1. EMPA Statistical Analysis
Week 3

5. Mean and Standard Deviation
We know everything about the distribution of a normally distributed variable if we know the mean and the standard deviation.
Using these two values, we can figure out how likely we are to observe any value.
This is very useful when we need to standardize variables for decisionmaking

6. z-score
Let’s call a one standard deviation distance from the mean “z”.
When we count distances from the mean in “z,” this is called a z-score or a standard normal score.

7. Counting z’s
You can figure out how many “z”s away from the mean any number is.
This is equivalent to counting how many standard deviations away from the mean the number is.
z = (raw value - mean)/standard deviation

8. Meaning of z A negative z-score indicates a value below the mean.
A positive z-score indicates a value above the mean.
The normal curve is symmetric, so a z-score of –z has the same probability of occurring as a score of z.

9. Using z-scores
Option 1: Create an index variable (weighted or unweighted)
Option 2: Identify a cutoff point based on a probability
Option 3: Identify a probability associated with a cutoff point (or points)

10. Excel Functions
=standardize (x, mean, standard deviation) returns the z-score for a value in your distribution
=normdist (x, mean, standard deviation, TRUE) returns the probability of returning any number up to and including x
=norminv(probability, mean, standard deviation) returns the value associated with a particular cumulative probability
=normsdist(z) returns the cumulative probability associated with a particular z-score
=normsinv(probability) returns a z-score associated with a particular cumulative probability

11. Creating z-scores in R
Data/manage variables in active data set/standardize variables
R automatically creates z-scores for selected variables and names them with the prefix z.

12. Creating indexes in R
Standardize the variables you want to include in the index
Create a new variable by going to data/manage variables in active data set/compute new variable
Enter a formula for the new index
For a simple additive index, add the component variables together.
You can also weight the variables

13. Creating indexes in R
Your new index variable has no inherent meaning—it only has relative meaning, with values close to 0 being average, positive numbers being above average, and negative numbers being below average.
An index is best described using descriptive statistics like maximum, minimum, and range.
You can standardize the index, in which case the index will be in terms of standard deviations above or below the mean index score (recommended!!)

14. Example: Garbage collection
Crew
Tons collected
Complaints against A
127 6 B
132 8 C
118 4 D
170 9 E
123 3
Step 1: Calculate mean and standard deviation

15. Example: Garbage collection
Crew
Tons collected
Complaints against A
127 6 B
132 8 C
118 4 D
170 9 E
123 3
Mean
134
Standard deviation
18.6
2.3
Step 2: Calculate z-scores

16. Example: Garbage collection
Crew
Tons collected
z-score (T)
Complaints against
Complaints
z-score A
127
-.38 6 B
132
-.11 8
.87 C
118
-.86 4
-.87 D
170
1.94 9
1.30 E
123
-.59 3
-1.30
Mean
134
Standard deviation
18.6
2.3
Step 3: Add weights, if desired

17. Example: Garbage collection
Crew
Tons collected
z-score (T)
Complaints against
Complaints (reversed)
z-score (C) A
127
-.38 6 B
132
-.11 8
-.87 C
118
-.86 4
.87 D
170
1.94 9
-1.30 E
123
-.59 3
1.30
Mean
134
Standard deviation
18.6
2.3
Step 4: Add the z-scores to create a composite index

18. Example: Garbage collection
Crew
Tons collected
Tons collected z-score (T)
Complaints against
Complaints (reversed)
z-score (C)
Equally weighted performance score (T+C) A
127
-.38 6 B
132
-.11 8
-.87
-.98 C
118
-.86 4
.87
.01 D
170
1.94 9
-1.30
.64 E
123
-.59 3
1.30
.71
Mean
134
Standard deviation
18.6
2.3
Ta-da! Use these scores to evaluate

19. Creating a cutoff point
Step 1: Calculate the mean and standard deviation of the population or sample
Step 2: Define the problem in terms of cutoff points and probabilities.
Step 3: Identify the probability associated with the desired cutoff point.
Step 4: Use Excel to identify the raw score associated with the desired cutoff point.

