Download 214 Lab 4: Overview

PSYC 214
NORMAL DISTRIBUTIONS AND STANDARD SCORES
Agenda
Percentiles
Distributions
Standard (-ization of) Scores
Appendix B
Examples
Few Reminders
Rounding
◦ Usually round to the hundredth’s place (two numbers after decimal point
e.g. 1.234 = 1.23, 1.235 = 1.24)
◦ Below 5 round down
◦ 5 and above round up
Greater than or less than
>
or
◦ Greater than > (e.g. 6 > 5)
◦ Less than < (e.g. 1 < 4)
◦ Alligator eats the biggest number
<
Percentile Rank
Percentage of scores in the distribution that are at or below a
specific value; Pxx
Describes the position of a score
◦ In relation to other scores
◦ What type of measurement is this?
Example: Baby at 90th percentile for height & weight is tall &
heavy: 90% of other babies are shorter/lighter and only 10%
are longer/heavier
p. 101-105
th
95
percentile Baby
Another View (7 months)
My Nieces
70th Height and 57th Weight (4
yrs)
55th Height and 32nd Weight
(18mths)
2 years
3 years
5 years
9 years, 11 years, and 7 years
How to find Pxx
A score’s percentile: “If you were 45th from the bottom of 60
people, you’d be at the _____ percentile.”
◦ Score position/total number of scores X 100 = percentile
◦ 45/60 X 100 = 75th percentile or P75
◦ If you scored 50th from the bottom of a class of 160 people, at what
percentile would your score be?
Practice
What’s the percentile rank for a score that’s:
◦ 23th from the bottom of 375 scores?
◦ 7th from the end out of 34 scores?
◦ 95th from the bottom of 180 scores?
What’s the position (from the bottom) for a score with a percentile rank of:
◦ P25 among 775 scores?
◦ P90 among 425 scores?
Percentile Rank as a z-Score
• But remember:
o
o
o
o
Pxx can reflect an ordinal measure of relative standing
(position relative to the group of scores)
As a rank it does not take into consideration means or
standard deviations (no math!)
Standardized scores are often used to present percentile
ranks, also referred to as z-scores
They describe how far a given score is from the other
scores, in terms of the z distribution
Purpose of z-Scores
Identify and describe location of every
score in the distribution
Standardize an entire distribution
◦ Takes different distributions and makes them equivalent and
comparable
Two distributions of exam scores
Locations and Distributions
Exact location is described by z-score
◦ Sign tells whether score is located above or below the
mean
◦ Number tells distance between score and mean in
standard deviation units
Relationship of z-scores and locations
Traits of the normal
distribution
Sections on the left side of the distribution have the same
area as corresponding sections on the right
Because z-scores define the sections, the proportions of
area apply to any normal distribution
◦ Regardless of the mean
◦ Regardless of the standard deviation
Normal Distribution with z-scores
z-Scores for Comparisons
All z-scores are comparable to each other
Scores from different distributions can be converted to
z-scores
The z-scores (standardized scores) allow the comparison
of scores from two different distributions along
Properties of z scores
Mean ALWAYS = 0
Standard deviation ALWAYS = 1
Positive z score is ABOVE the mean
Negative z score is BELOW the mean
Standard Scores
Give a score’s distance above or below the mean in terms of
standard deviations
Positive z-scores are always above the mean
Negative z-scores are always below the mean
Negative & positive z-scores only indicate directionality on
the x-axis, not necessarily a change in the value of the score
Standard Scores
• An index of a score’s relative standing
Standard (converted) scores are referred
to as z-scores
• x = raw score
• µ = population mean
• σ = population SD
• Backward formula: x = z (σ) + µ
▫ Use this one to solve for x
p.105
Equation for z-score
z
X 

