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