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2.6 Numerical Measures of Relative Standing

Teaching Tip

Use the SAT and ACT college entrance examinations as an example of the need for measures of relative standing. Note that all test scores reported contain a percentile measurement for use in comparison.

We’ve seen that numerical measures of central tendency and variability describe the general nature of a quantitative data set (either a sample or a population). In addition, we may be interested in describing the relative quantitative location of a particular measurement within a data set. Descriptive measures of the relationship of a measurement to the rest of the data are called measures of relative standing.

One measure of the relative standing of a measurement is its percentile ranking, or percentile score. For example, suppose you scored an 80 on a test and you want to know how you fared in comparison with others in your class. If the instructor tells you that you scored at the 90th percentile, it means that 90% of the grades were lower than yours and 10% were higher. Thus, if the scores were described by the relative frequency histogram in Figure 2.26, the 90th percentile would be located at a point such that 90% of the total area under the relative frequency histogram lies below the 90th percentile and 10% lies above. If the instructor tells you that you scored in the 50th percentile (the median of the data set), 50% of the test grades would be lower than yours and 50% would be higher.

Figure 2.26

Location of 90th percentile for test grades

Percentile rankings are of practical value only with large data sets. Finding them involves a process similar to the one used in finding a median. The measurements are ranked in order, and a rule is selected to define the location of each percentile. Since we are interested primarily in interpreting the percentile rankings of measurements (rather than in finding particular percentiles for a data set), we define the pth percentile of a data set as follows.

For any set of n measurements (arranged in ascending or descending order), the pth percentile is a number such that p% of the measurements fall below that number and $(100 - p) %$ $(100 - p) %$ fall above it.

Teaching Tip

Use the median to illustrate that students have already been exposed to the 50th percentile of a data set.

WLTASK Example 2.14 Finding and Interpreting Percentiles—The “Water-Level Task”

Problem

Refer to the water-level task deviations for the 40 subjects in Table 2.4. An SPSS printout describing the data is shown in Figure 2.27. Locate the 25th percentile and 95th percentile on the printout, and interpret the associated values.

Solution

Both the 25th percentile and 95th percentile are highlighted on the SPSS printout. The values in question are 2.0 and 24.8, respectively. Our interpretations are as follows: 25% of the 40 deviations fall below 2.0 and 95% of the deviations fall below 24.8.

Figure 2.27

SPSS percentile statistics for water-level task deviations

Look Back

The method for computing percentiles with small data sets varies according to the software used. As the sample size increases, the percentiles from the different software packages will converge to a single number.

Now Work Exercise 2.115

Percentiles that partition a data set into four categories, each category containing exactly 25% of the measurements, are called quartiles. The lower quartile (Q_L) is the 25th percentile, the middle quartile (M) is the median or 50th percentile, and the upper quartile (Q_U) is the 75th percentile, as shown in Figure 2.28. Therefore, in Example 2.14, we have (from the SPSS printout, Figure 2.27) $Q_{L} = 2.0, M = 7.0$ $Q_{L} = 2.0, M = 7.0$ , and $Q_{U} = 10.0$ $Q_{U} = 10.0$ . Quartiles will prove useful in finding unusual observations in a data set (Section 2.7).

Figure 2.28

The quartiles for a data set

The lower quartile (Q_L) is the 25th percentile of a data set. The middle quartile (M) is the median or 50th percentile. The upper quartile (Q_U) is the 75th percentile.

Another measure of relative standing in popular use is the z-score. As you can see in the following definition, the z-score makes use of the mean and standard deviation of a data set in order to specify the relative location of the measurement.

The sample z-score for a measurement x is

[&z|=|*frac*{x|-|*orule*{x}}{s} &]

z = \frac{x - \bar{x}}{s}

$z = \frac{x - \bar{x}}{s}$

The population z-score for a measurement x is

[&z|=|*frac*{x|-||mu|}{|sig|} &]

z = \frac{x - μ}{σ}

$z = \frac{x - μ}{σ}$

Teaching Tip

The z-score is the measure of relative standing that will be used extensively with the normal distribution later. It is helpful if the student becomes familiar with the z-score concept now.

Note that the z-score is calculated by subtracting $\bar{x}$ $\bar{x}$ (or $μ$ $μ$ ) from the measurement x and then dividing the result by s (or $σ$ $σ$ ). The final result, the z-score, represents the distance between a given measurement x and the mean, expressed in standard deviations.

