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**Problem H1
**We used a metric tape measure to measure the lengths of 20 pine needles to the nearest millimeter:

**a. **Think of a question that collecting this data might answer.

**b. **Notice that the 20 pine-needle measurements are not all the same. What is the source of this variation?

**c. **If you have access to some pine needles, measure the length (to the nearest millimeter) of 10 different needles with a metric ruler. If you do not have access to pine needles, measure the length (to the nearest millimeter) of 10 different pieces of spaghetti with a metric ruler. Are the lengths the same?

**Problem H2
1. Ask a Question
**Where were my nickels minted?

A nickel’s mint location (if present) is located on the head side of the coin below Jefferson’s ponytail. We recorded the mint locations and years for 100 nickels (“S” indicates San Francisco, “D” indicates Denver, “P” indicates Philadelphia, and “N” indicates that no mint location was present):

**a. **There are differences in the mint locations. What is the source of this variation?

**b. **There are differences in the mint years. What is the source of this variation?**
c. **What observations can you make about this data? For example, which location appears most frequently? Which decade?

View the Session 2 video to see how the onscreen participants solved this problem.

**Problem H3**

**1. Ask a Question**

What is your pulse rate?

**2. Collect Data**

Take your pulse rate, in beats per minute, seven times during the same day.

**a. **Are the measurements of your pulse rate the same? If not, what is the source of this variation?

**b. **Suppose you wanted to estimate your typical pulse rate. How could you answer this statistical question?

Try this experiment on yourself. You might take your pulse rate under several different conditions, for example, early in the morning, late in the day, and after exercising. Are your measurements the same?

**Suggested Readings:**

**Krus, David and Webb, James (Autumn, 1997). “Demonstrating Variance Using the Muller-Lyer Illusion.” Teaching Statistics, 19 (3), 72-76.**

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Demonstrating Variance Using the Muller-Lyer Illusion

Continued

*Principles and Standards for School Mathematics.* (Reston, VA: National Council of Teachers of Mathematics, 2000) Standards on Data Analysis and Probability by grade level: K-2, 108-115; 3-5, 176-181; and 6-8, 248-255.

Reproduced with permission from the publisher. Copyright © 2000 by the National Council of Teachers of Mathematics. All rights reserved.

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Data Analysis and Probability Standards for Grades Pre-K-2

Continued

Data Analysis and Probability Standards for Grades 3-5

Continued

Data Analysis and Probability Standards for Grades 6-8

Continued

- One question might be, “What is the average length of a pine needle from a fully grown pine tree?”
- Here are some possible sources of the variation in this data:
- Pine needles are inherently different and come in different sizes.
- Measurement errors may have occurred.
- The needles may have come from different places on the tree, which could affect their lengths.
- The needles may have come from different trees, whose different locations could affect the needles’ lengths.
- Some of the needles may have come from one type of pine tree and some from another.

- They probably will not all be identical, due to the many possible sources of variation.

- Here are some possible sources of the variation in this data:
- Coins are minted in different places.
- Measurement errors may have occurred, including recording errors of the mint location.
- These coins were obtained from different locations, some with more of a certain type of coin. For example, on the East Coast, you’re more likely to see nickels with a P marking.

- This variation is due to the fact that the same mints produce many different coins each year, and they all remain in circulation.
- The location D appears most frequently; roughly half the data (51 coins out of 100) come from this location. The 1990s is the most frequently appearing decade, with 40 coins; the 1980s is a close second, with 37.
- The coins with no mint location were minted between 1972 and 1978. One new question might be, “Where were the coins with no mint location minted?”
- The data should be roughly comparable but not identical, due to variation. Some causes of this variation are the location in which the coins were obtained, the condition of the coins that are kept in circulation, and measurement error.
- It seems clear that the coins with no mint location were minted in Philadelphia. The range of dates for the N coins is 1972-1978, and the earliest date for a P coin is 1981. In contrast, there are many coins from Denver minted through the 1970s. The implication is that Philadelphia didn’t begin imprinting its location until after 1978. (There are not enough coins with the S mint mark to suggest that the larger number of coins with no mint location came from San Francisco.)

- The measurements are very unlikely to be the same. Here are some possible sources of the variation in this data:
- Measurement errors may have occurred.
- The measurements are taken at different times of the day, which affects pulse rate.
- The measurements are taken under different conditions, for example, at different levels of physical activity or stress.

- This is difficult, since there is a lot of variation in pulse-rate data. You might take a large number of readings, then partition them into activities (working out, sleeping, eating, etc.) to find a typical pulse rate for each activity. Any single number to estimate overall pulse rate would be subject to a lot of variation, and any range that contains all possible pulse rates for one day would be a very wide one.