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The Normal Curve, Standardization, and z Scores NOTE TO INSTRUCTORS A number of essential statistical concepts are covered in this chapter, including the two most important concepts: the standardization process using z scores and the central limit theorem. In order to get students more comfortable with standardization and z scores, it is essential that they have sufficient practice transforming raw scores. Make sure you go through several examples from the textbook during your lecture, and use Classroom Activity 6-3, “Comparing Apples and Oranges,” to help teach the material. Students can find that the central limit theorem is a little tricky to grasp conceptually. Again, use the examples from the textbook and the Classroom Activity 6-5, “Distribution of Scores vs. Means,” to illustrate this theorem.

OUTLINE OF RESOURCES I.

The Normal Curve ■ Classroom Activity 6-1: Make It Your Own: Normal Distributions (p. 50) ■ Classroom Activity 6-2: Observing the Normal Curve (p. 50)

II.

Standardization, z Scores, and the Normal Curve ■ Discussion Question 6-1 (p. 51) ■ Classroom Activity 6-3: Comparing Apples and Oranges (p. 51) ■ Discussion Question 6-2 (p. 51) ■ Classroom Activity 6-4: Make It Your Own: z Scores (p. 51)

III.

The Central Limit Theorem ■ Discussion Question 6-3 (p. 52) ■ Classroom Activity 6-5: Distribution of Scores vs. Means (p. 52) ■ Discussion Question 6-4 (p. 52) ■ Discussion Question 6-5 (p. 53) ■ Discussion Question 6-6 (p. 53) 49

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Classroom Activity 6-6: Make It Your Own: Normal Distributions (p. 53)

Next Steps: The Normal Curve and Catching Cheaters ■ Additional Readings (p. 54) ■ Online Resources (p. 54) Handouts Handout 6-1: Observing the Normal Curve (p. 55) ■ Handout 6-2: Comparing Apples and Oranges (p. 56) ■ Handout 6-3: Distribution of Scores vs. Means (p. 57) ■

CHAPTER GUIDE I.

The Normal Curve 1. The normal curve is a specific bell-shaped curve that is unimodal, symmetric, and defined mathematically.

Classroom Activity 6-1 Make It Your Own: Normal Distributions ■ Use the height and weight data that were anonymously collected previously (see Chapter 3, Classroom Activity 3-1). ■ Using the graph function in SPSS, generate histograms for each of these variables. ■ Ask your students to describe the distribution. Classroom Activity 6-2 Observing the Normal Curve ■ Have students observe the normal curve in the world around them. ■ For this activity, it would be best if students continue to try to find normal curves in their everyday life outside of the classroom and bring in their observations to discuss in class. II.

Standardization, z Scores, and the Normal Curve 1. Standardization converts individual scores to standard scores for that which we know the mean, standard deviation, and percentiles. To standardize, we convert individual scores from different normal distributions to a shared normal distribution with a known mean, standard deviation, and percentiles. 2. To do this, we will convert our raw scores to z scores. A z score is the number of standard deviations a particular score is from the mean. 3. The z score is part of the z distribution, a normal distribution of z scores or standardized scores. 4. The standardized z distribution allows us to transform raw scores into standardized scores called z scores. We can also transform z scores back into raw scores. The standardization also allows us to compare z scores

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to each other even when the z scores represent raw scores on different scales. In addition, we can use the z distribution to transform z scores into percentiles that are more easily understood. > Discussion Question 6-1 What is the purpose of standardization? How is standardization useful? Your students’ answers should include: ■ The purpose of standardization is to compare scores from different normal distributions to a shared normal distribution with a known mean, standard deviation, and percentiles. ■ Standardization is useful because it allows us to compare the raw scores on two different scales by converting them to z scores. Classroom Activity 6-3 Comparing Apples and Oranges ■ Have students use Handout 6-2, found at the end of this chapter, to examine the usefulness of standardization. ■ Have them also use standardization and the handout to compare amounts that use different units. 5. The normal curve is symmetric such that 50% of scores lie below the mean and 50% lie above the mean. 6. We also know that 34% of scores fall between the mean and a z score of 1 as well as between the mean and a z score of –1. 7. Looking between z scores of 1 and 2 (or between –1 and –2), we know that 14% of scores will lie between these z scores. 8. Lastly, 2% of scores fall between the z scores of 2 and 3 (or –2 and –3). 9. Putting this together, we know that 68% of scores lie within 1 standard deviation of the mean, 96% lie within 2 standard deviations of the mean, and nearly all scores lie within 3 standard deviations of the mean. > Discussion Question 6-2 What percentage of scores lies within 1 standard deviation of the mean? within 2? within 3? Your students’ answers should include: ■ The percentage of scores that lie within 1 standard deviation of the mean is 34%. ■ The percentage of scores that lie within 2 standard deviations of the mean is 14%. ■ The percentage of scores that lie within 3 standard deviations of the mean is 2%. Classroom Activity 6-4 Make It Your Own: z Scores ■ Use the data from Classroom Activities 6-1 and 6-6 (height, weight, SAT, and GPA data).

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Assume that your class is the population and provide the students with the population mean. Then have the students calculate their own z scores.

The Central Limit Theorem 1. The central limit theorem refers to how a distribution of sample means is a more normal distribution than a distribution of scores, even when the population distribution is not normal. As sample size increases, a distribtuion of sample means more closely approximates a normal curve. 2. The central limit theorem demonstrates two important principles. First, repeated sampling will approximate a normal curve, even when the original population is not normally distributed. Second, a distribution of means is less variable than a distribution of individual scores. It is a distribution composed of many means that are calculated from all possible samples of a given size, all taken from the same population.

