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

• A transformation of a raw score in a distribution using the mean and standard deviation of the distribution
• Tells us the distance between the mean and any particular raw score expressed in standard deviation units
• The sign of a z score tells us the position of the score relative to the mean
• Positive z scores:  raw scores above the mean
• Negative z scores:  raw scores below the mean
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?

• Tell us the relative position of a raw score in its distribution
• Allow us to compare scores obtained from different measures
• z scores better describe the place of the raw score in the distribution
• z scores also allow us to compare scores obtained from different measures
Senator Klott is up for re-election. We are interested in how liberal his voting record is.
Example:  How liberal is Sen. Klott?
• He voted yes on 5 out of 12 environmental protection bills during his last term.
• If liberals tend to vote for more environmental protection bills than conservatives, is Sen. Klott more or less liberal than other Senators?
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:
• -2 on environmental bills
• +1 on tax bills
• If we had compared raw scores, we might have concluded that Sen. Klott voted consistently conservatively
Standard normal curve
• The distribution of z scores is described by the standard normal curve, which is a special example of the normal curve
• Symmetrical
• Bell-shaped
• Unimodal
• Mean = Median = Mode
• Not skewed
• Normal Curve
Describes many naturally occurring phenomena:
• height
• weight
• blood pressure
• IQ scores
• There are many possible normal curves, each having a different mean and standard deviation
• Normal curves will have different means if they are centered around different values
• Normal curves will have different standard deviations if values in the distribution cluster around the mean differently
• As S increases, the distribution flattens
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

• Standard normal curve tablein TABLE  A of APPENDIX  D on p. 590

Finding areas (or proportions) under the curve

• IQ Scores
• Example 1:  Finding a proportion below a score when the score is to the right of the mean
• What percentage of people have IQs below 132?
• 1.  Sketch normal curve and find target area
• 2.  Convert raw score (132) to z score
• z  =  (132-100)/16  =  2.00
• 3.  Find z of 2.00 in table
• 4.  Determine columns you need to use
• A:  for positive z score
• B:  proportion from mean to z score
• 5.  Area between mean and z = 2 is
• .4772
• 47.72% of scores fall between the mean (100) and 132
• 6.  Determine total area below 132:
• 47.72% + 50.00% = 97.72%
• Example 1b:  Alternative solution
• What percentage of people have IQs below 132?
• 1.  Sketch normal curve and find target area
• 2.  z  =  (132-100)/16  =  2.00
• 3.  Find z of 2.00 in table
• 4.  Use col. C to determine proportion above 132
• 5.  Area beyond z = 2 is
• .0228
• 6.  We want everything but this area:
• 1.00 - .0228 = .9772
• Example 2:  Finding a proportion below a score when the score is to the left of the mean
• What percentage of people have IQs below 91?
• Area below z = -.56 is
• .2877
• 28.77% of scores fall below 91
• Example 3:  Finding a proportion above a score when the score is to the right of the mean
• What percentage of people have IQs above 120?
• 1.  Sketch normal curve and find target area
• 2.  Convert raw score to z score
• z  =  (120 -100)/16  =  1.25
• 3.  Find z of 1.25 in table
• 4.  Determine column:
• A:  z score (positive)
• C:  area above z score
• 5.  Area above z = 1.25 is
• .1056
• 10.56% of scores fall above 120
• Example 4:  Finding a proportion above a score when the score is to the left of the mean
• What percentage of people have IQs above 80?
• Area between mean and z = -1.25 is
• .3944
• 39.44% of scores fall between the mean (100) and 80
• Determine the total area above 80
• 39.44% + 50.00% = 89.44%
• Example 5:  Finding an area between two scores
• What percentage of people have IQs between 90 and 120?
• 1.  Sketch normal curve and find target area
• 2.  Convert raw scores to z score
• z90  =  (90-100)/16  =  -.625
• z120  =  (120-100)/16  =  1.25
• 3.  Find zs of -.625 and 1.25 in table
• 4.  Determine columns:
• For z90 = -.625
• A:  z score (negative)
• B:  area between score and mean
• Area between -.625 (or -.63) and mean is
• For z120 = 1.25
• A:  z score (positive)
• B:  area between score and mean
• Area between and mean and 1.25 is
• .3944
• 5.  Add proportions:  .2357 + .3944
• 63.01% of scores fall between 90 and 120
• Example 6:  Finding an area between two scores on the same side of the mean.
• What percentage of people have IQs between 110 and 125?
• 20.49% of scores fall between 110 and 125
• Example 7:  Finding scores cutting off areas
• If a persons IQ is at the 75th percentile, what is her/his IQ score?
• Sketch the normal curve and find the target area
• Remember: The mean is at the 50th percentile
• Find the z score that corresponds to the 75th percentile
• Everything below the mean = .5
• Find the z score that is .25 above the mean
• .5 + .25 = .75
• Determine column:
• B:  area between mean and score
• Find the number under column B that cuts off .25 of the curve between mean and score:
• .2486
• What z score corresponds to .2486?
• Convert z score to raw score:
• The IQ score at the 75th percentile is 110.72