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Sampling Distributions of the Sample Mean and Sample Proportion

Douglas M. Andrews
Department of Mathematics and Computer Science
Wittenberg University
Springfield, OH 45501

Statistics Teaching and Resource Library, June 30, 2003

© 2003 by Douglas M. Andrews, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

In these activities designed to introduce sampling distributions and the Central Limit Theorem, students generate several small samples and note patterns in the distributions of the means and proportions that they themselves calculate from these samples. Outside of class, students generate samples of dice rolls and coin spins and draw random samples from small populations for which data is given on each individual. Students report their sample means and proportions to the instructor who then compiles the results into a single data file for in-class exploration of sampling distributions and the Central Limit Theorem.

Key words: Sampling distribution, sample mean, sample proportion, central limit theorem

## 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.

1. What do we mean by a “sampling distribution”?
2. Which of the following have sampling distributions: variables, parameters, statistics, data, individuals?
3. What does the Central Limit Theorem tell you about the sampling distributions of the sample mean and sample proportion?
4. 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.
5. 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.
6. 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.