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Statistics
Variability
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Variability
What does it mean?
• Variability:
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Degrees of Variability
Variability
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Variability
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Traits of good measures of variability
Variability
1.)
2.)
3.)
4.)
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4 types:
Measures of variability
1.) Range
2.) Interquartile Range
3.) Standard Deviation*
4.) Variance*
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The range
Formula
Range =
Measures of variability
ex. Calculate the range for the following ages:
2, 4, 28, 34, 35, 59, 61, 70, 73, 83
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The Interquartile Range & Semi-Interquartile Range
• Quartile sounds like what?
Distribution slicing
Measures of variability
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• Why do we use? How do we find the quartiles?
First Quartile (Q1):
Second Quartile (Q2):
Third Quartile (Q3):
The Interquartile Range & Semi-Interquartile Range
Measures of variability
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Interquartile range:
Semi-Interquartile Range:
Measures of variability
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Measures of variability
Figure 4.2, p. 108
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Q1=
24, 24, 25, 26, 26, 26, 27, 27, 30, 33, 33, 35, 35, 36, 43
• SIQR=
• IQR=
Find the Interquartile & Semi-Interquartile Ranges
Q3=Median=
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Standard Deviation & Variance
• The mean is the reference point for calculating these
measures
• Goal:
Measures of variability
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The mean and standard deviation: Population
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Standard Deviation & Variance
Step 1: find the deviation of
each score from the mean
• Formulas
Step 2a: Calculate the mean
of the deviation scores
Deviation score = X - µ
X
6
10
4
4
6
7
11
7
3
11
6
12
X - µ
-1.25
2.75
-3.25
-3.25
-1.25
-0.25
3.75
-0.25
-4.25
3.75
-1.25
4.75
Step 2b: To get rid of the
negative signs square
each deviation score first
(X - µ)²
1.56
7.56
10.56
10.56
1.56
0.06
14.06
0.06
18.06
14.06
1.56
22.56
Measures of variability
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Measures of variability
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Standard Deviation & Variance
Step 3: Compute the mean
squared deviation “Variance”
• Formulas
X
6
10
4
4
6
7
11
7
3
11
6
12
X - µ
-1.25
2.75
-3.25
-3.25
-1.25
-0.25
3.75
-0.25
-4.25
3.75
-1.25
4.75
(X - µ)²
1.56
7.56
10.56
10.56
1.56
0.06
14.06
0.06
18.06
14.06
1.56
22.56
Mean squared deviation =∑ (X - µ)²
OR
N
Mean squared deviation =
Variance (²) =
SS
N
=
Measures of variability
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Standard Deviation & Variance
• Formulas
Step 4: “Unsquare” to correct for
the squaring of all the
individual distances (i.e., take
the square root).
X
6
10
4
4
6
7
11
7
3
11
6
12
X - µ
-1.25
2.75
-3.25
-3.25
-1.25
-0.25
3.75
-0.25
-4.25
3.75
-1.25
4.75
(X - µ)²
1.56
7.56
10.56
10.56
1.56
0.06
14.06
0.06
18.06
14.06
1.56
22.56Standard Deviation ( ) = Variance =
Measures of variability
Standard Deviation ( ) =
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Standard Deviation & Variance
• Quick Steps for computing
Step 1: Find the distance from
Step 2: Square
Step 3: Find the Sum
Step 4: Find the mean
Step 5: Take the square root
Measures of variability
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Population vs. Sample Variability
Figure 4.6, p. 117
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Standard Deviation & Variance
• Steps for computing for a sample
Step 1: Find the distance from the mean for each individual
Step 2: Square each distance
Step 3: Sum the Squared distances (SS)
X - X
(X – X)²
SS = ∑(X – X)² Definitional formula
Measures of variability
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Standard Deviation & Variance
• Steps for computing for a sample
Step 4: Find the mean of the squared distances (Sample Variance)
Step 5: Take the square root of the sample variance (Std. Deviation)
• Must correct for bias in sample variability
Sample variance =
Sample standard deviation = S = S² =
Measures of variability
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In class exercise (Part 1)
1.) Calculate the mean (by hand)
2.) Calculate the variance (by hand)
3.) Calculate the standard deviation (by hand)
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Degrees of freedom (df)
n - 1df=
Measures of variability
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Biased vs. unbiased statistics
• Biased statistic
• Unbiased statistic
Measures of variability
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Importance of Variance and Std. Deviation
• Provides information
• Large variance:
• Small variance:
Measures of variability
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Factors that Affect Variability