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PSY 201 Lecture Notes
Measures of Variability and Shape
Variability ‘
Variability refers to differences between score values
The larger the differences, the greater the variability
Example of low variability: Costs of year old Toyota Camrys in $1000s in a small town
25
27
25
26
26
26
27
Example of large variability: Costs of year old Toyota Camrys in $1000s in a large city
26
24
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22
30
32
20
Dot plots
Small town prices
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
25
30
35
40
Big city prices
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
25
30
35
40
After examination of the numbers, we can see that there are bigger differences between the 2nd than
between the 1st. How should those differences be summarized?
The possible measures:
1. Range: Difference between largest score and smallest score.
2. Interquartile Range: Difference between the score at the third quartile and score at the 1st
quartile.
3. Variance
4. Standard Deviation
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
4/29/2017
The Range
Range = Largest value in the collection minus smallest value.
Problems with the Range
1. Quite variable from sample to sample, even if all samples are from the same population.
How ironic that a measure of variability would be too variable.
2. May be restricted by ceiling or floor of the scale.
Much psychological measurement comes from scales to which persons respond on a 1-5
scale, often labeled 1=Strongly disagree, 2=Disagree, 3=Neutral, 4=Agree, 5=Strongly Agree.
“I support Richard M. Nixon.” in 1950s
Value Freq
5
1
4
3
3
100
2
3
1
1
“I support Richard M. Nixon.” in 1970s
Value
Freq
5
100
4
50
3
20
2
50
1
100
Range:
Range: 5-1=4
5-1=4
So the range is not generally useful, although it is often reported.
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The Interquartile Range
Quartiles:
Points identifying "quarters" of a distribution.
Conceptual Definitions
Q4
Fourth Quartile
The value below which 4/4th's of the distribution falls.
Q3
Third Quartile
The value below which 3/4ths of the distribution falls.
Q2
Second Quartile
The value below which 2/4ths of the distribution falls.
Q1
First Quartile
The value below which 1/4th of the distribution falls.
Q0
"Zeroth" Quartile
The value below which 0/4th's of the distribution falls.
Operational Definitions
Q4
The largest score value in the distribution.
Q3
The median of the scores in the upper half of the distribution.
(If N is odd, include the overall median in the upper half.)
Q2
The overall median of the collection. Compute using the median formula.
Q1
The median of the scores in the lower half of the distribution..
(If N is odd, include the overall median in the lower half.)
Q0
The smallest score value in the distribution.
Interquartile Range: The distance (on the number line) between the Q1 and Q3 - between the first
quartile and the third quartile.
IQR = Q3 - Q1
Interpretation
The distance or interval size required to contain the middle 50% of the scores.
If the middle 50% is contained in a small area, the distribution is quite "crowded" - the
scores are close to each other; the distribution has little variability.
If the middle 50% is contained in a wide area, the distribution is sparse - the scores are far
from either other; the distribution has much variability.
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
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Example - A distribution with an even number of scores. (But you don’t have to compute by hand.)
Upper half of distribution
75
65
50
45
40
40
35
35
30
30
30
25
25
10
So the interquartile range (IQR) for this distribution is 45 – 30 = 15.
Example - A distribution with an odd number of scores.
Note that 35, the overall median is included
in both the lower and upper halves.
Upper half of distribution
Lower half of distribution
65
50
45
40
35
35
30
25
25
20
15
So, the interquartile range (IQR) for this distribution is 42.5 – 25 = 17.5
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
4/29/2017
Comparing Variability of ACTComp scores of Females and Males
Q1
Q2
Q3
Females
Males
Interestingly, Q1, Q2, and Q3 are identical for Males and Females in this sample.
The interquartile range for each distribution is 25 – 19 = 6 for both distributions. Go figure.
Males and females are about equally variable on most characteristics.
Showing the interquartile range in a boxplot.
The IQR is simply the distance between
the top and bottom of the box.
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The Variance
The variance is the “average” of the squared differences of the scores from the mean.
Group
Symbol
Formula
Population
σ2 (sigma squared)
Σ(X-µ)2 / N
Sample
S2 or s2 (ess squared)(
Σ(X-X-bar)2 / (N-1)
So it’s a mean square.
Note that the formula for the sample variance is different from the formula for the population
variance.
The sample variance requires dividing the sum of squared differences by N-1, not N.
For this reason, the sample variance is almost the average of squared differences when computed
from a sample.
It is exactly the average when computed from a population.
Hence the quotes around average in the definition above.
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
4/29/2017
Computing the Variance using paper and pencil
The first set of Camry prices above
(X-Mean)2
1
1
Σ(X-X-bar)2 (called SS by Corty)
1
0
0
0
1
4
Sum of squared differences (Σ(X-X-bar)2) = 4
2
Population Variance = Σ(X-µ) / N = 4/7 = .57 We will never compute a population variance.