Example 2.15 Finding a `z`-Score

Problem

Suppose a sample of 2,000 high school seniors’ verbal SAT scores is selected. The mean and standard deviation are

[&*orule*{x}|=|550|em| s|=|75 &]
$\bar{x} = 550 s = 75$ $\bar{x} = 550 s = 75$

Suppose Joe Smith’s score is 475. What is his sample z-score?

Solution

Joe Smith’s verbal SAT score lies below the mean score of the 2,000 seniors, as shown in Figure 2.29.

Figure 2.29

Verbal SAT scores of high school seniors

We compute

[&z|=|*frac*{x|-|*orule*{x}}{s}|=|*frac*{475|-|550}{75}|=||minus|1.0 &]
$z = \frac{x - \bar{x}}{s} = \frac{475 - 550}{75} = - 1.0$ $z = \frac{x - \bar{x}}{s} = \frac{475 - 550}{75} = - 1.0$

which tells us that Joe Smith’s score is 1.0 standard deviation below the sample mean; in short, his sample z-score is $- 1.0$ $- 1.0$ .

Look Back

The numerical value of the z-score reflects the relative standing of the measurement. A large positive z-score implies that the measurement is larger than almost all other measurements, whereas a large negative z-score indicates that the measurement is smaller than almost every other measurement. If a z-score is 0 or near 0, the measurement is located at or near the mean of the sample or population.

Now Work Exercise 2.117

We can be more specific if we know that the frequency distribution of the measurements is mound shaped. In this case, the following interpretation of the z-score can be given:

Interpretation of `z`-Scores for Mound-Shaped Distributions of Data

Approximately 68% of the measurements will have a z-score between $- 1$ $- 1$ and 1.
Approximately 95% of the measurements will have a z-score between $- 2$ $- 2$ and 2.
Approximately 99.7% (almost all) of the measurements will have a z-score between $- 3$ $- 3$ and 3.

Teaching Tip

Draw a picture of a mound-shaped distribution and locate the z-scores −3, −2, −1, 0, 1, 2 and 3 on it to help students understand what the z-score measures.

Note that this interpretation of z-scores is identical to that given by the empirical rule for mound-shaped distributions (Rule 2.2). The statement that a measurement falls into the interval from $(μ - σ)$ $(μ - σ)$ to $(μ + σ)$ $(μ + σ)$ is equivalent to the statement that a measurement has a population z-score between $- 1$ $- 1$ and 1, since all measurements between $(μ - σ)$ $(μ - σ)$ and $(μ + σ)$ $(μ + σ)$ are within one standard deviation of $μ$ $μ$ . These z-scores are displayed in Figure 2.30.

Figure 2.30

Population `z`-scores for a mound-shaped distribution

Exercises 2.114–2.131

Understanding the Principles

2.114 For a quantitative data set
1. What is the 50th percentile called?
  
  Median
2. Define Q_L.
3. Define Q_U.
2.115 Give the percentage of measurements in a data set that are above and below each of the following percentiles:
1. 75th percentile
  
  25%, 75%
2. 50th percentile
  
  50%, 50%
3. 20th percentile
  
  80%, 20%
4. 84th percentile
  
  16%, 84%
2.116 For mound-shaped data, what percentage of measurements have a z-score between $- 2$ $- 2$ and 2?

$\approx 95 %$ $\approx 95 %$

Learning the Mechanics

2.117 Compute the z-score corresponding to each of the following values of x:
1. $x = 40, s = 5, \bar{x} = 30$ $x = 40, s = 5, \bar{x} = 30$
  
  2
2. $x = 90, μ = 89, σ = 2$ $x = 90, μ = 89, σ = 2$
  
  .5
3. $μ = 50, σ = 5, x = 50$ $μ = 50, σ = 5, x = 50$
  
  0
4. $s = 4, x = 20, \bar{x} = 30$ $s = 4, x = 20, \bar{x} = 30$
  
  $- 2.5$ $- 2.5$
5. In parts a–d, state whether the z-score locates x within a sample or within a population.
6. In parts a–d, state whether each value of x lies above or below the mean and by how many standard deviations.
2.118 Compare the z-scores to decide which of the following x values lie the greatest distance above the mean and the greatest distance below the mean:
1. $x = 100, μ = 50, σ = 25$ $x = 100, μ = 50, σ = 25$
  
  2
2. $x = 1, μ = 4, σ = 1$ $x = 1, μ = 4, σ = 1$
  
  $- 3$ $- 3$
3. $x = 0, μ = 200, σ = 100$ $x = 0, μ = 200, σ = 100$
  
  $- 2$ $- 2$
4. $x = 10, μ = 5, σ = 3$ $x = 10, μ = 5, σ = 3$
  
  1.67
2.119 Suppose that 40 and 90 are two elements of a population data set and that their z-scores are $- 2$ $- 2$ and 3, respectively. Using only this information, is it possible to determine the population’s mean and standard deviation? If so, find them. If not, explain why it’s not possible.

