampling distribution of the sample
proportion into a more meaningful context, students are then asked to do
repeated sampling from a tangible population. Given is the home state for
each of the 610 new students entering Wittenberg University in 1995.
Before looking at the distribution of their sample proportions, students
should be asked to use their new-found theoretical result to anticipate
the shape, center, and spread of this sampling distribution, and to sketch
this distribution. To make these calculations, students will need to know
that 383 of all 610 students are from Ohio, so that the population
proportion is 0.628, and hence the sampling distribution of the sample
proportions should be approximately N(0.628, 0.108). Students then check
the accuracy of these predictions. Below is a histogram based on 117
samples collected by 39 students:
The shape is clearly normal, and the
mean and standard deviation of these particular 117 values of the sample
proportion are 0.633 and 0.106, respectively, which match the theoretical
predictions almost perfectly.
Teacher notes
Most students will not understand the
idea of a sampling distribution unless they themselves carry out repeated
sampling and calculate the desired statistic from each of several samples.
Merely presenting results collected by the instructor or results simulated
by computer will not reach many students. Hence students should be
required to do the sampling themselves, ideally in some familiar context.
Unfortunately, it often takes several dozen samples before a statistic’s
sampling distribution emerges recognizably; students are understandably
reluctant to believe that a generated sampling distribution is normally
distributed when looking at the statistic’s values from a dotplot of a
mere 10 or 15 samples. It helps, then, to pool the sampling energies of
the entire class. Even if the class section is large (say, over 100
students) and a single sample’s result from each student would suffice to
make a convincingly smooth sampling distribution, however, there is
pedagogical benefit to requiring each student to generate more than one
sample each: students then get first-hand experience with the variability
in a statistic’s values in repeated sampling. Students will, of course,
justifiably resent the tedium if we force them to generate a large number
of samples or if the measurement process is very time-consuming, so these
activities require only three samples each of 4 and 10 dice rolls, 10 and
20 penny spins, and 20 or 25 individuals from a known population of a few
hundred individuals. And in each case, the statistic calculated from each
sample takes only seconds to calculate.
To save valuable contact time for getting insight from the sample results,
the actual sampling should be done outside of class, and the instructor
should combine the results into a single data file in preparation for
class as well. The assignment can be given two class sessions before the
target class session and students can hand in their results at the session
before the target session, or the assignment can be given in the
penultimate session and the results submitted electronically by a deadline
chosen to give the instructor time to consolidate the results.
Assessment
Students should be able to articulate
what is meant by a sampling distribution, both in the abstract and in a
given context. Moreover, students should be able to use the Central Limit
Theorem to predict the sampling distribution of the sample mean and sample
proportion, both in the abstract and in a given context. As an optional
reinforcing activity, students can examine the sampling distribution of a
simple statistic from repeated sampling on some aspect of their class.
-
What do we mean by a “sampling
distribution”?
-
Which of the following have sampling
distributions: variables, parameters, statistics, data, individuals?
-
What does the Central Limit Theorem tell you
about the sampling distributions of the sample mean and sample
proportion?
-
The weight of chicken eggs varies with a
mean of 56g and a standard deviation of 6g. Eggs are packed in cartons
by the dozen. Describe in context the relevant sampling distribution.
What does the Central Limit Theorem predict for this sampling
distribution? Sketch the distribution of individual egg weights, and
sketch the relevant sampling distribution on a separate graph using the
same scale.
-
Suppose 15% of the incoming students at this
college are left-handed, and that students are assigned in groups of 25
to freshman advisors. Describe in context the relevant sampling
distribution. What does the Central Limit Theorem predict for this
sampling distribution? Sketch the relevant sampling distribution.
-
Gather the heights (in cm) of all students
in the class. Make a visual display of this distribution, and report
measures of center and spread. Take 100 samples of 5 students each and
calculate the mean height of each sample. Make a visual display of these
sample mean heights, and report measures of center and spread. What does
the Central Limit Theorem predict for this sampling distribution? How
close are the its predictions to your actual sampling distribution?
References
Richardson, M., Curtiss, P., and Gabrosek, J. (2002). “What is the
Significance of a Kiss?” Statistics Teaching and Resource Library
[on-line]. March 17.
Scheaffer, R., Gnanadesikan, M., Watkins, A., and Witmer, J. (1996).
Activity-Based Statistics. New York: Springer.
