Download Chapter 4 - Bakersfield College

Chapter 4
SUMMARIZING SCORES WITH
MEASURES OF VARIABILITY
Going Forward
Your goals in this chapter are to learn:
• What is meant by variability
• What the range indicates
• What the standard deviation and variance are
and how to interpret them
• How to compute the standard deviation and
variance when describing a sample, when
describing the population, and when
estimating the population
Measures of Variability
Measures of variability describe the extent to
which scores in a distribution differ from each
other.
Three Samples
Three Variations of the Normal Curve
Measures of Variability
• Smaller variability indicates
– Scores are consistent
– