Scanning the body of the normal distribution table, we fi nd that the closest probability to .3000, without going under it, is .3023. The z score associated with this probability is .85 and is the one we seek. (If we had chosen a z score with an associated probability less than .3000, we would cut off slightly more than the top 20% of scores on the police examination.)
Step 2: Convert this z score to a raw score. Recall that a z score is the number of standard deviations a score of interest is from the mean. We want the raw score associated with a z score of .85. We multiply this z score times the standard deviation (10) to get 8.5; our score of interest is, thus, 8.5 above the mean. To complete the conversion, add this number to the mean to get 100 1 8.5 5 108.5. If the police academy class is limited to the top 20% of all applicants, only those who score 108.5 or above on the examination should be admitted.
An example from nonprofi t management is in order. Th e Gold Star Mentoring Program selects peer mediators from among Cub Scouts in the area. To be eli- gible, a Cub Scout must score in the top 75% of applicants on a test of social knowledge and interest in helping other people. Because the Gold Star Mentor- ing Program would like to involve as many Cub Scouts as possible, it accepts the top 75% of all those taking the exam. Last year, the mean score on this exam was 80, with a standard deviation of 6. Exam scores were normally distributed. At what value should the minimum acceptable score be set to admit only the top 75% of Cub Scouts who apply?
Because we know 50% of the applicants will score above 80 (mean score), we need to know the score below the mean that will contain 25% of the scores between it and the mean. Looking up a probability of .25 in the normal table, we fi nd a value of .2486 associated with a z score of .67. Because we are interested in scores below the mean, this z score is 2.67. Converting a z score of 2.67 to a raw score, we get a minimum acceptance score of 76 [( 2.67 3 6) 1 80 5 76].
Th us, the Gold Star Mentoring Program should accept Cub Scouts who score 76 or above on the test of social knowledge and interest in helping other people.
A Measurement Technique Based
of greater than 100,000 population, you could use the normal distribution table to place the cities in percentiles on this variable.
You can also use normal scores to combine variables into scales or indices. A scale or index is a composite measure combining several variables into a single, unifi ed measure of a concept. Because variables are usually measured on diff er- ent scales, you should not simply add scores together to create a new scale. For example, if you needed to create and report an overall measure of the physical fi t- ness of military troops, their height (inches), weight (pounds), age (years), speed (time to run a mile), and personal habits (frequency of smoking, drinking, and so on) might all be important—but simply summing scores across the diff erent variables would be meaningless. Th e measurement scales for the variables are not comparable.
You can use standard normal z scores to overcome the problem of diff erent scales of measurement. A z score converts any variable to the same unit of mea- surement; hence the name standard normal score. Regardless of the initial scale of measurement of a variable, all z scores are measured on the same normal curve or distribution and have the same probability interpretation. Th us, if variables are converted fi rst to z scores to place them on the same measurement scale and then summed, the resulting scales or indices will not be distorted by any initial diff erences in measurement: Each variable will contribute appropriately, with the correct infl uence or weighting, to the composite scale or index.
Let’s look at an example. John Pelissero, a data analyst for the city of Evans- ville, feels that excellent performance by city garbage collection crews can be sum- marized by two variables: tons of trash collected and the number of complaints received from citizens. Mr. Pelissero would like to combine these data into a single measure of performance, but he realizes that the variables are based on very dif- ferent units of measurement (tons of trash and number of complaints). Because the tons of trash are so much larger in magnitude, they will dwarf the number of complaints, which are relatively small numbers. He knows that to create a mean- ingful index of performance, something must be done to place the two variables on the same measurement scale. Obviously, Mr. Pelissero has read this book and will go a long way in public or nonprofi t management. In Table 8.1 he has as- sembled the data from each of the fi ve city work crews on the number of tons of trash collected and the number of complaints lodged against them by citizens.
Table 8.1 Performance of City Garbage Collection Crews in Evansville
Crew Tons Collected Complaints Against
A 127 6
B 132 8
C 118 4
D 170 9
E 123 3
Mr. Pelissero wants to combine these two variables into a single measure of performance, so he follows these steps.
Step 1: Calculate the mean (m) and standard deviation (s) for each variable.
Th ey are shown in Table 8.2.