$μ = 60, σ = 10$ $μ = 60, σ = 10$

Applying the Concepts—Basic

2.120 Math scores of twelfth graders. According to the National Center for Education Statistics (2013), scores on a mathematics assessment test for United States 12th graders have a mean of 153, a 25th percentile of 111, a 75th percentile of 177, and a 90th percentile of 197. Interpret each of these numerical descriptive measures.
2.121 Drivers stopped by police. According to the Bureau of Justice Statistics (Sept. 2013), 79% of all licensed drivers stopped by police are 25 years or older. Give a percentile ranking for the age of 25 years in the distribution of all ages of licensed drivers stopped by police.

21st percentile
2.122 Stability of compounds in new drugs. Refer to the Pfizer Global Research and Development study (reported in ACS Medicinal Chemistry Letters, Vol. 1, 2010) of the metabolic stability of drugs, Exercise 2.35 (p. 49). Recall that the stability of each of 416 drugs was measured by the fup/fumic ratio.
1. In Exercise 2.35, you determined the proportion of fup/fumic ratios that fall above 1. Use this proportion to determine the percentile rank of 1.
  
  96.6
2. In Exercise 2.35, you determined the proportion of fup/fumic ratios that fall below .4. Use this proportion to determine the percentile rank of .4.
  
  69.5
SANIT 2.123 Sanitation inspection of cruise ships. Refer to the sanitation levels of cruise ships presented in Exercise 2.41 (p. 51).
1. Give a measure of relative standing for the Nautilus Explorer’s score of 74. Interpret the result.
  
  z = −3.83
2. Give a measure of relative standing for the Star Princess’ score of 92. Interpret the result.
  
  z = −.46
2.124 Motivation of drug dealers. Refer to the Applied Psychology in Criminal Justice (Sept. 2009) study of convicted drug dealers’ motivations, Exercise 2.102 (p. 77). Recall that the sample of drug dealers had a mean Wanting Recognition (WR) score of 39 points, with a standard deviation of 6 points.
1. Find and interpret the z-score for a drug dealer with a WR score of 30 points.
  
  $- 1.5$ $- 1.5$
2. What proportion of the sampled drug dealers had WR scores below 39 points? (Assume the distribution of WR scores is mound shaped and symmetric.)
  
  .50

Applying the Concepts—Intermediate

NZBIRDS 2.125 Extinct New Zealand birds. Refer to the Evolutionary Ecology Research (July 2003) study of the patterns of extinction in the New Zealand bird population, first presented in Exercise 2.24 (p. 42). Again, consider the data on the egg length (measured in millimeters) for the 132 bird species.
1. Find the 10th percentile for the egg length distribution and interpret its value.
  
  23
2. The moas (P. australis) is a bird species with an egg length of 205 millimeters. Find the z-score for this species of bird and interpret its value.
  
  3.28
2.126 Lead in drinking water. The U.S. Environmental Protection Agency (EPA) sets a limit on the amount of lead permitted in drinking water. The EPA Action Level for lead is .015 milligram per liter (mg/L) of water. Under EPA guidelines, if 90% of a water system’s study samples have a lead concentration less than .015 mg/L, the water is considered safe for drinking. I (coauthor Sincich) received a report on a study of lead levels in the drinking water of homes in my subdivision. The 90th percentile of the study sample had a lead concentration of .00372 mg/L. Are water customers in my subdivision at risk of drinking water with unhealthy lead levels? Explain.

No
2.127 Voltage sags and swells. The power quality of a transformer is measured by the quality of the voltage. Two causes of poor power quality are “sags” and “swells.” A sag is an unusual dip, and a swell is an unusual increase in the voltage level of a transformer. The power quality of transformers built in Turkey was investigated in Electrical Engineering (Vol. 95, 2013). For a sample of 103 transformers built for heavy industry, the mean number of sags per week was 353 and the mean number of swells per week was 184. Assume the standard deviation of the sag distribution is 30 sags per week and the standard deviation of the swell distribution is 25 swells per week. Suppose one of the transformers is randomly selected and found to have 400 sags and 100 swells in a week.
1. Find the z-score for the number of sags for this transformer. Interpret this value.
  
