Intent

Students return to their work in Penny Weight, bringing to bear their new tool for judging ordinary and rare data.

Mathematics

In Penny Weight, students addressed these important mathematical questions: Given the weights of a collection of real coins, is another coin of a particular weight real or counterfeit? Is the weight of the suspect coin what you would expect, given the variability in the weights of real coins? At that time, students had few tools at their disposal to address these questions. Now they have standard deviation.

Progression

Students will work on the activity individually and then use their group members’ work to evaluate their calculations and draw conclusions. After this activity, students will rely on their calculators, or software such as Fathom, to calculate statistics such as mean and standard deviation. [link to statistics on the calculator.doc]

Approximate Time

30 minutes for activity (at home or in class)

45 minutes for discussion

Classroom Organization

Individuals, then groups, followed by whole-class discussion

Materials

Transparency of the graph of the data (optional) [Key: Link to a blackline master of this new graph (below).]

Doing the Activity

Advise students to keep track of all their computations so they can compare their results with others in their groups and locate any computational errors fairly easily.

Discussing and Debriefing the Activity

Have students work in their groups to achieve consensus on the answer to Question 1 and to begin sharing ideas about Question 2.

Once students agree on the value of the mean (2500 mg) and the standard deviation (approximately 9.88 mg), turn to answers and explanations for Question 2. The goal is for students to recognize that the weight of the uncle’s penny is more than two standard deviations from the mean and that the chance of getting a result this far (or farther) from the mean is less than 5%. That is, if a coin were selected at random from a set of fair coins, the probability that it would be this far or farther from the mean is less than .05.

Note that the percentages for the normal distribution are essentially probability values. For instance, one could restate the condition given in Standard Deviation Basics that “Approximately 68% of all results are within one standard deviation of the mean” as “If a data value is chosen at random, the probability that it is within one standard deviation of the mean is approximately .68.” Using the language of probability may help students connect the ideas in this unit to their work in The Game of Pig.

Ask the class what assumption they are making in using this value of 5%. They should see that they are assuming that the distribution of weights among pennies is approximately normal. You can assure them that this is a reasonable assumption. Remind them that the percentages in Standard Deviation Basics refer only to normal distributions.

Ask students, Does 9.9 mg seem reasonable for the standard deviation?

As needed, use the question to review the idea that standard deviation measures, more or less, a kind of average distance from the mean (although it is not literally an average). Students should agree that, because some pennies are less than 10 mg from the mean and others are more than 10 mg from the mean, a standard deviation of 9.9 mg seems about right. You might ask them to actually calculate the average distance from the mean—that is, the mean absolute deviation—which is a variation on Kai’s method from Kai and Mai Spread Data. (For the pennyweight data, the value of mean absolute deviation is 8.2 mg.)

Ask students to sketch a normal distribution with a mean of 2500 and a standard deviation of about 9.9 and to shade the 5% area that they are discussing—that is, the area that represents results that are more than two standard deviations from the mean. The diagram should look something like this.

Figure 1

Have students point out the regions on their sketches where 95% of all measurements of real pennies should fall. [link to math maps]

Ask students to discuss their answers to Question 2b and to compare their thinking now with their conclusions in Question 2 of Penny Weight. Specifically, ask how helpful standard deviation was in making a decision. They may say the decision was just as easy to make without the use of standard deviation but that they are more confident or more knowledgeable about their decision when they use standard deviation.

The reasoning students are using here is at the heart of inferential statistics. It goes something like this: Given what we know about real pennies, how likely is it that this suspect penny is also real? To make this inference, the sample of pennies is treated as an approximation of the collection of all possible pennies. In this case, statisticians use a slightly different calculation of standard deviation. Students will be confronted with the following two different ways to calculate standard deviation when they learn to use their calculators to do this. You may want to discuss population standard deviation versus sample standard deviation with them now. [link to statistics on the calculator.doc]

Standard Deviation: Population ( σ ) or Sample (s)

The computation students did for Penny Weight Revisited gave them the standard deviation of their sample data. However, in evaluating the authenticity of the supposedly counterfeit penny, what they really should consider is the standard deviation of the set of weights of all pennies in the world. If students respond that this is impossible, ask what they suggest instead. Try to draw out the idea of estimating the standard deviation of a population from a sample.

Technically, the best estimate of the standard deviation of a population, based on data from a sample, is given by a slightly different calculation than the one they used. Specifically, rather than divide the sum of the squares by n (the number of data items), one divides by n – 1, a smaller number. The statistic calculated in this way is usually represented by the letter s and is called the sample standard deviation.

If n is reasonably big, σ and s will differ only slightly. Using s means dividing by a slightly smaller number, so that you get a slightly larger estimate for the standard deviation of the distribution. One way to make sense of this is to consider that the sample of data only approximates the population. This adds an extra source of variability to the data, variability due to sampling error. Thus we would expect a measure of the variability in the sample to be slightly larger than the variability in the population. [link to math maps]

Key Question

Does 9.9 seem about right for the standard deviation?