Step 2: Convert each of the raw scores of trash collected and complaints re- ceived into z scores by taking the raw score, subtracting the mean, and dividing that diff erence by the standard deviation. Th e calculations are shown in Tables 8.3 and 8.4. Because complaints indicate poor perfor- mance (two complaints are worse, not better, than one complaint), the z scores for complaints were multiplied by 21 to reverse the scale.
Step 3: For each crew, add the z scores for trash collected and complaints re- ceived, as shown in Table 8.5, to obtain performance scores. Th e per- formance scores can then be used to compare the trash collection crews with each other on a single measure. Th e z score procedure simply con- verts each variable to a common base (standard deviations from the mean) so that the variables can be added appropriately. Note that, as Table 8.3 shows, the highest-performing trash crew was fourth in total tons of trash collected but received the fewest complaints from citizens.
Th is procedure weights tons of trash collected and complaints received equally. If Mr. Pelissero felt that tons of trash collected was twice as important as complaints received, he would multiply the z scores for tons of trash by 2 before adding them to the z scores for complaints, as shown in Table 8.6. Note that some of the trash crews change ranking when tons of trash collected is given a higher weight- ing (twice as important).
Table 8.2 Mean and Standard Deviation for Each Variable
Crew Tons Complaints Against
m 134. 6
s 18.6 2.3
Table 8.3 z Scores for Tons of Trash Collected
Tons 2 m (X 2 m) 4 s 5 z
127 2 134 5 27 4 18.6 5 2.38 132 2 134 5 2 2 4 18.6 5 2.11 118 2 134 5 216 4 18.6 5 2.86 170 2 134 5 36 4 18.6 5 1.94 123 2 134 5 211 4 18.6 5 2.59
A useful equal-weighting scheme is to weight the z scores for tons and the z scores for complaints by one-half, or .5. Th is procedure is equivalent to calcu- lating the mean performance score for each garbage collection crew. Th e resulting performance scores retain the value of a z score in indicating how many units (standard deviations) a score is above or below the mean. Positive scores indicate performance above the mean, and negative scores indicate performance below the mean. Performance scores calculated in this manner, as displayed in Table 8.7, show that crews A and B are below the mean in performance, crew C is at the mean, and crews D and E are above the mean.
A public or nonprofi t manager can assign any weights to the variables so long as the variables have been converted to z scores fi rst and the weights can be justifi ed. For example, the board of directors of a nonprofi t food bank feels that dispensing meals is by far its most important activity. In comparing its perfor- mance with that of other food banks, it might weight meals served by a factor of 3.0, monetary donations by 2.0, and volunteer hours by 1.5. Th e number of variables that can be combined using the z score method has no limit. A manager can use 2, 4, 6, or 50 or more indicators and combine them by adding z scores.
Of course, the variables or indicators should have relevance for the concept the manager wants to measure.
Table 8.4 z Scores for Complaints Received
Complaints 2 m 5 (X 2 m) 4 s 5 z 2 z
6 2 6 5 0 4 2.3 5 0 0
8 2 6 5 2 4 2.3 5 .87 2.87
4 2 6 5 22 4 2.3 5 2.87 .87
9 2 6 5 3 4 2.3 5 1.30 21.30 3 2 6 5 23 4 2.3 5 21.30 1.30
Table 8.5 Performance Scores for Garbage Collection Crews
Crew Tons Complaints Performance
A 2.38 0 2.38
B 2.11 2.87 2.98
C 2.86 .87 .01
D 1.94 2.30 .64
E 2.59 1.30 .71
A note of caution is in order. Th e z score technique should be used only when the manager believes that the concept being measured has only one dimension.
If the concept has two or more dimensions (as, for example, the eff ectiveness of a receptionist is a function of effi ciency and courtesy), a more sophisticated statisti- cal technique called factor analysis should be used to combine several indicators or variables. When you have multiple indicators of a multidimensional concept and want to combine some of the indicators, consult a statistician.
Chapter Summary
Th e normal distribution is a very common and useful probability distribution in public and nonprofi t administration. Th e general bell shape of the curve de- scribes the distribution of many phenomena of interest to the public or nonprofi t manager.
Th e normal distribution is measured in terms of its standard deviation. In a normal distribution, 68.26% of all values lie within one standard deviation of the mean in either direction, 95.44% of all values fall within two standard deviations of the mean, and 99.72% of all values fall within three standard deviations.
