Homework answers / question archive / The Standard Normal Distribution and z Scores 3 Keren Su/Corbis Chapter Learning Objectives After reading this chapter, you should be able to do the following: 1

The Standard Normal Distribution and z Scores 3 Keren Su/Corbis Chapter Learning Objectives After reading this chapter, you should be able to do the following: 1

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The Standard Normal Distribution and z Scores 3 Keren Su/Corbis Chapter Learning Objectives After reading this chapter, you should be able to do the following: 1. Identify the characteristics of the standard normal distribution. 2. Demonstrate the use of the z transformation. 3. Determine the percent of a population above a point, below a point, and between two points on the horizontal axis of a normal distribution. 4. Calculate z scores using Excel. 5. Describe alternative standard scores. 6. Demonstrate the use of the modified standard score. 61 © 2016 Bridgepoint Education, Inc. All rights reserved. Not for resale or redistribution. tan82773_03_ch03_061-090.indd 61 3/3/16 9:56 AM Introduction ? Introduction The data that describe characteristics of groups come from either samples or populations, explained in the first two chapters. By way of reminder, recall that populations include all possible members of any specified group. All university students, all psychology majors, all residents of Orange County, and all left-handed male tennis players in their 20s are each descriptions of a population. We rely on Greek letters, such as µ for the mean and σ for the standard deviation, to distinguish population parameters from the statistics that describe samples. (The word parameter indicates a characteristic of a population.) Remove one or more individuals from any population, and the resulting group is a sample. As we were describing populations, we noted that some are “normally distributed.” These characteristics indicate normality: (a) data distributions are symmetrical, (b) all the measures of central tendency have very similar values, and (c) the value of the standard deviation is about one-sixth of the range. Data normality does not simply mean that the frequency distribution will appear as a bellshaped curve; it means that predictable proportions of the entire population will occur in specified regions of the distribution, and this holds for all normal data distributions. For example, the region under a normal curve from the mean of the population to one standard deviation below the mean always includes 34.13% of the area under the curve. Because ­normal ­distributions are symmetrical, from the mean to one standard deviation above the mean also includes 34.13%, so from 11σ or 21σ includes about 68.26% of the area under the curve in any normally distributed population. As long as the data are normally distributed, those percentages hold true. Since many mental characteristics are normally distributed, researchers can know a good deal about such a characteristic without actually gathering the data and doing the analysis. Whether the characteristic is intelligence, achievement motivation, anxiety, or any other normally distributed characteristics, the proportion of the distribution within 11 or 21 standard deviation from the mean will be the same: • If a particular intelligence scale has µ 5 100 and σ 5 15, about 68% of any general population will have intelligence scores between 85 and 115. • Likewise, if an achievement motivation scale has µ 5 40 and σ 5 8, about 2/3 of any population will have achievement motivation scores from 32 to 48. • And for an anxiety measure with µ 5 25 and σ5 5, about 68% of any general population will have scores between 20 and 30. The consistency in the way so many characteristics are distributed affords a good deal of interpretive power. Anyone who needs information about the likelihood of individuals scoring in certain areas of a distribution has an advantage when data are normally distributed. In addition to the 68% of any general population likely to score between 11σ and 21σ, • from µ to 12σ is about 47.72% of the population, so about 95% (2 3 47.72) of the people in any general population will have intelligence scores between 70 (100 2 30) and 130 (100 1 30). • from 13σ (49.87%) to 23σ includes nearly everyone in any normally distributed population (2 3 49.87 5 99.74). These observations emphasize that, sometimes, isolated bits of data can be quite informative. When a 12-year-old with an intelligence score of 170 pops up on YouTube, it is immediately © 2016 Bridgepoint Education, Inc. All rights reserved. Not for resale or redistribution. tan82773_03_ch03_061-090.indd 62 3/3/16 9:56 AM Section 3.1 A Primer in Probability apparent that this is a very unusual child. An intelligence score of that magnitude is about 4.667σ (170 2 100 5 70; 70 4 15 5 4.667) beyond the mean of the general population. If from 13σ to 23σ includes more than 99% of the population, from 14.667σ to 24.667σ must include all but the utmost extreme scores. We obtain an even better context for how common (or uncommon) particular measures may be when we can determine the precise probability of their occurrence. 