Numerator is a deviation score
Denominator expresses deviation in standard deviation units
Determining raw score
from z-score
z
X 

so X    z
Numerator is a deviation score
Denominator expresses deviation in standard deviation units
Order of Operation
Backward formula
Regular formula
Raw score (x)
% or Proportion
Std. score (z)
Std. score (z)
% or Proportion
Raw score (x)
Raw (X)
z
% or Proportion
May not go directly from X to % or % to X – always go through z
Standard Normal Distribution
50%
of total area under the curve
1)
Symmetrical
μ=0
2)
Asymptotic
σ = 1.0
+1 σ is where the
line turns from
concave to convex
p.110, Fig. 4.1
Distributions
Normal distribution: all normal distributions
are symmetrical and bell shaped. Thus, the
mean, median & mode are all equal!
“Ideal world” bell curve
◦μ = 0
◦ σ = 1.0
◦ Completely symmetrical
◦ Asymptotic
Standard Normal Distribution
Fig 4.1
IQ Standard Normal Distribution
95%
68%
95%
99%
99%
Genius
Gifted
Above average
Higher average
Lower average
Below average
Borderline low
Low
>144
130-144
115-129
100-114
85-99
70-84
55-69
<55
0.13%
2.14%
13.59%
34.13%
34.13%
13.59%
2.14%
0.13%
Adapted from http://en.wikipedia.org/wiki/IQ_test
Shape of a Distribution
Researchers describe a distribution’s shape in
words rather than drawing it
Symmetrical distribution: each side is a mirror
image of the other
Skewed distribution: scores pile up on one side and
taper off in a tail on the other
◦ Tail on the right (high scores) = positive skew
◦ Tail on the left (low scores) = negative skew
Skewed Distributions
Mean, influenced by extreme scores, is found far toward the
long tail (positive or negative)
Median, in order to divide scores in half, is found toward the
long tail, but not as far as the mean
Mode is found near the short tail.
If Mean – Median > 0, the distribution is positively skewed.
If Mean – Median < 0, the distribution is negatively skewed
Distributions
• Positively skewed distribution: tendency for
scores to cluster below the mean
p.90
Distributions
Negatively skewed distribution: tendency for
scores to cluster above the mean
p.90
p.617
Appendix B
Used to find proportions under the normal curve
We only use Columns 1, 3 and 5
Column 3 gives proportion from the z-score towards either tail
Z or
z
Column 5 gives proportion between z-score and the mean
z
μ or μ
Z
Standard Scores
• Standard scores = z scores (in pop)
• µ = mu = population mean
• σ = sigma = population standard
deviation
• x = z (σ) + µ (Backward formula)
Finding a proportion from a raw score
1: Sketch the normal distribution
2: Shade the general region
corresponding to the required
proportion
3: Using the “forward” formula
compute the corresponding z-score
from the raw score (x).
4: Locate the proportion in the
correct column of the table
What proportion of
people have an IQ of 85
or less?
What about x < 70?
Step 1: draw curve
Step 2: use formula
z=x–μ
σ
Step 3: Appendix B
◦ Which column do you use?
For IQ scores
µ = 100 & σ =15
What proportion of people have an IQ of
130 or above?
Of x > 120?
Step 1: draw curve
Step 2: use formula
z=x–μ
σ
Step 3: Appendix B
◦ Which column do you use?
For IQ scores
µ = 100 & σ =15
Finding a z-score from a proportion
1: Sketch the normal distribution
2: Shade the general region
corresponding to the required
proportion
3: Locate the proportion in the correct
column of the table
4: Identify corresponding z-score in Col.1
5: Calculate raw score (x) with
“backwards” formula
What is the lowest IQ score
you can earn and still be in
the top 5% of the
population?
Step 1: draw curve
Step 2: Appendix B
◦ Which column do you use?
Step 3: use formula
X = (z)(σ) + µ
For IQ scores
µ = 100 & σ =15
• What proportion of
test takers score
between 300 and 650
on the SAT?
• Step 1: draw curve
• Step 2: use formula
z=x–μ
σ
• Step 3: Appendix B
▫ Which column do you use?
For SAT scores
µ = 500 & σ =100
Applications of z-Scores
Any observation from a normal distribution can be converted
to a z-score
If we have an actual, theoretical, or hypothesized normal
distribution of many means we can determine the position of
one particular mean relative to the other means in this
distribution
This logic is the basis of many hypothesis tests
Z-score Practice
Lecture Handout