Descriptive Statistics: Standard  Scores and the Normal Distribution
I) z scores
A) What are they?
B) How do we compute them?
C) How do we compute a "raw score" given a z score?
D) What good are they?
II) Normal Curves
A) Review characteristics of normal distribution
B) Standard normal curve
1) "standard" mean
2) "standard" S (standard deviation)
3) What good are they?
a) Provide information about distribution of data (area/proportions)
b) Steps to find area under curve
i) above a certain point
ii) below a certain point
iii) between 2 points
 
 
 
Standard Scores and Normal Distributions

Standard score or z score

Example
Consider a distribution of raw scores with: Mean = 10, s.d., = 2.

Formula for converting raw scores to z scores

X = 8
z = (8-10)/2 = -2/2 = -1
8 is 1 standard deviation unit below the mean
 

X = 10
z = (10-10)/2 = 0/2 = 0
10 is 0 standard deviation units away from the mean
 

X = 16?
z = (16-10)/2 = 6/2 = 3
16 is 3 standard deviation units above the mean
 
 

Formula for converting z scores to raw scores
z = -2?
X = (-2)(2) + 10 = 6
6 is 2 standard deviation units below the mean
 

z = 0?
X = (0)(2) + 10 = 10
10 is 0 standard deviation units below the mean
 

z = 1?
X = (1)(2) + 10 = 12
12 is 1 standard deviation unit above the mean

What can z scores do for us?

Senator Klott is up for re-election. We are interested in how liberal his voting record is.
Example:  How liberal is Sen. Klott? In order to answer this question, we need to know how Sen. Klott voted relative to the rest of the Senate.
How does his voting record on environmental bills compare to other Senators?

If he is above the mean, Sen. Klott has a liberal voting record on environmental issues.
If he is below the mean, Sen. Klott has a conservative voting record on environmental issues.

On the average, Senators voted for 9 out of 12 environmental protection bills, with a standard deviation of 2
mean = 9
standard deviation = 2

Convert Sen. Klott’s voting record to a z score to determine how liberal or conservative he is on environmental issues:
z = (5 - 9)/2 = -2

Sen. Klott falls 2 standard deviations below the average Senator, indicating that he is probably a conservative

Sen. Klott is a conservative on environmental issues, but does he always vote conservatively?

We examined his voting record on bills to raise taxes.
He voted for 5 out of 12 bills to raise taxes.
Is this a conservative or liberal record?

If we compared the raw number of votes for environmental issues to the raw number of votes on tax bills, what we would conclude?

5 “yes” votes out of 12 on the environment
2 standard deviation units below the mean
5 “yes” votes out of 12 on taxes
But let’s compare z scores, NOT raw scores.
Mean “yes” votes on tax bills: 3
Standard deviation:   2
Sen. Klott’s z score on tax bills?
z = (5 - 3)/2 = +1
Sen. Klott’s z scores: Standard normal curve Standard Normal Curve
Special example of the normal curve with mean = 0, sd = 1.

z scores allow us to use the standard normal curve table to answer questions about distributions of data


Finding areas (or proportions) under the curve