# Using Your Hair to Understand Descriptive Statistics

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

#### Summary

## Learning Goals

Students who complete this activity will

- discover the concepts of a mean, median, outlier, and distance-to-mean with minimal use of numbers, and
- gain a visual understanding of descriptive measures used in statistics.

## Context for Use

This activity can be used when discussing descriptive statistics in your course. We would suggest using this activity before the formal discussion of descriptive statistics. This activity has been used by middle and high school teachers as well as colleagues who teach introductory statistics in a university setting.

This activity will take about 45 minutes to complete in its entirety.

This activity will take about 45 minutes to complete in its entirety.

## Description and Teaching Materials

The first step is to divide your class into groups of size 5. Obtain a piece of hair from each group member and measure the length of your hair (in millimeters) using the ruler provided. Have each group member cut a piece of white string that matches their hair length. Using a piece of tape, label each string with your name. Students are asked to discover how to compute the median hair length using their strings. [Note: Groups of size 5 make finding the median easier.]

Next, have each group cut a single piece of red string that has a length equal to the sum of all hair lengths. Students are asked to discover how to compute the average or mean hair length using this piece of string. This is done by folding this long peice of string into equal lengths. Cut the red string into equal lengths and give each group member a piece. [Note: Groups of size 5 make folding the string into equal lengths more challenging.]

Finally, students can easily determine distance-to-mean by comparing their white string to their piece of red string. Students should cut whichever string is necessary so that the two pieces are of the same length. Students are asked to discuss this residual piece of string in detail (e.g. some students will have red residual string and other will have white).

Next, have each group cut a single piece of red string that has a length equal to the sum of all hair lengths. Students are asked to discover how to compute the average or mean hair length using this piece of string. This is done by folding this long peice of string into equal lengths. Cut the red string into equal lengths and give each group member a piece. [Note: Groups of size 5 make folding the string into equal lengths more challenging.]

Finally, students can easily determine distance-to-mean by comparing their white string to their piece of red string. Students should cut whichever string is necessary so that the two pieces are of the same length. Students are asked to discuss this residual piece of string in detail (e.g. some students will have red residual string and other will have white).

**Required materials: Two different colors of string, scissors, masking tape, and a ruler for each group of 5 students****Handouts:**Student Version (Word) (Microsoft Word 344kB May18 07) | Student Version (PDF) (Acrobat (PDF) 32kB May18 07) | Instructor Version (Word) ( 361kB May18 07) | Instructor Version (PDF) (Acrobat (PDF) 47kB May18 07)## Teaching Notes and Tips

This activity is appropriate for various grade levels; however, the activity does contain some challenge questions that may not be appropriate for all students. Some additional questions (not included in the activity) to direct discussions are included below.

The following questions allow for comparisons to be made between genders.

The following questions are more challenging and may not be appropriate for all grade levels.

The following questions allow for comparisons to be made between genders.

- What would likely happen to the white pieces of string if your group consisted of all boys? How about a group with all girls?
- Having all boys or girls in a group would also affect the red pieces of string. What differences would we expect in the red string for a group of all boys versus a group of all girls?

The following questions are more challenging and may not be appropriate for all grade levels.

- Would a group of all boys have mostly red residual string? Would a group with all girls have mostly white residual strings? Why or why not?
- Align the white residual strings end-to-end. Do the same for the red residual strings. Are these two sets the same length? What are the consequences of this result when attempting to measure distance-to-mean?

## Assessment

This activity compliments a student's understanding of descriptive measures. Some example assessment items are given here.

Assessment items that are centered around various group characteristics.

Other possible assessment items for students of various mathematical abilities.

Assessment items that are centered around various group characteristics.

- What would likely happen to the white pieces of string if your group consisted of all boys? How about a group with all girls?
- Having all boys or girls in a group would also affect the red pieces of string. What differences would we expect in the red string for a group of all boys versus a group of all girls?

Other possible assessment items for students of various mathematical abilities.

- Give a definition of an average or mean without using a formula.
- Would a group of all boys have mostly red residual string? Would a group with all girls have mostly white residual strings? Why or why not?
- Align the white residual strings end-to-end. Do the same for the red residual strings. Are these two sets the same length? What is the consequence of this result when attempting to measure distance-to-mean.
- Ask students to give a visual proof as to why the Sum of (Data-Mean) over all observations is equal to 0.
- Ask students to visually explain why the standard deviation cannot be less than 0.

## References and Resources

Understanding Mean Activity, TeacherVision. [Online: www.teachervision.fen.com/statistics/mathematics/4913.html]