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Scales of Measurement
S1-1. Scales of Measurement: important for selecting stat's (later on)
1. Nominal Scale: number is really a name!
1 = male 2 = female
2. Ordinal Scale: number represents degree of an attribute: (order things!)
take all exams and rank: 1,2,3,4,5,
but 5-1 = 4 no information!
3. Interval Scale:
A. different degrees of an attribute are indicated
B. different levels, or degree numbers are equally spaced
C. zero point is arbitrary (doesn't indicate absence of variable)
4. Ratio Scale:
A. numbers are equidistant (as in interval)
B. numbers indicate diff. degree of attribute
C. O indicates absence of attribute
Which one is nominal?
1.
The number of cigarettes a person
smokes in a day
2.
Level in school: freshman= 1,
sophomore=2, etc.
3.
Age in years
4. How happy are you?
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Which one is ratio?
12% 12% 12% 12% 12% 12% 12% 12%
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6.
7.
8.
Blood pressure level
Heart beat rate
Age in years
Number of dates one had last week
The number of hours of TV watched in a
month
All of the above
None of the above
I haven’t the faintest!
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1.
2.
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4.
5.
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Shavelson – Descriptive Statistics
Chapter 3 Objectives
(Much of this will also be conducted using SPSS in the lab)
Given data, be able to:
create a data matrix
create a frequency distribution
make sure you brush up on graphing in general (54) then,
given data, be able to create a histogram, and frequency
polygon (54-57)
know the difference between symmetric and skewed
distributions (positively and negatively), as well as
unimodal and bimodal distributions (60)
calculate and create a relative and cumulative frequency
distribution (and graph it). This requires that you can
calculate relative frequency (which is a proportion) and
cumulative proportions and percentages (62-66).
Given a percentile score, be able to explain what it means;
given a cumulative percentage distribution (e.g. figure 27), be able to tell what percentile a raw score lies at, or
given a percentile, give the raw score correlated to that
percentile (67-68).ge in a single-participant design (398401).
Shavelson – Descriptive Statistics
Shavelson Chapter 4
Mode: most frequently occurring score
Median: the score dividing the distribution in half (50th
percentile)
Mean: the “center of gravity”; scores above mean
balances with those below
Relationship among measures of Central Tendency
•
Mode: least stable. Good for quick estimates
•
Median: use for asymmetric distributions (best
measure of central tendency on skewed)
•
Mean most commonly used; useful for inferential
stats (but extreme scores influences mean more)
What is the mean of
22, 18, 24, 26, 10
1.
2.
3.
4.
5.
27
19
21
20
24
20%
20%
20%
20%
20%
1
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27
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What is the median of
33, 17, 25, 20, 35
1.
2.
3.
4.
5.
33
17
25
20
35
20%
20%
20%
20%
20%
1
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5
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27
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What is the mode of
19, 23, 54, 23, 29
1.
2.
3.
4.
5.
27.5
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26
24
23
20%
20%
20%
20%
20%
1
2
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5
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24
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27
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Shavelson – Descriptive Statistics
Variability
Hi score – low score = Range
Variance (square of SD)
SD = the average deviation from the mean
∑(X – X)2
N-1
= standard deviation
score (x) mean(xbar)
x-xbar
(x-xbar)2
5
5.09
-0.09
0.008264
7
5.09
1.91
3.644628
3
5.09
-2.09
4.371901 90.909091
5
5.09
-0.09
0.008264
2
5.09
-3.09
9.553719
10
5.09
4.91
24.09917 square root of var is Std Dev
9
5.09
3.91
15.28099
5
5.09
-0.09
0.008264
7
5.09
1.91
3.644628
3
5.09
-2.09
4.371901
0
5.09
-5.09
25.91736
average
5.0909
sum
0.00
sum (x-xbar)2
90.90909
n-1
90.9
9.090909is variance
10
3.02is SD
Shavelson Chapter 5
S5-1. Define, be able to create and recognize graphic representations of a normal
distribution (115-121).
Normal distribution: Provides a good
model of relative frequency
distribution found in behavioral
research.
Shavelson Chapter 5
S5-2. Know the four properties of the normal distribution (120-121).
Unimodal, thus the greater the distance a
score lies from the mean, the less the
frequency of at score.
Symmetrical
Mean, mode, and median all the same
Aymptotic line never touches the
abscissa
Note that the mean and variance can
differ, thus “a family of normal
distributions”
Shavelson Chapter 5
S5-3. You should know what is meant by the phrase “a family of normal
distributions” (121,3). I will also cover in class the general issues of
“distributions” which are frequently used in statistical analyses.
From:http://www.gifted.uconn.edu/siegle/research/Normal/instructornotes.html
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These distributions have the same
mode, different median and SD
These distributions have different
mode, same median, different SD
These distributions have different
means, modes and variances
These distributions have the same
mode, mean and median, but
different SDs
Th
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These distributions have the same
mean, but different SDs
These distributions have a
different means and medians, but
the same modes and SDs
These distributions have different
means, modes, and
Nothing is the same with these
two!
Th
1.
45
Shavelson Chapter 5
S5-4. Know the areas under the curve of a normal distribution (roughly, e.g. 34.13%,
13.59%, 2.14 % and .13% on either side of the mean)
From:http://www.gifted.uconn.edu/siegle/research/Normal/instructornotes.html
Shavelson Chapter 5
Shavelson Chapter 5
S5-5a. What is a standard score (z-score) (123,3)? Be able to calculate the z-score,
given a raw score, mean, and standard deviation.
Z score = X-mean
S
X = raw score
Mean = mean of distribution
S = standard deviation
Notice that to calculate the Z score you need the mean
and S of a distribution of scores.
Shavelson Chapter 5
S5-5b. What two bits of information does the z-score provide us (125, 1-2)?
Z scores provides the following information:
1. Size of Z scores indicates the number of standard
deviations raw score is from the mean
2. Sign (+ or -) indicates if the raw score is above the
mean (+) or below the mean (-)
A z score of -1.8 means…
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4.
The mean of the distribution is 1.8
The distribution is skewed
The raw score lies 1.8 means above
the mean
The raw score lies 1.8 standard
deviations below the mean
The raw score 1.8 lies standard
deviations above the mean
Th
1.
2.
3.
45
Shavelson Chapter 5
•
Know the three characteristics of a
distribution of z-scores summarized
as (125):
–
–
–
Mean=0
Std. Dev. And Variance=1
The shape of a z-score distribution will be
the same as the original frequency
distribution.