Download Types of Measurement

Types of Measurement
• Continuous
– Underlying continuum
• Index of amount
• Intermediate numbers make sense
• Discreet
– Is or is not
– Usually use only whole numbers
Scales of Measurement
• What are the Rules for assigning numerals
to objects?
• Nominal
• Ordinal
• Interval
• Ratio
Representing Data
•
•
•
•
•
Frequency Distribution
Grouped Frequency Distribution
Histogram
Bar Graph
Frequency Distribution
Measures of Central Tendency
• Mean
 fX
n
• Median
• Mode
Measures of Variability
• Range
• Standard Deviation & Variance
(X  M)

n
2
Steps for Calculating
Standard Deviation
1. Calculate the mean
 fX
n
2. Calculate each deviation score
(X  M)
3. Square each deviation score
(X  M)
2
S.D. Calculation (cont.)
4. Sum the deviation scores
(X  M)
2
5. Divide the sum of the squared deviations
by the sample size. This is the variance
(X  M) 2
n
6. Calculate the square root of the variance.
This is the S.D.
(X  M)
n
2
Scatter Plot:
Sample Data Set 1
T est Score scatter plot
5
nu mber of cases
4
3
2
1
0
0
1
2
3
4
5
6
tes t scor es
7
8
9
10
11
12
Shapes of Distributions
• Skewness
– The extent to which symmetry is absent (the tail
tells the tale)
• Positively skewed
– M>Md>Mode
• Negatively skewed
– Mode>Md>M
Shapes of Distributions (cont.)
• Whose class would you take?
– The mean scores in both Prof. A’s & Prof. B’s
sections of PSY 3601 were 75.
– The scores in Prof. A’s class were distributed in
a positively skewed manner.
– The scores in Prof. B’s class were distributed in
a negatively skewed manner.
Shapes of Distributions
• You represent the teachers’ union in contract
negotiations. The mean number of years of
experience for your teachers is 18 (a veteran
group). Teacher pay goes up with years of
experience.You want to make a case that your
teachers are underpaid.
• What measure of central tendency would you
report as average?
Shapes of Distributions
• Kurtosis
– The steepness of a distribution in its center
• Leptokurtic
• Mesokurtic
• Platykurtic
Normal Curve
• Bell-shaped, smooth, highest at center and tapers
on both sides
• Range from positive infinity to negative infinity
• Symmetrical
• Many variables, psychological & otherwise,
distributions approximate normal distribution
• Scores distributed normally are much easier to
interpret than scores from other distributions.
Characteristics of the normal
distribution
• 50% of scores above and 50% below the
mean
• Mean, Median, & Mode are the same point
• 68% of cases between 1 S.D. above and 1
S.D. below the mean
• 95% of cases between 2 S.D. above and 2
S.D. below the mean
Normal curve (cont.)
• Cumulatively
–
–
–
–
–
–
2 % of cases < 2 S.D. below the mean
16% of cases < 1 S.D. below the mean
50% of cases < Mean
84% of cases < 1 S.D. above the mean
98% of cases < 2 S.D. above the mean
99.9% of cases < 3 S.D. above the mean
Making sense of scores on
psychological tests
•
•
Convert “raw” score to another type of score
Why are converted or transformed scores
important?
1. Variability in tests: their format, length, people for
whom they are intended.
2. Psychological tests may or may not have “right” &
“wrong” answers.
•
Relative standing of individual in comparison to
group
Percentiles
• An expression of the percentage of people
whose score on a test falls below a
particular raw score; a converted score that
refers to a percentage of test takers
• Rank order information
nL
P  100
N
Percentiles (cont.)
• Advantages
–
–
–
–
Direct & clear picture
Easily communicated
Not influenced by the shape of the distribution
Always between 0 & 100
• Disadvantages
– Ordinal scale-can’t do arithmetic operations
– Not proportional in relationship to raw scores
– Tend to bunch up in the middle
• Overemphasize differences in the middle of the distribution
• Underemphasize differences in the extremes
Standard Scores
• Definition: A raw score that has been
converted from one scale into another, the
latter scale (1)having some arbitrarily set
mean and standard deviation and (2) being
more widely used and readily interpretable
• Different systems of standard scores
• Deviation scores
z Scores
• Tells how many standard deviations a score
is from the mean
• The difference between a particular raw
score and the mean divided by the standard
deviation
X X
z
s
z Scores (cont.)
• Ease comparing performance on different
tests
• Can manipulate them mathematically
(interval(?)) scale
• Scores can range from approximately -3 to
+3; M=0 & S.D.=1
z Scores (cont.)
• Problems with z Scores:
– Awkwardness of 0 & negative numbers
– Dealing with decimals
• Solution:
– Use z Scores as building block for other
Standard Scores
– Different tests use different standard score
systems. All based on z scores; tell you how
many S.D.s from the mean.
z Score Conversion
Formula:
Xnew  S.D.new (z) Mnew
Conversions (cont.)
• Xnew = new standard score
• S.D.new = desired & arbitrarily chosen
standard deviation of new standard score
• Mnew = desired & arbitrarily chosen mean of
new standard score
Xnew  S.D.new (z) Mnew
T Scores
•
•
•
•
•
•
Mean = 50
Standard Deviation = 10
T 60 = z 1(1 S.D. > M)
T 65 = z 1.5 (1.5 S.D.>M)
T 40 = z ?
T 35 = z ?
Other Standard Scores
• SAT & GRE
– M=500, S.D.=100
• Deviation “IQ” Scores
– Wechsler, etc.
• M=100, S.D.=15
– Binet
• M=100, S.D.=16
Stanines
• Standard 9
–
–
–
–
Divide normal distribution into 9 segments
Represented as numerals 1 through 9
Each stanine 1/2 S.D. in width
Stanine 5 straddles mean from 1/4 S.D. to 1/4
S.D. above
Transformed scores
• Linear Transformation: retains a direct linear
relationship to original raw score; differences
among standard scores parallels differences among
raw scores
• Nonlinear: when distribution is not normal; data
manipulated to approximate normal curve:
“normalized”; for comparing with tests from
which scores may be normally distributed