20. Example: MPA Admissions
There are 268 applicants to the MPA program. I can only admit 80 students to the program, or about 30 percent of all applicants. I can offer scholarships to the top 5 percent of applicants.
Step 1: Calculate the mean and standard deviation of the population or sample.

21. Example: MPA Admissions
There are 268 applicants to the MPA program. I can only admit 80 students to the program, or about 30 percent of all applicants. I can offer scholarships to the top 5 percent of applicants. The mean GRE score is with a standard deviation of
Step 2: Define the problem in terms of cutoff points and probabilities.

22. There are 268 applicants to the MPA program
There are 268 applicants to the MPA program. I can only admit 80 students to the program, or about 30 percent of all applicants. I can offer scholarships to the top 5 percent of applicants. The mean GRE score is with a standard deviation of
What GRE score must a person have to be in the top 30 percent of applicants?
What GRE score must a person have to be in the top 5 percent of applicants?
Step 3: Identify the probability associated with the desired cutoff point.

23. There are 268 applicants to the MPA program
There are 268 applicants to the MPA program. I can only admit 80 students to the program, or about 30 percent of all applicants. I can offer scholarships to the top 5 percent of applicants. The mean GRE score is with a standard deviation of
What GRE score must a person have to be in the top 30 percent of applicants, or to have 70 percent of applicants below his/her score?
What GRE score must a person have to be in the top 5 percent of applicants, or to have 95 percent of applicants below his/her score?
Step 3: Use Excel to identify the raw score associated with the desired cutoff point.

24. The mean GRE score is 1120.56 with a standard deviation of 158.36.
What GRE score must a person have to be in the top 30 percent of applicants, or to have 70 percent of applicants below his/her score?
=NORMINV(0.7, , ) =
What GRE score must a person have to be in the top 5 percent of applicants, or to have 95 percent of applicants below his/her score?
=NORMINV(0.95, , ) =
We can admit students with GRE scores above 1204 and offer scholarships to students with GRE scores above 1381.

25. Converting a score to a probability
Step 1: Calculate the mean and standard deviation of the population or sample
Step 2: Identify the score of interest
Step 3: Use Excel to identify the z-score associated with the raw score of interest.
Step 4: Use Excel to determine what percent of observations in the sample or population have scores below the score of interest

26. Example: The Applicant
I am an MPA applicant and have a GRE score of I want to know how that compares with other applicant’s scores.
Step 1: Calculate the mean and standard deviation of the population or sample

27. I am an MPA applicant and have a GRE score of 1310
I am an MPA applicant and have a GRE score of I want to know how that compares with other applicant’s scores. I discover that among applicants, the mean GRE score is with a standard deviation of
Step 2: Identify the score of interest

28. I am an MPA applicant and have a GRE score of 1310
I am an MPA applicant and have a GRE score of I want to know how that compares with other applicant’s scores. I discover that among applicants, the mean GRE score is with a standard deviation of
Step 3: Use Excel to identify the z-score associated with the raw score of interest.

29. My score is about 1.2 standard deviations above the mean.
I am an MPA applicant and have a GRE score of I want to know how that compares with other applicant’s scores. I discover that among applicants, the mean GRE score is with a standard deviation of
=STANDARDIZE(1310, , )
=1.196
My score is about 1.2 standard deviations above the mean.
Step 4: Use Excel to determine what percent of observations in the sample or population have scores below the score of interest

30. My score is about 1.2 standard deviations above the mean.
I am an MPA applicant and have a GRE score of I want to know how that compares with other applicant’s scores. I discover that among applicants, the mean GRE score is with a standard deviation of
=STANDARDIZE(1310, ,158.36)
=1.196
My score is about 1.2 standard deviations above the mean.
=NORMDIST(1310, , , TRUE)
=.884
About 88.4 percent of all other applicants have scores below mine.