Table 8.6 Performance Scores when Tons Are Weighted Twice as Heavily as Complaints
Crew Tons 3 2 Complaints Performance A 2.38 3 2 5 2.76 0 2 .76 B 2.11 3 2 5 2.22 2 .87 2 1.09 C 2.86 3 2 5 21.72 .87 2 .85 D 1.94 3 2 5 3.88 2 1.30 2.58 E 2.60 3 2 5 21.20 1.30 .10
Table 8.7 Performance Scores when Tons and Complaints Are Weighted by One-Half (.5)
Crew Tons 3 .5 Complaints 3 .5 Performance A 2 .38 3 .5 5 2.19 0 3 .5 5 0 2.19 B 2 .11 3 .5 5 2.055 2 .87 3 .5 5 2.435 2.49 C .86 3 .5 5 2.43 .87 3 .5 5 .435 .005 D 21.94 3 .5 5 .97 21.30 3 .5 5 .65 .32 E 2.59 3 .5 5 2.295 1.30 3 .5 5 .65 .355
To use the normal distribution, the manager must calculate and interpret a z score. A z score is defi ned as the number of standard deviations a score of interest lies from the mean of the distribution. It is used to convert raw data values into their as- sociated probabilities of occurrence with reference to the mean. Denoting the score of interest X, the mean m, and the standard deviation s, the formula for the z score is
z5X2 m s
Th e normal distribution presented in Table 1 in the Statistical Tables at the end of this book displays the percentage of cases falling between any z score and the mean of the distribution (or the probability that falls under this region of the curve). Using this information, the public or nonprofi t manager can answer a variety of important questions, such as the probability of observing a score less than, or greater than, the score of interest; or the probability of a score falling between two scores of interest. She or he can also identify scores that will cut off portions of the curve of special interest, such as the top 5% of qualifi ers.
One can also use standard normal scores to combine variables into a com- posite scale or index. Th is technique is very useful to public and nonprofi t man- agers, who often have several indicators of a concept (for example, performance) and need to combine them into a single measure.
Problems
8.1 Vince Yosarian works for the Bureau of Forms. His manager has reprimanded Vince because she feels that he has not performed up to standard for processing forms at the bureau, the crucial part of the job. At the bureau, the number of forms processed by employees is normally distributed, with a mean of 67 forms per employee per day, with a standard deviation of 7. His manager has calcu- lated that Vince’s average rate is 50 forms processed per day. What percentage of employees at the bureau process fewer forms than Vince? What percentage of employees at the bureau process more forms than Vince? Does the manager’s complaint seem justifi ed or not?
8.2 The average (mean) amount of time that a manager at the Mount Pleasant Department of Human Services spends in the annual performance review with an employee is 27.2 minutes, with a standard deviation of 4.9 minutes (normal distribution). What percentage of the annual performance reviews in the department take between 22.3 and 32.1 minutes? Between 17.4 and 37 minutes?
Between 12.5 and 41.9 minutes? Explain your answers.
8.3 According to records kept by Compton County, the average (mean) amount of time that it takes for employees to be reimbursed for professional expenses incurred in service to the county is 36 days, with a standard deviation of 5 days.
Th e distribution is normal. About 2 months ago, Marissa Lancaster attended a training conference for her job in the County Recreation Department. She fi led
for reimbursement of expenses that she incurred at the conference 42 days ago, but she has not received payment. What is the probability of receiving reimburse- ment within 42 days of fi ling? After 42 days? At this point, should Ms. Lancaster be apprehensive about receiving reimbursement?
8.4 Refer to Problem 8.3. Th e head of the Compton County Accounting Depart- ment wants to establish a standard regarding the length of time that employees can expect to wait to receive reimbursement for professional expenses. She wants to publish a standard that states the number of days it takes the department to process 95% of the claims fi led. Help the department head fi nd that standard.