> Discussion Question 6-3 What is the central limit theorem? What are two principles of the central limit theorem? Your students’ answers should include: ■ The central limit theorem states that regardless of the shape of the population, the shape of the sampling distribution of the mean approximates a normal curve if the sample size is adequately large. ■ Two principles of the central limit theorem are that repeated sampling will more closely resemble a normal distribution, and that a distribution of means is less variable than a distribution of raw scores. Classroom Activity 6-5 Distribution of Scores vs. Means ■ Have students work in small groups, and give each group two six-sided dice. ■ Have students observe how the distribution changes from rolling the two dice a different number of times. ■ Next, have students take a distribution of means to further observe how distribution changes. 3. The distribution of means (which is symbolized as μM) is a distribution composed of many means that are calculated from all possible samples of a given size, all taken from the same population. 4. When creating a distribution of means rather than a distribution of scores, the spread of the distribution decreases. > Discussion Question 6-4 What is a distribution of means, and why does the spread of the distribution decrease with the distribution of means? Your students’ answers should include:

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A distribution of means is composed of the means of all possible random samples, and the mean of the sampling distribution of the mean always equals the mean of the population. ■ The spread of the distribution decreases with the distribution of means because extreme scores are balanced and averaged out by other scores. 5. In addition, larger sample sizes will shift a skewed distribution to a normal distribution of means from that population. 6. The standard error is the name for the standard deviation of a distribution of means and is symbolized by σM. 7. The standard error is calculated by the formula: σM = (σ/√N). ■

> Discussion Question 6-5 What is the standard error? How is it calculated? Your students’ answers should include: ■ The standard error is the standard deviation of the distribution of means. ■ The standard error is calculated by dividing the standard deviation of the population by the square root of the sample size, N. 8. As sample size increases, the mean of a distribution of means approaches the mean of the population of individual scores. 9. The standard error is smaller than the standard deviation of a distribution of scores. As sample size increases, the standard error becomes even smaller. 10. The shape of the distribution of means will have an approximately normal shape if the distribution of the population of individual scores has a normal shape or if the size of each sample that comprises the distribution is sufficiently large (at least 30). > Discussion Question 6-6 Why is it that the shape of the distribution of means will have an approximately normal shape if the size of each sample that comprises the distribution is sufficiently large? Your students’ answers should include: ■ The shape of the distribution of means will have an approximately normal shape if the size of each sample that comprises the distribution is sufficiently large because of the central limit theorem, which indicates that the shape of the sampling distribution of the mean approximates a normal curve if the sample size is adequately large. 11. The formula for the z statistic for a distribution of means is: z = (M – μM)/σM. Classroom Activity 6-6 Make It Your Own: Normal Distributions ■ Anonymously collect SAT and GPA data.

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Graph the data and have the students discuss the distributions. This activity can also be used as a homework assignment.

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Next Steps: The Normal Curve and Catching Cheaters 1. Statisticians can use principles based on the normal curve to determine when certain patterns are extreme as in the case of standardized testing in the Chicago Public School system or researchers who “play” with their data to beat the 5% cutoff. Additional Readings Jegerski, J. (1999). Probability distributions with real social judgment data. In Ludy, B., Nodine, B., Ernst, R. M., & Broeker, C. B. (Eds.), Activities handbook for the teaching of psychology (Vol. 4, pp. 77-79). Washington, DC: American Psychological Association. This chapter provides some useful background information on teaching the normal curve along with a classroom activity. Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses (Springer Tests in Statistics). New York: Springer Science + Business Media Inc. First published in 1959, this is a classic text on hypothesis testing, providing the fundamentals of the theory of hypothesis testing. The book is geared toward graduate students and researchers. Online Resources For demonstrations of the normal distribution using different means and standard deviations, see: http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity .html or http://psych.colorado.edu/~mcclella/java/zcalc.html

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HANDOUT 6-1: OBSERVING THE NORMAL CURVE Directions: For this activity, you will observe the normal curve in the world around you. Use the textbook and your creativity to observe specific instances of the normal curve, and use the chart below to help you keep track of your observations. For each observation, take note of when it occurred and where. Then, briefly describe what you observed, and explain why it would be an example of the normal curve. Observation Number

When

Where

Describe and Explain

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HANDOUT 6-2: COMPARING APPLES AND ORANGES Directions: The purpose of this exercise is not to compare apples and oranges, per se, but instead to use the technique of standardization to compare two different values. In this case, we will be comparing whether the New York Yankees baseball team is better at baseball than the New York Giants football team is at football. To explore this concept, answer each of the following questions.

1. In words, describe the steps you would need to take if you were given raw scores for each of the teams along with descriptive statistics for the leagues as a whole.

2. For the 2009 season, imagine the Yankees had a mean batting average of .285 with a standard deviation of .062, whereas the mean for all teams in major league baseball was a batting average of .262 with a standard deviation of .014. Imagine that for the New York Giants, they had a mean of 22.2 points per game with a standard deviation of 5.13. In comparison, the rest of the league had a mean of 20.67 points per game with a standard deviation of 4.27. With this information, determine which team is better.

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HANDOUT 6-3: DISTRIBUTION OF SCORES VS. MEANS Directions: For this activity, your instructor should have given you two dice for your group. Answer each of the questions below to observe the difference between a distribution of scores and a distribution of means. 1. Roll the two dice 10 times, recording the sum of the two dice each time. Afterward, roll the two dice again 30 times, recording the sum of the two dice each time. For each set of rolls, create a histogram to illustrate the patterns of rolls.

2. How are the two histograms different? Why do you think they look different?

3. For each of the two sets of rolls, take the mean of three random sums 10 times and record these means. Then, repeat this, but take the mean of three random sums 30 times and record these sums. For each set of means, draw a histogram.

4. How are the histograms different? Do you observe a difference from taking the mean 10 times compared to taking it 30 times? Why do you think there is a difference?