Probability &
Using Frequency Distributions
Chapters 1 & 6
Homework: Ch 1: 9-12
Ch 6: 1, 2, 3, 8, 9, 14
Probability: Definitions
Chapter 1, pp. 8-10
 Experiment:
 controlled operation
 yields 1 of several possible outcomes
 e.g., drawing a card from deck
 Event
 a set of possible outcomes

e.g. draw a heart
13 possible outcomes ~
Probability: Definitions

Probability(P) of an event (E)
 Assuming each outcome equally likely
P(E) = # outcomes favorable to E
total # possible outcomes
P(drawing ) =
P(7 of ) =
P(15 of ) =
P(or or or ) =
Probability:
3 important characteristics
1. Probability event cannot occur is 0
2. P(E) that must occur = 1
3. 0 P(E) 1, probabilities lie b/n 0 & 1 ~
Determining Probabilities
Must count ALL possible outcomes
 A fair die: P(1) = P(2) = … = P(6)
 P(4) =
 Event = sum of two fair dice
 P(4) =
 36 possible outcomes of rolling 2 dice
 Sum to 4:
 __ possible outcomes favorable to E ~

Determining Probabilities
Single fair die
 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 ~
Points in Distributions
Up to now describing distributions
 Comparing scores from different
distributions
 Need to make equivalent comparisons
 Percentile rank & standard scores
 z scores ~

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
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 Equations
Sample: z =
Population: z =
X-X
s
X-m
s
z Score Computation

e.g., 90th percentile = (A) 56; (B) 46
 convert to z scores
 A: s = 5; m = 50
 B: s = 10; m = 29
Areas Under Distributions
Area = frequency
 Relative area
 total area = ____
= proportion of individual values in
area under curve ~

10
20
30
40
50
60
70
80
90
Total area under curve = 1.0
Using Areas Under Distributions
Relative area is independent of shape of
distribution
 Given value, what is relative frequency?
 Question: what % of days is the
temperature over 60 o?
o
 Or P(temperature > 60 ) ~

% of days the temperature is above 60 o
10
20
30
40
50
60
70
80
90
Average Daily Temperature (oF)
% of days temperature is between 30 & 50o?
10
20
30
40
50
60
70
80
90
Average Daily Temperature (oF)
Using Areas Under Distributions

Given relative frequency, what is value?
 e.g., What is temperature on the
hottest 10% of days
 find value of X at border ~
temperature on the hottest 10% of days
10
20
30
40
50
60
70
80
90
Average Daily Temperature (oF)
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 A.1 ~

Areas Under Normal Curves
f
-2
-1
0
1
standard deviations
2
Calculating Areas from Tables
Table A.1 (in our text)
“Proportions of areas under the
normal curve”
 3 columns
 z
 (A) Area between mean and z
 (B) Area beyond z (in tail)
 Negative z: area same as positive ~

Calculating Areas from Tables
Area between mean and z=1
 0 < z < 1 =
(from A)
 beyond z=1:
(from B)
 A + B = .5
 Area: 1 < z < 2
 find z=2;
0<z<2=
 subtract area for z=1
~

Calculating Areas from Tables

Area between z=-2 and z=1
 add areas for z=-2 and z=1
 -2 < z < 0 =
 0 < z < 1 =
~
Calculating Areas from Tables
Area between ...
z=0 and z=1.34
 0 < z < 1.34
z=1.5 and z=1.92
 1.5 < z < 1.92
z=-1.37 and z=.23
 1.37 < z < .23

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
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
Samples:
 X = zs + X
X-X
z =
s
Know/want Diagram
X=zs+m
Raw Score (X)
z =
Table: column A or B
z score
X-m
s
area under
distribution
Table: z column
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 ~
IQ Scores
f
IQ
70
85
100
115
130
z scores
-2
-1
0
1
2
percentile
rank