3.1 A Primer in Probability Scholars, data analysts, and in fact people on the whole are rarely interested in outcomes that occur every time. If everyone had an intelligence score of 170, no one would pay any attention to someone with such a score. The fact that we know it to be uncommon is what piques our curiosity. If we are not interested in events that always occur, neither do we closely follow events that never occur. If no one had ever had an intelligence score of 170, probably no one would wonder about what such a score means for the person who has it. The things that occur some of the time, however, intrigue us. The “some of the time” indicates that the event has some ­probability, or likelihood, of occurrence. • What is the probability that those newlyweds will divorce? • How likely is Germany to win the World Cup? • What is the probability that an earthquake will occur on a particular day for someone who lives near the San Andreas Fault? • What is the probability of an IRS audit for one taxpayer? Because all of the items listed have happened in the past and because their occurrence is important to at least someone, people are interested in the probability of those occurrences whether or not they use the language of probability. When stated numerically, probability values range from 0 to 1.0. Something with a probability of zero (p 5 0) never occurs. On the other hand, p 5 1.0 indicates that the event occurs every time, and p 5 0.5 indicates that the event occurs 50% of the time. As that last point indicates, percentages can be converted to probability values. Dividing the percentage of times an event occurs by 100 indicates the associated probability of the event. Returning to the intelligence scores, we see that because about 68% of the population has intelligence scores between 85 and 115, the probability (p) that someone selected at random from the general population will have a score somewhere between 85 and 115 is 0.68 (68.26/100, if the result is rounded to two decimal places). Joseph Sohm/Visions of America/Corbis The probability that something will occur, such as how likely it is that our favorite baseball team will win the World Series, intrigues us and is an important component in the decisionmaking process. © 2016 Bridgepoint Education, Inc. All rights reserved. Not for resale or redistribution. tan82773_03_ch03_061-090.indd 63 3/3/16 9:57 AM Section 3.2 The Standard Normal Distribution What is the probability that someone selected at random from the general population will have an intelligence score of 100 or lower? Because 100 is the mean for intelligence scores, and because 50% of the population occur at the mean or below, p 5 0.5. What is the probability that someone selected at random will have an intelligence score higher than 115? First, we noted earlier that 34.13% of the population falls between the mean, µ, and one standard deviation above the mean at σ 5 11.0 in any normally distributed population. In terms of intelligence score values, that is the region between scores of 100 and 115. Since 50% of any normally distributed population will occur at the mean and above, if we subtract from 50% that portion between the mean and one standard deviation above the mean, the remainder will be the portion of the distribution above 115: 50% 2 34.13% 5 15.87%; that is, 15.87% of all intelligence scores in a normally distributed population will occur above 115. Dividing by 100 (15.87/100 5 0.1587) and rounding the result to two decimal places produces the probability p 5 0.16. By the same logic, because a score of 85 is one standard deviation below the mean, the probability p 5 0.16 means that someone selected at random from the population will score below 85. If we combine the two outcomes, the probability is p 5 0.32 that someone from the population will score either below 85 or above 115. Consider the number line shown in Figure 3.1. Figure 3.1: Standard deviations for intelligence scores The number line shows the portion of scores that fall within two standard deviations above and below the mean. If M 5 100, we can know the probability of someone scoring below 85 or above 115. 34% 16% –2σ –1σ 70 85 34% M 100 Intelligence scores 16% + 1σ + 2σ 115 130 If this number line represents all intelligence scores ranging from two standard deviations below to two standard deviations above the mean, we can see the percentages of the population that will have scores in the designated areas. Using the percentages and dividing by 100 indicates the probability of a score in any of the designated areas. Recall that the lowest probability for any value is zero (p 5 0). If p 5 0, then the event or outcome never occurs. There is no such thing as a negative probability. 