Sample Variance = Σ(X-X-bar)2 / (N-1) = 4/6 = .67 (Note – this is what SPSS computes.)
X
25
27
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27
Mean
26
26
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26
26
26
26
X-Mean
-1
1
-1
0
0
0
1
Graphical representation of the (X-Mean) differences for the first set of prices.
O
O
O
O
O
O
O
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Now the second set of Camry Prices
X
26
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30
32
20
Mean
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26
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26
X-Mean
0
-2
2
-4
4
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-6
(X-Mean)2
0
4
4
16
16
36
36
112
Sum of squared differences = 112
Population Variance = Σ(X-µ)2 / N
=112/7 = 16.
2
Sample Variance
= Σ(X-X-bar) / (N-1) = 112/6 = 18.67
Graphical representation of the (X-Mean) differences for the second set of prices
O
O
O
O
O
O
O
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Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
4/29/2017
What’s good and what’s bad about the variance
What’s good
1) The variance is “connected” to every score in the collection. This is generally regarded as a plus
– changing the value of ANY score will change the variance.
2) The variance has good lineage – it’s part of the formula for the Normal Distribution.
3) The variance is a key quantity in many inferential statistics.
What’s bad
1) The variance is in squared units. So its value is not easily related to the individual score values.
So the variance is not a good DESCRIPTIVE measure of variability.
Computing the Variance using SPSS
Enter the data
Analyze -> Descriptives
Choose all the Options you wish
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
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The Standard Deviation
The standard deviation is the square root of the variance.
Memorize this
formula
Group
Symbol
Formula
Population
σ (sigma)
Σ(X-µ)2 / N
Sample
S
Σ(X-X-bar)2 / (N-1)
Note that as was the case for the variance the formula for the sample standard deviation is different
from the formula for the population standard deviation. The sample standard deviation requires
dividing the sum of squared differences by N-1, not N.
FYI – Most computer programs automatically compute the “dividing by N-1” standard deviation.
This is what SPSS does.
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
4/29/2017
Computing the Standard Deviation using paper and pencil
Computation of the standard deviation involves
1) computing the variance,
2) taking the square root of the variance.
Consider the first set of Camry prices above
X
25
27
25
26
26
26
27
X-bar
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X-X-bar
-1
1
-1
0
0
0
1
(X-X-bar)2
1
1
1
0
0
0
1
We did this above computing
the variance.
Sum of squared differences = 4
Population VarianceN = 4/7 = .57
Sample VarianceN-1 = 4/6 = .67
Population standard deviation = sqrt(.57) = .75
Sample standard deviation = sqrt(.67) = .82
New stuff
Now the second set of Camry Prices – the big city prices
X
26
24
28
22
30
32
20
X-bar
26
26
26
26
26
26
26
X-X-bar
0
-2
2
-4
4
6
-6
(X-X-bar)2
0
4
4
16
16
36
36
We did this above computing
the variance.
Sum of squared differences = 112
Population VarianceN = 112/7 = 16
Sample VarianceN-1 = 112/6 = 18.67
Population standard deviation = sqrt(16) = 4.00
Sample standard deviation = sqrt (18.67) = 4.32
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
New stuff
4/29/2017
What’s good and what’s bad about the standard deviation
What’s good:
1) Connected to every score;
2) Good lineage;
3) Fits the data – the values of the standard deviation make sense.
What’s bad:
1) Inflated by skewness, outliers
2) What does the standard deviation mean????
Unfortunately, there is no simple, easy to digest, description of what the standard deviation
represents.
If anything, it might be thought of as the “average” of the differences of the scores from
the mean. If someone not familiar with statistics asks me what it is, that’s what I tell
them.
But, lack of an interpretation of the number doesn’t prevent us from using that number.
Computing the Standard Deviation using SPSS
Enter the data
Analyze -> Descriptives
Choose the Options you wish to have SPSS compute
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
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Computing the Standard Deviation using Excel
Enter the data into Excel
Highlight a cell.
Formula -> More Functions -> Statisticsl -> STDEV.S
Put the coordinates of the cells containing the data into one of the fields.
Marvel at your handiwork
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
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Two key facts about the standard deviation
1. For large (N bigger than 30) unimodal, symmetric (US) distributions, with no outliers . . .
About 2/3 of the scores will be within one standard deviation of the mean,
that is, between Mean-1SD and Mean+1SD.
<------About 2/3 of scores ----->
Mean - SD
Mean
Mean + SD
2. For large unimodal, symmetric (US) distributions, with no outliers . . .
About 95% of the scores will be within two standard deviations of the mean,
that is, between Mean–2SD and Mean+2SD.