8.5 Parnelli Jones, a vehicle manager for the northeast region of the forestry service, is charged with purchasing automobiles for service use. Because forestry service employees often have to drive long distances in isolated regions, Parnelli is very concerned about gasoline mileage for the vehicle fl eet. One automobile manufac- turer has told him that the particular model of vehicle that interests him has aver- aged 27.3 miles per gallon in road tests with a standard deviation of 3.1 (normal distribution). Parnelli would like to be able to tell his superiors at the forestry service that the cars will get at least 25 miles to the gallon. According to the road test data, what percentage of the cars can be expected to meet this criterion?
Parnelli also thinks that his superiors might settle for cars that got 24 miles to the gallon or better. What percentage of the cars can be expected to meet this criterion?
8.6 Refer to Problem 8.5. During the negotiations with the automobile manufacturer, Parnelli gets a sinking feeling that some of the cars he is thinking of purchasing for the forestry service might average less than 20 miles to the gallon of gasoline.
According to the road test data, what percentage of the cars can be expected to fall below this mileage level? Should Parnelli be very worried?
8.7 Butterworth, Missouri, is trying to increase its tourism, a great source of revenue.
Among other things, Butterworth prides itself on its temperate climate. Th e tour- ism department would like to include in its glossy new brochure the middle range of temperatures that occur on 90% of the days there. A check with the National Weather Service fi nds that the average temperature in Butterworth is 76 degrees with a standard deviation of 5.7 (the temperatures were normally distributed).
Help the tourism department by fi nding the two temperatures that cut off the middle 90% of temperatures in Butterworth.
8.8 Th e average (mean) paid by the Bureau of Indian Aff airs (BIA) for a 5-day training workshop contracted with the private sector is $1,500 (standard deviation 5 $280;
normal distribution). Perk Morgan, a BIA employee, would like to attend a sensi- tivity training workshop in Bermuda (he feels that he must go to an exotic loca- tion to get in touch with his true feelings); the government rate for the training workshop is $2,400. What percentage of 5-day training workshops funded by the BIA exceed this amount? Perk has also inquired about a sensitivity training workshop to be held in Sausalito, California (less exotic location, but Perk will still be able to get in touch with feelings); the cost for this workshop is $2,100.
What percentage of BIA-sponsored training workshops exceed this cost? What would you advise Perk to do?
8.9 Ima Fastrack is a data analyst at the Oakmont Civil Service Commission. When she was fi rst hired 3 months ago, she was satisfi ed with her salary of $24,832.
In the time since she was hired, Ima has talked with several others in similar positions with the Oakmont city government; all earn more than she does. Ima has obtained data on the salaries of data analysts with the city. Th e distribution of salaries is normal, with an average salary of $25,301 and a standard deviation of
$986. Ima has come to believe that she is seriously underpaid relative to others in her position. Do you agree or disagree with her? Use the normal curve to support your answer.
8.10 Th e director of development at the Midfi eld Wisconsin Museum of the Arts has recently collected data on donations for the past several years. She fi nds that the data are normally distributed with a mean of $23 and a standard deviation of
$3.57. Th e director’s ideal point for a minimum donation is $25. What percentage of individual donations are $25 or more? What percentage of individual donations are less than $25? Th e director’s long-term goal for an average individual donation is at least $40. Based on the current data, does achieving this goal seem reason- able in the immediate future?
8.11 Th e Warrenville Empowerment Center has begun to track how many days of employment counseling its clients receive. Th e center’s intern, Amanda Strom, finds that the average client receives 17 days of counseling with a standard deviation of 3.1 days (the data are normally distributed). Th e center’s director has become more concerned about managing costs, so she asks Amanda to generate some information from these data. Because the director wonders if a target of 15 or fewer days of counseling is realistic, she asks Amanda to determine what percentage of clients receive 15 or fewer days of counseling. Th e director is es- pecially concerned about clients needing 25 or more days of counseling and wonders what percentage of the current client base fi ts this profi le. What should Amanda tell the director after analyzing the data?
8.12 Administrators at the national headquarters of the Center for the Display of Visual Arts are authorized to make small work-related purchases using purchas- ing cards. Th e average amount administrators charge to their cards per month is
$278 with a standard deviation of 33 dollars. Th e data are normally distributed.
Th e auditing staff at the center is concerned with the spending patterns of an administrator who has averaged $349 per month in charges. What percentage of balances are between $278 and $349 per month? What percentage of balances exceed $349 per month? Should the auditing staff be concerned about the spend- ing habits of this particular administrator?