EMPA Statistical Analysis Week 3

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

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.

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

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.

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)

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

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.

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

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!!)

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

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

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

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

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

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.

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.

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 1120.56 with a standard deviation of 158.36. Step 2: Define the problem in terms of cutoff points and probabilities.

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 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? 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.

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 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? 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.

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, 1120.56, 158.36) =1203.604 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, 1120.56, 158.36) =1381.039 We can admit students with GRE scores above 1204 and offer scholarships to students with GRE scores above 1381.

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

Example: The Applicant I am an MPA applicant and have a GRE score of 1310. 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

I am an MPA applicant and have a GRE score of 1310 I am an MPA applicant and have a GRE score of 1310. I want to know how that compares with other applicant’s scores. I discover that among applicants, the mean GRE score is 1120.56 with a standard deviation of 158.36. Step 2: Identify the score of interest

I am an MPA applicant and have a GRE score of 1310 I am an MPA applicant and have a GRE score of 1310. I want to know how that compares with other applicant’s scores. I discover that among applicants, the mean GRE score is 1120.56 with a standard deviation of 158.36. Step 3: Use Excel to identify the z-score associated with the raw score of interest.

My score is about 1.2 standard deviations above the mean. I am an MPA applicant and have a GRE score of 1310. I want to know how that compares with other applicant’s scores. I discover that among applicants, the mean GRE score is 1120.56 with a standard deviation of 158.36. =STANDARDIZE(1310, 1120.56,158.3555) =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

My score is about 1.2 standard deviations above the mean. I am an MPA applicant and have a GRE score of 1310. I want to know how that compares with other applicant’s scores. I discover that among applicants, the mean GRE score is 1120.56 with a standard deviation of 158.36. =STANDARDIZE(1310, 1120.56,158.36) =1.196 My score is about 1.2 standard deviations above the mean. =NORMDIST(1310, 1120.56, 158.36, TRUE) =.884 About 88.4 percent of all other applicants have scores below mine.