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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

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points in 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
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 ~
areas under distributions
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 ~
slide5

0.5

0.5

10

20

30

40

50

60

70

80

90

Total area under curve = 1.0

using areas under distributions
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
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 curves8
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 curves9

.34

.02

.02

.14

-2

-1

0

+1

+2

Areas Under Normal Curves

f

.34

.14

standard deviations

calculating areas from tables
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 tables11

f

-2

-1

0

+1

+2

z

Calculating Areas from Tables
  • Finding proportions
    • z < 1 = (from B)
    • z > 1: (from C) ~
calculating areas from tables12

f

-2

-1

0

+1

+2

z

Calculating Areas from Tables
  • Area: 1 < z < 2
    • find proportion for z = 2;
    • subtract proportion for z = 1 ~
other standardized distributions
Other Standardized Distributions
  • Normal distributions,
    • but not unit normal distribution
  • Standardized variables
    • normally distributed
    • specify m and sinadvance
  • e.g., IQ test
    • m = 100; s = 15 ~
other standardized distributions14

-2

85

-1

100

0

115

+1

130

+2

70

z scores

Other Standardized Distributions

m = 100

s= 15

f

IQ Scores

transforming to from z scores

X - m

s

z =

Transforming to & from z scores
  • From z score to standardized score in population

X = zs + m

  • Standardized score ---> z score
normal distributions percentiles percentile rank
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
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
Percentiles

A

58

56

54

54

52

50

48

46

44

42

B

50

46

32

30

30

23

23

22

21

20

  • E.g., test scores
    • 30th percentile = (A) 46; (B) 22
    • 90th percentile = (A) 56; (B) 46 ~
percentile rank
Percentile Rank

A

58

56

54

54

52

50

48

46

44

42

B

50

46

32

30

30

23

23

22

21

20

  • e.g., Percentile rank for score of 46
    • (A) 30%; (B) = 90%
  • Problem: equal differences in % DO NOT reflect equal distance between values ~
slide20

.34

.34

.02

.02

.14

.14

-2

85

-1

100

0

115

+1

130

+2

70

z scores

IQ Scores

f

IQ

percentile

rank

2d

16th

50th

84th

98th

determining probabilities
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 probabilities23
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
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

Table: column B or C

X = z s + m

area under

distribution

Raw Score (X)

z score

X - m

s

z =

Table: z - column A

Know/want Diagram
percentage raw score
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