3.2 The Standard Normal Distribution Not all populations are normally distributed. Home sales are usually reported in terms of the median price of a home, and salary data are likewise reported as median values. Those cases © 2016 Bridgepoint Education, Inc. All rights reserved. Not for resale or redistribution. tan82773_03_ch03_061-090.indd 64 3/3/16 9:57 AM Section 3.2 The Standard Normal Distribution use the medians because the related populations are very unlikely to be normally distributed and, as a measure of central tendency, medians are less affected by extreme values than are means. A few very high salaries or home values create positive skew in the resulting distribution. In contrast, when it comes to, say, mental characteristics such as intelligence, achievement motivation, problem-solving ability, verbal aptitude, reading comprehension, and so on, population data are often normally distributed. Hello Lovely/Corbis Although there are many normal distributions all When evaluating information about having the same proportions, each has different people’s characteristics, keep in descriptive values. An intelligence test might have mind that data are often normally µ 5 100 and σ 5 15 points. A nationally adminisdistributed. tered reading test might have a mean of 60 and a standard deviation of 8. These different parameters can make it difficult to compare one individual’s performance across multiple measures. As one author noted regarding scores from the Wechsler Intelligence Test for Children (WISC), “A raw score of 5 on one [sub]test will not have the same meaning as a raw score 5 on another [sub]test” (Brock, 2010). One way to resolve this interpretation problem is to convert the scores from different distributions into a common metric, or measurement system. If researchers alter scores from different distributions so that they both fit the same distribution, they can compare scores directly. A researcher can compare them directly to determine, for example, on which test an individual scored highest. Such comparisons are one of the purposes of the standard normal distribution. The standard normal distribution looks like all other normal distributions—from the mean to 11 standard deviation includes 34.13% of the distribution, for example. What separates it from the others is that in the standard normal distribution, the mean is always 0, and the standard deviation is always 1.0 (Figure 3.2). Other distributions may have fixed values for their means and standard deviations, but here µ is always 0 and σ is always 1.0. Figure 3.2: The standard normal distribution In the standard normal distribution, the mean is always 0, and the standard deviation is always 1.0. The Mean = 0 The Standard Deviation = 1.0 –3 –2 –1 0 +1 +2 +3 © 2016 Bridgepoint Education, Inc. All rights reserved. Not for resale or redistribution. tan82773_03_ch03_061-090.indd 65 3/3/16 9:57 AM The Standard Normal Distribution Section 3.2 The Standard Normal, or z, Distribution Although various normal distributions have different means and standard deviations, they all mirror each other in terms of how much of their populations occur in particular regions. The standard normal distribution’s advantage is that the proportions of the whole that occur in the various regions of the distribution have been calculated. That means that if data from any normal distribution are made to conform to the standard normal distribution, we can answer questions about what is likely to occur in virtually any area of the distribution, such as how likely it is to score 2.5 standard deviations below the mean on a particular test, or what percentage of the entire population will likely occur between two specified points. All such questions can be answered when adapting normal data to the characteristics of the standard normal distribution. Individual scores in the standard normal distribution are called z scores, which is why the standard normal distribution is often called “the z distribution.” The formula used to turn scores from any normal distribution into scores that conform to the standard normal distribution is the z transformation: Formula 3.1 z5 x2M s where z is a score in the standard normal distribution, x is the score from the original distribution (often called a “raw” score), M is the mean of the scores before the original distribution, and s is the standard deviation of the scores from the original distribution. Because normality is characteristic of only very large groups, samples will rarely be normal. However, we can apply the z transformation to sample data when there is reason to believe that the population from which the sample was drawn is normally distributed. This is what Formula 3 reflects. The M and s indicate that the data involved are sample data. In those situations where an analyst has access to population data—a social worker has all the data for those served by Head Start in a particular county, for example—µ replaces M and σ replaces s in the formula. With either sample or population data, the transformation is from data that can have any mean and standard deviation to a distribution where the mean will always equal 0 and the standard deviation will always equal 1.0. To turn raw scores into z scores, perform the following steps: 1. Determine the mean and standard deviation for the data set. 2. Subtract the mean of the data set from each score to be transformed. 