Points in Distributions

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# Points in Distributions - PowerPoint PPT Presentation

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

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Presentation Transcript
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 ~
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 ~

.34

.02

.02

.14

-2

-1

0

+1

+2

Areas Under Normal Curves

f

.34

.14

standard deviations

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 ~

f

-2

-1

0

+1

+2

z

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

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
• Normal distributions,
• but not unit normal distribution
• Standardized variables
• normally distributed
• e.g., IQ test
• m = 100; s = 15 ~

-2

85

-1

100

0

115

+1

130

+2

70

z scores

Other Standardized Distributions

m = 100

s= 15

f

IQ 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
• 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
• Percentile rank
• Per cent of scores £ a given score
• Score: a value of any variable ~
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

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 ~

.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

### Supplementary Material

Determining Probabilities
• Must count ALL possible outcomes
• e.g. of flipping 2 coins

outcomes

1

2

3

4

coin A:

tail

tail

coin B:

tail

tail

Determining Probabilities
• Single fair die
• P(1) = P(2) = … = P(6)
• 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 ~

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
• 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