<------95% of scores ----->
Mean - 2SD
Mean - SD
Mean
Mean + SD
Mean + 2SD
This means that if you know three things about the distribution:
1) that it’s unimodal and symmetric,
2) the mean, and
3) the standard deviation,
you can tell pretty much how an individual score placed in that distribution.
For example, Joe scores two standard deviations above the mean on a test.
What percent of the persons taking the test scored worse than Joe?
Fact 2 above says that 95% of the scores are below Joe’s.
And of the remaining 5%, ½ of that, or 2 ½ % would be in the left hand tail of the distribution, way
below Joe’s score and the other 2 ½% would be in the upper tail, above Joe’s score.
Joe
2 ½%
2 ½%
95%
So the answer is approximately 97.5% of the scores would be below Joe’s.
TEST QUESTION.
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
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Why do we care about Variability
1. It may be important to identify situations for which variability of opinion is high.
Example: Attitudes toward abortion.
Example: Attitudes toward parking meters in the Fort Wood area – large variability from ++ to --.
2. There may be situations in which it is important to have low (or high) variability.
For example, most teachers prefer low variability of entering ability when teaching classes such as
this.
If variability is low, this means that the teacher’s presentation will likely be understood by
everyone, if it’s chosen appropriately.
If variability is high, some may not understand and some may be bored.
The average depth of a river you must cross is 3’. If the standard deviation of depths is 6”, you’re
OK. But if the standard deviation of depths is 2’, then it’s quite likely you’ll hit a hole that it 7’
deep.
3. Variability is part of individual differences that must be explained by psychology.
There is variability in almost every human characteristic. The discovery of explanations for that
variability occupies much of the time of research psychologists.
Example
We measure Extraversion. Some people score high – they’re always “on” at parties and
functions. Others score low – they’d rather be alone. Why?
If you’re an introvert, see the Susan Cain You-Tube TED Talk on introverts.
By the way, Susan Cain has a best-selling book about introverts.
4. Variability, as measured by the standard deviation, is used assess the size of differences in
means.
Conventions concerning the size of mean differences
.2 Standard deviations = Small difference
.5 Standard deviations = Medium difference
.8 Standard deviations = Large difference
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
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Measures of distribution shape
Measures of skewness
A popular measure of skewness is the following, given by
Kirk, R. (1999). Statistics: An introduction. 4th Ed. New York: Harcourt Brace.
Skewness = (Σ(X-Mean)3 / N ) / S3
In English: The sum of the cubed deviations of scores from the mean divided by N, then divided by
the cube of the standard deviation.
Interpretation of values
Value of Skewness measure
Interpretaton
Larger than 0
Positively skewed distribution
0
Symmetric distribution
Less than 0
Negatively skewed distribution
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
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Example of the skewness statistic
1. Salaries from the Employee Data file.
2. Extroversion scores of 109 UTC students
Sta tistic s
sal ary Curren t Sa lary
N
Va lid
47 4
Mi ssing
Ske wne ss
2.1 25
Std . Erro r of S kewness
Sta tistic s
0
.11 2
he xt
N
Va lid
10 9
Mi ssing
1
Ske wne ss
Histogram
-.2 20
Std . Erro r of S kewness
.23 1
120
Histogram
100
14
12
60
10
40
20
0
$0
Mean = $34,419.57
Std. Dev. =
$17,075.661
N = 474
$40,000
$80,000
$120,000
$20,000
$60,000
$100,000
$140,000
Frequency
Frequency
80
8
6
4
Current Salary
2
Mean = 4.4582
Std. Dev. = 0.95104
N = 109
0
0.00
2.00
4.00
6.00
8.00
hext
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Kurtosis
Kurtosis refers to the relationship of the shape of a distribution to the shape of the Normal
Distribution.
Kirk gives the following measure of Kurtosis
Kurtosis = ( (Σ(X-Mean)4 / N ) / S4 ) - 3
In English: The sum of the deviations of scores from the mean raised to the fourth power divided by
N, then divided by the standard deviation raised to the fourth power minus 3.
Interpretation
Value of Kurtosis measure
Interpretaton
Larger than 0
More peaked than the Normal distribution
0
Same peakedness as the Normal distribution.
Less than 0
Less peaked (flatter) than the Normal distribution.
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
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Examples of two distributions with different Kurtosis.
1. Scores of 1000 values from a uniform distribution.
According to the Kurtosis measure the distribution is less peaked – flatter - than the Normal
Distribution.
2. Conscientiousness scores of 547 UTC students . . .
The Conscientiousness scores are slightly more peaked than the normal distribution.
Biderman’s 201 Handouts Topic 4 (Numeric Measures II) -14
4/29/2017