3. Divide the difference by the standard deviation of the data set. For example, consider a psychologist interested in the level of apathy among potential voters regarding mental health issues that affect the community. Scores on the Summary of WHo’s Apathetic Test (the SoWHAT for short), an apathy measure, are gathered for 10 registered voters: 5, 6, 9, 11, 15, 15, 17, 20, 22, 25 What’s the z score for someone who has an apathy score of 11? • Verify that for these 10 scores, M 5 14.5 and s 5 6.737. © 2016 Bridgepoint Education, Inc. All rights reserved. Not for resale or redistribution. tan82773_03_ch03_061-090.indd 66 3/3/16 9:57 AM Section 3.2 The Standard Normal Distribution • The z score equivalent for an apathy score of 11 is z5 x2M 11 2 14.5 5 5 20.5195 s 6.737 An apathy score of 11 translates into a z score of 20.5195. Because the mean of the z distribution is 0 and the standard deviation in the z distribution is 1.0, where would a score of 20.5195 occur on the horizontal axis of the data distribution? It would be a little over half a standard deviation below the mean, right? Figure 3.3 shows the z distribution and the point about where a raw score of 11 occurs in this distribution once it is transformed into a z score. It is important to know that the z transformation does not make data normal. Calculating z scores does not alter the distribution; it just makes them fit a distribution where the mean is 0 and the standard deviation is 1.0. Evaluating skew and kurtosis must allow the analyst to assume that the data are normal before using the z transformation. With a mean of 0 in the standard normal distribution, half of all z scores—all the scores below the mean—are going to be negative. A raw score of 11 from the SoWHAT data is lower than the mean, which was M 5 14.5, so it has a negative z value (20.5195). Besides indicating by its sign whether the z score is above or below the mean, the value of the z score indicates how far from the mean the z score is in standard deviations. If a score had a z value of 1.0, it would indicate that the score is one standard deviation above the mean. The z score for the raw score of 11 was 20.5195, indicating that it is just over half a standard deviation below the mean. This ease of interpretation is one of Try It!: #1 the great values of z scores: the sign of the score indiHow many standard deviations from cates whether the associated raw score was above or the mean of the distribution is a z score below the mean, and the value of the score indicates of 1.5? how far from the mean the raw score falls, in standard deviation units (Fischer and Milfont, 2010). Figure 3.3: Location of a score on the z distribution Half of all z scores will fall below the mean, resulting in a negative value. A score of z 5 20.5195 is slightly less than one-half a standard deviation below the mean. –3z –2z –1z 0 +1z +2z +3z z = – 0.5195 © 2016 Bridgepoint Education, Inc. All rights reserved. Not for resale or redistribution. tan82773_03_ch03_061-090.indd 67 3/3/16 9:57 AM Section 3.2 The Standard Normal Distribution Comparing Scores from Different Instruments Consider another application of the standard normal distribution. A counselor has intelligence and reading scores for the same person and wishes to know on which measure the individual scored higher. Table 3.1 shows the data for the two tests. On the intelligence test, the individual scored 105, and on the reading test, the individual scored 62. Table 3.1: Reading and intelligence test results Test Intelligence Reading Mean Standard deviation 100 15 60 8 If the counselor transforms both scores to make them fit the standard normal distribution, they can be compared directly. The z for the intelligence score is z5 x2M 105 2 100 5 5 0.333 s 15 The z for the reading test score is z5 x2M 62 2 60 5 5 0.250 s 8 The intelligence score of 105 and the reading score of 62 are difficult to compare because they belong to different distributions with different means and standard deviations. When both are transformed to fit the standard normal distribution, an analyst can directly compare scores. The larger z value for intelligence makes it clear that individual scored higher in intelligence than in reading. Expanding the Use of the z Distribution Because the standard normal distribution is a normal distribution, we know that predictable proportions of its population will occur in specific areas. As we noted earlier, however, those proportions are known in great detail for the z distribution because this population is so often used to answer detailed questions about the likelihood of particular outcomes. Table 3.2 indicates how much of the entire population is above or below all of the most co...