Points in Distributions
Up to now describing distributions
 Comparing scores from different
distributions
 Need to make equivalent comparisons
 z scores

standard scores

Percentile, Percentile rank ~
Standard Scores
Convert raw scores to z scores
 raw score: value using original scale of
measurement
 z scores: # of standard deviations score
is from mean
 e.g., z = 2
= 2 std. deviations from mean
 z = 0 = mean ~

z Score Equation
z =
X-m
s
Areas Under Distributions
Area = frequency
 Relative area
 total area = 1.0
= proportion of individual values in
area under curve
 Relative area is independent of
shape of distribution ~

0.5
0.5
10
20
30
40
50
60
70
80
90
Total area under curve = 1.0
Using Areas Under Distributions

Given relative frequency, what is value?
 e.g., the hottest 10% of days the
temperature is above ____?
 find value of X at border ~
Areas Under Normal Curves
Many variables  normal distribution
 Normal distribution completely
specified by 2 numbers
 mean & standard deviation
 Many other normal distributions
 have different m & s ~

Areas Under Normal Curves
Unit Normal Distribution
 based on z scores
m =0
s =1
 e.g., z = -2
 relative areas under normal distribution
always the same
 precise areas from Table B.1 ~

Areas Under Normal Curves
f
.34
.34
.02
.02
.14
-2
.14
-1
0
+1
standard deviations
+2
Calculating Areas from Tables
Table B.1 (in our text)
 The Unit Normal Table
 Proportions of areas under the
normal curve
 3 columns
 (A)
z
 (B)
Proportion in the body
 (C)
Proportion in the tail
 Negative z: area same as positive ~

Calculating Areas from Tables

Finding proportions
 z < 1 =
(from B)
 z > 1:
(from C) ~
f
-2
-1
0
z
+1
+2
Calculating Areas from Tables

Area: 1 < z < 2
 find proportion for z = 2;
 subtract proportion for z = 1
~
f
-2
-1
0
z
+1
+2
Other Standardized Distributions
Normal distributions,
 but not unit normal distribution
 Standardized variables
 normally distributed
 specify m and s in advance
 e.g., IQ test
 m = 100; s = 15 ~

Other Standardized Distributions
m = 100
s = 15
f
z scores
70
85
100
115
130
-2
-1
0
+1
+2
IQ Scores
Transforming to & from z scores

From z score to standardized score
in population
X = zs + m
Standardized
score ---> z score
z =
X-m
s
Normal Distributions:
Percentiles/Percentile Rank

Unit normal distributions
 50th percentile = 0 = m
 z = 1 is 84th percentile
50% + 34%

Relationships
 z score & standard score linear
 z score & percentile rank nonlinear ~
Percentiles & Percentile Rank
Percentile
 score below which a specified
percentage of scores in the
distribution fall
 start with percentage ---> score
 Percentile rank
 Per cent of scores  a given score
 start with score ---> percentage
 Score: a value of any variable ~

Percentiles

E.g., test scores


30th percentile =
(A) 46; (B) 22
90th percentile =
(A) 56; (B) 46 ~
A
58
56
54
54
52
50
48
46
44
42
B
50
46
32
30
30
23
23
22
21
20
Percentile Rank
e.g., Percentile rank for
score of 46
 (A) 30%; (B) = 90%
 Problem: equal differences
in % DO NOT reflect equal
distance between values ~

A
58
56
54
54
52
50
48
46
44
42
B
50
46
32
30
30
23
23
22
21
20
IQ Scores
f
.34
.34
.02
.02
.14
.14
IQ
70
85
100
115
130
z scores
-2
-1
0
+1
+2
16th
50th
percentile
rank
2d
84th
98th
Supplementary Material
Determining Probabilities
Must count ALL possible outcomes
 e.g. of flipping 2 coins

outcomes
1
2
3
4
coin A:
head
tail
tail
head
coin B:
head
tail
head
tail
Determining Probabilities

Single fair die
P(1) = P(2) = … = P(6)
Addition rule
 keyword: OR
 P(1 or 3) =
 Multiplication rule
 keyword AND
 P(1 on first roll and 3 on second roll) =
 dependent events ~

Conditional Probabilities

Put restrictions on range of possible
outcomes
 P(heart) given that card is Red
 P(Heart | red card) =

P(5 on 2d roll | 5 on 1st roll)?
 P =
 1st & 2d roll independent events ~
Know/want Diagram
X=zs+m
Raw Score (X)
z =
Table: column B or C
z score
X-m
s
area under
distribution
Table: z - column A
Percentage  raw score
Percentile rank  percentile
 Or probability  raw score
 What is the 43d percentile of IQ scores?
 1. Find area in z table
 2. Get z score
 3. X = zs + m