  1.57
2. Find the z-score for the number of swells for this transformer. Interpret this value.
  
  −3.36
2.128 Blue versus red exam study. In a study of how external clues influence performance, psychology professors at the University of Alberta and Pennsylvania State University gave two different forms of a midterm examination to a large group of introductory psychology students. The questions on the exam were identical and in the same order, but one exam was printed on blue paper and the other on red paper (Teaching Psychology, May 1998). Grading only the difficult questions on the exam, the researchers found that scores on the blue exam had a distribution with a mean of 53% and a standard deviation of 15%, while scores on the red exam had a distribution with a mean of 39% and a standard deviation of 12%. (Assume that both distributions are approximately mound shaped and symmetric.)
1. Give an interpretation of the standard deviation for the students who took the blue exam.
2. Give an interpretation of the standard deviation for the students who took the red exam.
3. Suppose a student is selected at random from the group of students who participated in the study and the student’s score on the difficult questions is 20%. Which exam form is the student more likely to have taken, the blue or the red exam? Explain.
  
  Red
ECOPHD 2.129 Ranking Ph.D. programs in economics. The Southern Economic Journal (Apr. 2008) published a guide to graduate study in economics by ranking the Ph.D. programs at 129 colleges and universities. Each program was evaluated according to the number of publications published by faculty teaching in the Ph.D. program and by the quality of the publications. Data obtained from the Social Science Citation Index (SSCI) were used to calculate an overall productivity score for each Ph.D. program. The mean and standard deviation of these 129 productivity scores were then used to compute a z-score for each economics program. Harvard University had the highest z-score $(z = 5.08)$ $(z = 5.08)$ and, hence, was the top-ranked school; Howard University was ranked last since it had the lowest z-score $(z = - 0.81)$ $(z = - 0.81)$ . The data (z-scores) for all 129 economic programs are saved in the ECOPHD file.
1. Interpret the z-score for Harvard University.
2. Interpret the z-score for Howard University.
3. The authors of the Southern Economic Journal article note that “only 44 of the 129 schools have positive z-scores, indicating that the distribution of overall productivity is skewed to the right.” Do you agree? (Check your answer by constructing a histogram for the z-scores in the ECOPHD file.)
  
  Yes

Applying the Concepts—Advanced

2.130 Ranking Ph.D. programs in economics (cont’d). Refer to the Southern Economic Journal (Apr. 2008) study of Ph.D. programs in economics, Exercise 2.129. The authors also made the following observation: “A noticeable feature of this skewness is that distinction between schools diminishes as the rank declines. For example, the top-ranked school, Harvard, has a z-score of 5.08, and the fifth-ranked school, Yale, has a z-score of 2.18, a substantial difference. However, . . . the 70th-ranked school, the University of Massachusetts, has a z-score of $- 0.43$ $- 0.43$ , and the 80th-ranked school, the University of Delaware, has a z-score of $- 0.50$ $- 0.50$ , a very small difference. [Consequently] the ordinal rankings presented in much of the literature that ranks economics departments miss the fact that below a relatively small group of top programs, the differences in [overall] productivity become fairly small.” Do you agree?

Yes
2.131 GPAs of students. At one university, the students are given z-scores at the end of each semester, rather than the traditional GPAs. The mean and standard deviation of all students’ cumulative GPAs, on which the z-scores are based, are 2.7 and .5, respectively.
1. Translate each of the following z-scores to corresponding GPA scores: $z = - 2.5.$ $z = - 2.5.$
  
  3.7; 2.2; 2.95; 1.45
2. Students with z-scores below $- 1.6$ $- 1.6$ are put on probation. What is the corresponding probationary GPA?
  
  1.90
3. The president of the university wishes to graduate the top 16% of the students with cum laude honors and the top 2.5% with summa cum laude honors. Where (approximately) should the limits be set in terms of z-scores? In terms of GPAs? What assumption, if any, did you make about the distribution of the GPAs at the university?

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Table of Contents for 2.6 Numerical Measures of Relative Standing

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Table of Contents for
2.6 Numerical Measures of Relative Standing