Transcript Using Frequency Distributions
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 -
s m
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 s
= 0 = 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
-2 .14
-1 0 +1 standard deviations .14
+2 .02
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 = z > 1: (from B) (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 ~
f
Other Standardized Distributions m s
= 100 = 15 z scores 70 -2 85 -1 100 0 IQ Scores 115 +1 130 +2
Transforming to & from z scores
From z score to standardized score in population X = z
s
+
m
Standardized score ---> z score z = X -
s m
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
30 th percentile = (A) 46; (B) 22
90 th 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
.02
IQ z scores percentile rank 70 -2 2 d .14
85 -1 16 th 100 0 50 th .34
115 +1 84 th .14
130 +2 98 th .02
Supplementary Material
Determining Probabilities
Must count ALL possible outcomes
e.g. of flipping 2 coins coin A: 1 head outcomes 2 tail 3 tail 4 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 = z
s
+
m
Table: column B or C
Raw Score (X) z score area under distribution
z = X -
s m
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 = z
s
+
m