# Introduction to Science - Penny Histograms

#### Summary

Overview: This exercise serves as an introduction to histograms. Students are given a short introduction to histograms in lecture and reading. They are then broken up into different groups of 3-5 students. Students start by making a prediction about the distribution of the weights of the pennies they have been given. Each group then weighs their pennies and: 1) enter their data into a spreadsheet of results for the entire class and 2) each student must draw a histogram of their own results. Each group must then interpret their graphs and each student must explain in writing how their graphs either confirms or disproves their initial predictions. While students are working, the instructor creates a histogram of the data from the entire class. After students have completed the written part of the assignment, each group gives a short summary of their predictions, their results, and their conclusions.

Results: The students find that 2 sets of pennies have overlapping weights (mean = 2.5 g, stdev = 0.2 g), while the third set is slightly different (mean 3.1 g, stdev = 0.2 g). The instructor can then lead a discussion around the topic of variability and how to statistically evaluate whether or not different data sets are statistically similar. This discussion should include how to account for instrumental error. This leads into a discussion of the meaning of mean values and standard deviation. This discussion can also include a comparison of how the results from an individual group compares with the data set from the whole class: this serves as a good place to discuss how scientists determine how much data to collect for a given project. This and similar exercises are thus needed before students start to interpret their own data. Expected Outcomes: Students will gain an understanding of how to read a histogram and how to evaluate the meaning of mean values and standard deviations.

Results: The students find that 2 sets of pennies have overlapping weights (mean = 2.5 g, stdev = 0.2 g), while the third set is slightly different (mean 3.1 g, stdev = 0.2 g). The instructor can then lead a discussion around the topic of variability and how to statistically evaluate whether or not different data sets are statistically similar. This discussion should include how to account for instrumental error. This leads into a discussion of the meaning of mean values and standard deviation. This discussion can also include a comparison of how the results from an individual group compares with the data set from the whole class: this serves as a good place to discuss how scientists determine how much data to collect for a given project. This and similar exercises are thus needed before students start to interpret their own data. Expected Outcomes: Students will gain an understanding of how to read a histogram and how to evaluate the meaning of mean values and standard deviations.

## Context

#### Audience

Interdisciplinary Sciences 101. This is one of the first exercises that students complete in a class that is an introduction to science and scientific methods. Target Audience: General education students who have had no previous science classes.

#### Skills and concepts that students must have mastered

Basic arithmetic.

#### How the activity is situated in the course

This is part of a sequence of introductory exercises at the beginning of class. It is the third in a series of "introduction to science" exercises.

## Goals

#### Content/concepts goals for this activity

They should understand the statistical meaning of mean values and standard deviation.

#### Higher order thinking skills goals for this activity

Students students should gain an understanding on how to analyze variability and how to statistically evaluate whether or not different data sets are statistically similar.

#### Other skills goals for this activity

Other skills include working in groups, drawing graphs (histograms), writing interpretations, and oral presentation of results.

## Description and Teaching Materials

Materials needed:

colored pencils

graph paper

scales

and for each group of students:

2 sets of 20 "new" pennies pennies with composition = 97.5% zinc, 2.5% copper (core: 99.2% zinc, 0.8% copper; plating: pure copper) = 1983 and later. The average weight of these "new" coins is 2.5g

1 set of pennies with "old" composition of brass (95% copper, 5% zinc) = 1962-1981. The average weight of these "old" coins is 3.11g

Note: coins from 1982 may have both compositions - a scape across the side or weights will reveal whether or not a coin has a zinc core

Penny Histograms (Acrobat (PDF) 197kB Aug8 14)

colored pencils

graph paper

scales

and for each group of students:

2 sets of 20 "new" pennies pennies with composition = 97.5% zinc, 2.5% copper (core: 99.2% zinc, 0.8% copper; plating: pure copper) = 1983 and later. The average weight of these "new" coins is 2.5g

1 set of pennies with "old" composition of brass (95% copper, 5% zinc) = 1962-1981. The average weight of these "old" coins is 3.11g

Note: coins from 1982 may have both compositions - a scape across the side or weights will reveal whether or not a coin has a zinc core

Penny Histograms (Acrobat (PDF) 197kB Aug8 14)

## Teaching Notes and Tips

Note: While these concepts often seem obvious to experienced scientists, many introductory students have no idea how to interpret a histogram or how to evaluate the similarity of two data sets.

They often have particular trouble with choosing a bin size for histograms - I encourage them to start in pencil, and warn them that they may need to take several tries before they get a graph that is useful. I will give hints, but not the answers. This is where working in groups can be an advantage as different members of the same group can experiment with different bin sizes and they can compare answers to choose the best bin size for their group report.

Student are often intimidated by having to read and interpret graphs.

They often have particular trouble with choosing a bin size for histograms - I encourage them to start in pencil, and warn them that they may need to take several tries before they get a graph that is useful. I will give hints, but not the answers. This is where working in groups can be an advantage as different members of the same group can experiment with different bin sizes and they can compare answers to choose the best bin size for their group report.

Student are often intimidated by having to read and interpret graphs.

## Assessment

Participation in group discussion and class presentation.

Submission of histograms and completed assignment by all students

Submission of histograms and completed assignment by all students

## References and Resources

The following are not needed to complete the exercise, but students who need more help may benefit from the following websites:

Introduction to histograms: http://quarknet.fnal.gov/toolkits/ati/histograms.html

Histograms versus other types of plots: http://education.mit.edu/starlogo/graphing/graphing.html

An example of histograms: http://quarknet.fnal.gov/run2/histo.shtml

Selecting Bin size - adynamic histogram: http://quarknet.fnal.gov/run2/brian.html

Math is Fun: http://www.mathsisfun.com/data/histograms.html

Introduction to histograms: http://quarknet.fnal.gov/toolkits/ati/histograms.html

Histograms versus other types of plots: http://education.mit.edu/starlogo/graphing/graphing.html

An example of histograms: http://quarknet.fnal.gov/run2/histo.shtml

Selecting Bin size - adynamic histogram: http://quarknet.fnal.gov/run2/brian.html

Math is Fun: http://www.mathsisfun.com/data/histograms.html