It’s Elemental! Sampling from the

Periodic Table

Matth Malloure

Grand Valley State University

[email protected]m

Mary Richardson

Grand Valley State University

periodic table of elements, students will collect real data implementing simple random and

systematic sampling. With both samples collected, students will calculate appropriate descriptive

statistics and use the sampling distributions to compare the performance of the methods. They

will also determine how to set up a stratified random sample and a cluster sample, but will only

perform the cluster sample in this activity.

GAISE Components

This activity follows all four components of statistical problem solving put forth in the

Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report. The four

components are: formulate a question, design and implement a plan to collect data, analyze the

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

Common Core State Standard Grade Level Content (High School)

S-ID. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

S-ID. 2. Use statistics appropriate to the shape of the data distribution to compare center

(median, mean) and spread (interquartile range, standard deviation) of two or more different data

sets.

S-ID. 3. Interpret differences in shape, center, and spread in the context of the data sets,

margin of error through the use of simulation models for random sampling.

S-IC. 5. Use data from a randomized experiment to compare two treatments; use simulations to

decide if differences between parameters are significant.

NCTM Principles and Standards for School Mathematics

Data Analysis and Probability Standards for Grades 9-12

Formulate questions that can be addressed with data and collect, organize, and display

relevant data to answer them:

• understand the differences among various kinds of studies and which types of inferences

can legitimately be drawn from each;

• display and discuss bivariate data where at least one variable is categorical.

Develop and evaluate inferences and predictions that are based on data:

• use simulations to explore the variability of sample statistics from a known population

and to construct sampling distributions;

• understand how sample statistics reflect the values of population parameters and use

sampling distributions as the basis for informal inference.

Understand and apply basic concepts of probability:

• use simulations to construct empirical probability distributions.

Prerequisites

random, cluster, systematic, and stratified samples. In addition to the sampling methods, students

will also know how to calculate simple univariate descriptive statistics including the mean,

standard deviation, and the five number summary. Then, using stem plots, students will be able

to compare the sampling distributions for sample means to determine the optimal sampling

strategy.

Time Required

Materials Required

The students will only need to bring a pencil and a graphing calculator. The instructor will

provide the activity sheet, complete with a periodic table of elements.

Instructional Lesson Plan

activity is: After performing a simple random sample and a systematic sample on the periodic

table of elements, which of the two is the most appropriate method for this scenario? All other

aspects of this investigation will be based on this specific question.

II. Design and Implement a Plan to Collect the Data

The various sampling methods the students will implement in this activity are based on the

periodic table of elements, so give a basic description of the periodic table as this will be

important in the design. The following description is available in the Worksheet, but go ahead

and summarize it for the students. Dmitri Mendeleev created the periodic table in 1869 and the

table is structured in order to reflect the “periodic” trends in the elements. The most up to date

with similar physical properties due to the number of electrons in the respective valence shell.

The value that the students will be most interested in is the atomic weight of elements, which can

be determined using a weighted average of the weights for each elements various isotopes.

The periodic table the students will use comes from the National Institute of Standards and

Technology. This table only comes with 114 elements, so for the purposes of this activity the

population size is

grams per mole.

seed of 1967. This way, all of the students should get the same random sample, and hence, the

same sample mean. Using a TI-84 PLUS calculator, students should first set their seed to 1967.

To do this, students need to enter 1967  STO->  MATH  Scroll to PRB  Select 1:rand

 ENTER  ENTER. This will set the random integer value to 1967. Then they must go back

into MATH  PRB, but now select “5:randInt(”. In order to get a sample of 25 elements from

the 114 on the table, have the students enter randInt(1, 114, 25). The resulting elements in the

sample can be found in Table 1.

Table 1. Elements in the simple random sample with seed 1967.

60 64 33 111 83 6 56 34 27

65 57 69 46 79 18 81 32 75

108 59 90 61 91 103 49

Note: If an instance occurs where the sample produces duplicate elements, have the students

continue to sample elements until the sample is comprised of 25 unique elements.

Now, students should complete Strategy 2: finding a 1-in-5 systematic sample of elements using

the seed of 1985. Students should be able to set the seed using the same method as in Strategy 1.

Instead of sampling 25 from the 114 as in the simple random sample, the students will sample

one integer

and then select every 5th element from the ordered list of elements by

For Strategy 3 the students should explain how to take a stratified random sample of 25 elements

from the 114 using the following 4 strata: solid, liquid, gas, and artificial. Students are also

given that there are 77 solids, 2 liquids, 11 gases, and 24 artificial elements. In order to take a

stratified sample, students should divide each population stratum size by the population size and

roughly ordered by atomic weight, the systematic sample should be more representative of the

population of elements. The students don’t have to be correct in their response to this question

because they will know the answer at the conclusion of the activity.

Each student will use the last 4 digits of their phone number as their seed for the simple random

and systematic samples. Then, they will determine the mean atomic weight for both samples and

record their means on the board under the appropriate heading. So divide the board into two

sections such that students can compile a list of means needed for the sampling distribution

portion of the activity. Table 4 below contains example data from 30 students. Note that the

Table 4. Mean atomic weights for simple random and systematic samples for 30 students.

Figure 1. Stem plots for the sampling distributions of the sample mean atomic weights.

Table 5 shows descriptive statistics for the example data in Table 4.

Table 5. Descriptive statistics for the sampling distributions.

Statistic Simple Random Samples

Systematic Samples

Mean 141.50 141.30

IV. Interpret the Results

The students should be able to see right away that the standard deviation for the simple random

samples is quite large compared to that for the systematic samples. Also, the average mean

atomic weights for both types of samples are nearly equal at 141.50 and 141.30. Therefore,

students should conclude that the systematic sample produces a mean atomic weight that is more

accurate. In the periodic table, the elements are ordered according to the number of protons and

Assessment

A. Systematic Sample

B. Cluster Sample

C. Simple Random Sample

D. Stratified Random Sample.

Scenario 1: In a factory producing television sets, every 100

Scenario 2: A class of 200 students is numbered from 1 to 200, and a table of random digits is

used to choose 60 students from the class.

Scenario 4: An airline company randomly chooses one flight from a list of all international

flights taking place that day. All passengers on that selected flight are asked to fill out a survey

on meal satisfaction.

2. Suppose a state has 10 universities, 25 four-year colleges, and 50 community colleges, each

of which offer multiple sections of an introductory statistics course each year. Researchers want

to conduct a survey of students taking introductory statistics in the state. Explain a method for

collecting each of the following types of samples:

A. Stratified Random Sample

C. Simple Random Sample

Answers:

2. First, compile a list of all the introductory statistics courses taught in the state at each type of

learning institution.

(A) Randomly sample a representative proportion of introductory statistics courses from each of

the 3 strata: universities, four-year colleges, and community colleges.

(B) Randomly sample one of the 3 types of learning institutions and then take a census of all

introductory courses within that type of institution.

(C) Simply take a random sample of n introductory courses from the list of ܰ offerings without

considering the type of learning institution.

Possible Extensions

1. Carry out a cluster sample and stratified random sample to compare the variability and

precision of the mean atomic weight after repeated sampling.

2. Demonstrate the Central Limit Theorem by taking various sized samples of the simple

random sample. So repeatedly sample 5, 10, 25, and 50 elements and compare the sampling

distributions of the mean atomic weights.

3. Begin with the 114 elements and calculate the sample size needed to reach a specified margin

of error for the four sample methods.

3. Activity background adapted from: http://en.wikipedia.org/wiki/Periodic_table.