Download Lecture 4

Variability
Variability refers to the Spread or
Dispersion of the Distribution
Variability of the Distribution
(Common Statistics)

Range


Variance


Max - min
Average Squared Distance from Mean
Standard Deviation

Average Distance from Mean
(Verbal definitions of Variance and Standard Deviation are not
exactly right, but close enough to right and easy to remember.)
Range



Maximum – minimum. Quick, easy.
2, 3, 4, 5, 5, 5, 6, 7. Range is 7 – 2 = 5
2, 3, 4, 5, 5, 5, 6, 19. Range is 19 – 2 = 17
9
8
7
What is the range for the
distribution shown in the
boxplot?
6
5
4
3
2
1
N=
18
VAR00001
Variance
2 






2
(
X


)

N
Population Variance:
2

Where means population variance,
means population mean, and the other terms have their
usual meaning.
The variance is equal to the average squared deviation from
the mean.
To compute, take each score and subtract the mean. Square
the result. Find the average over scores. Ta da! The
variance.
Think of this as the average squared distance from the mean.
The farther scores are from the mean, the bigger the variance.

Computing the Variance
(N=5)
X
X
X 
2
X X
X
N
(X  X )


2
N
(X  X )
5
15
-10
100
10
15
-5
25
15
15
0
0
20
15
5
25
25
15
10
100
Total:
75
0
250
Mean:
Variance
Is 
50
2
Standard Deviation



Variance is average squared deviation from
the mean.
To return to original, unsquared units, we
just take the square root of the variance.
This is the standard deviation.
Population formula:

 ( X  )
N
2
Standard Deviation





Sometimes called the root-mean-square
deviation from the mean. This name says
how to compute it from the inside out.
Find the deviation (difference between the
score and the mean).
Find the deviations squared.
Find their mean.
Take the square root.
Computing the Standard Deviation 

(N=5)
X
( X  X )2
N
X  X (X  X )
X
2
5
15
-10
100
10
15
-5
25
15
15
0
0
20
15
5
25
25
15
10
100
Total:
75
0
250
Mean:
Variance
Is 
50
Sqrt
SD
Is 
 50  7.07
Two Population Distributions
Both distributions have a mean of
zero (mu). One has a standard
deviation of 1.6, the other has a
standard deviation of 10. The SD
can be considered the average
distance from the mean.
Example: Age Distribution
Distribution of Age
Central Tendency, Variability, and Shape
16
Median = 23
Mean=25.73
12
Frequency
Average Distrance from Mean
Mode = 21
SD = 6.47
What is the variance of
this distribution
(approximately)?
8
4
0
10
20
30
age
40
50
Heiman’s notation for Variance and
Standard Deviation
Sample Variance
S 
2
X
2
(
X

X
)

N
Estimate of Population
Variance
s X2 
2
(
X

X
)

N 1
Sample Standard
Deviation
SX 
2
(
X

X
)

N
Estimate of Population
Standard Deviation
sX 
2
(
X

X
)

Stats builds. You have to understand and remember these.
N 1
Uses



Range is used when a simple description is
desired or when the variance or standard
deviation cannot be computed.
Standard Deviation is used to communicate
the variability of the distribution. Has
applications we will cover later.
Variance is used in many statistical
calculations for significance tests.
Review


What is variability?
Define terms:



Range
Variance
Standard Deviation
Examples and Computation


Age from demographics
Grades from last semester
Computation

How do we compute the range statistic?

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
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1 99th percentile – 1st percentile
2 75th quartile – 25th quartile
3 maximum – minimum
4 minimum – maximum
Definition

The standard deviation is an index of a
distribution’s __________.

1
2
3
4



central tendency
kurtosis
skew
variability
Graphs

Which distribution appears to have the
largest variance?
7
7
6
6
5
5
4
4
3
3
2
2
1
1
N=
9
9
VAR00001
N=
9
VAR00001
7.0
7.0
6.5
6.5
6.0
6.0
5.5
5.5
5.0
5.0
4.5
4.5
4.0
4.0
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.0
1.5
1.0
N=
9
VAR00001
1
2
3
N=
9
VAR00001
4
Graphs
Which graph shows a higher mean?
a. Young
b. Old
c. Same
d. Cannot tell
Which graph shows a higher
standard deviation?
a. Young
b. Old
c. Same
d. Cannot tell
Graphs
Which graph shows a higher mean?
a. Young
b. Old
c. Same
d. Cannot tell
Which graph shows a higher
standard deviation?
a. Young
b. Old
c. Same
d. Cannot tell