In your opinion, does one do a better job of representing the proportion of the area under the standard curve? Give an example that illustrates your answer.
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4Applying z to Groups
Victor Faile/Corbis
Chapter Learning Objectives: After reading this chapter, you should be able to do the following:
1. Describe the distribution of sample means.
2. Explain the central limit theorem.
3. Analyze the relationship between sample size and confidence in normality.
4. Calculate and explain z test results.
5. Explain statistical significance.
6. Calculate and explain confidence intervals.
7. Explain how decision errors can affect statistical analysis.
8. Calculate the z test using Excel.
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Section 4.1 Distribution of Sample Means
Introduction As we noted at the end of Chapter 3, researchers are generally more interested in groups than in individuals. Individuals can be highly variable, and what occurs with one is not necessarily a good indicator of what to expect from someone else. What occurs in groups, on the other hand, can be very helpful in understanding the nature of the entire popu- lation. A Google search indicates that the suicide rate is higher among dentists than it is among those of many other professions. If we wanted to experiment with some therapy designed to relieve depressive symptoms among dentists, we would be more confident observing how a group of 50 dentists responds than in examining results from just one. This chapter will use the material from the first three chapters to begin analyzing people in groups.
Noting that many of the characteristics that interest behavioral scientists are normally dis- tributed in a population implies that some characteristics are not. Since samples can never exactly emulate their populations, it may not be clear in the midst of a particular study when data are normally distributed. This uncertainty potentially poses a problem: we may wish to use the z transformation and Table B.1 of Appendix B in our analysis, but Table B.1 is based on the normality assumption. If the data are not normal, where does that leave the related analysis?
4.1 Distribution of Sample Means What options do researchers have if they are suspi- cious about data normality? One important answer is the distribution of sample means, so named because the scores that constitute the distribution are the means of samples rather than individual scores.
Note that the descriptor population means all pos- sible members of a defined group. Recall that the frequency distribution—the bell-shaped curve rep- resenting the population—was a figure based on the individual measures sampled one subject at a time. In discussing the frequency distribution, we assumed that we would measure each individual on some trait, and then plot each individual score. Instead of selecting each individual in a population one at a time, suppose a researcher
1. selects a group with a specified size; 2. calculates the sample mean (M) for each
group; 3. plots the value of M (rather than the value
of each score) in a frequency distribution; 4. and continues doing this until the popula-
tion is exhausted.
iStockphoto/Thinkstock
A population is all members of a defined group, such as all voters in a county.
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Section 4.1 Distribution of Sample Means
How would plotting group scores rather than individual ones affect the distribution? Would the end result still be a population? The answer to the second question is yes: because every member is included, it is still a population. Whether a population is measured individually or as members of a group is incidental, as long as all are included.
Perhaps researchers are interested in language development among young children and wish to measure mean length of utterance (MLU) in a county population. Whether the researchers mea- sure and plot MLU for each child in a county’s Head Start program or plot the mean MLU for every group of 25 in the program, the result is population data for Head Start learners for that county.
The Central Limit Theorem The answer to the question “how would the distribution be affected?” is a little more involved, but it is important to nearly everything we do in statistical analysis. It involves what is called the central limit theorem:
If a population is sampled an infinite number of times using sample size n and the mean (M) of each sample is determined, then the multiple M measures will take on the characteristics of a normal distribution, whether or not the original population of individuals is normal.
Take a minute to absorb this. A population of an infinite number of sample means drawn from one population will reflect a normal distribution whatever the nature of the original distribu- tion. A healthy skepticism prompts at least two questions: 1) How would we prove whether this is true since no one can gather an infinite number of samples? and 2) Why does sampling in groups rather than as individuals affect normality?
Although prove is too strong a word, we can at least provide evidence for the effect of the central limit theorem using an example. Perhaps a psychologist is working with 10 people on their resistance to change, their level of dogmatism. Technically, because 10 constitutes the entire group, the population is N 5 10. Recall that N refers to the number in a population. Even with a small population we cannot have an infinite number of samples, of course, but for the sake of the illustration we will assume that
• dogmatism scores are available for each of the 10 people; • the data are interval scale; • the scores range from 1 to 10; and • each person receives a different score.
So with N 5 10, the scores are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Figure 4.1 depicts a frequency distribu- tion of those 10 scores.
The distribution in Figure 4.1 is not normal. With R 5 10 21 5 9 and s 5 3.028 (a calculation worth checking), the distribution is extremely platykurtic (i.e., flatter than normal); the range is less than 3 times the value of the standard deviation rather than the approximately 6 times associated with normal distributions. There is either no mode or there are 10 modes, neither of which suggests normality. We can illustrate the workings of the central limit theorem with a procedure Diekhoff (1992) used. We will use samples of n 5 2, and make the example man- ageable by using one sample for each possible combination of scores in samples of n 5 2 from the population, rather than an infinite number of samples.
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Section 4.1 Distribution of Sample Means
Table 4.1 lists all the possible combinations of two scores from values 1–10. Ninety combina- tions of the 10 dogmatism scores are possible. The larger the sample size, the more readily it demonstrates the tendency toward normality, but all combinations of (for example) three scores would result in a very large table.
Table 4.1: All possible combinations of the integers 1–10
1, 2 2, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, 1 9, 1 10, 1
1, 3 2, 3 3, 2 4, 2 5, 2 6, 2 7, 2 8, 2 9, 2 10, 2
1, 4 2, 4 3, 4 4, 3 5, 3 6, 3 7, 3 8, 3 9, 3 10, 3
1, 5 2, 5 3, 5 4, 5 5, 4 6, 4 7, 4 8, 4 9, 4 10, 4
1, 6 2, 6 3, 6 4, 6 5, 6 6, 5 7, 5 8, 5 9, 5 10, 5
1, 7 2, 7 3, 7 4, 7 5, 7 6, 7 7, 6 8, 6 9, 6 10, 6
1, 8 2, 8 3, 8 4, 8 5, 8 6, 8 7, 8 8, 7 9, 7 10, 7
1, 9 2, 9 3, 9 4, 9 5, 9 6, 9 7, 9 8, 9 9, 8 10, 8
1, 10 2, 10 3, 10 4, 10 5, 10 6, 10 7, 10 8, 10 9, 10 10, 9
Figure 4.1: A frequency distribution for the scores 1 through 10: Each score
occurring once
A frequency distribution of ten scores, each with a different value. This type of distribution, which is not normal, is highly platykurtic.
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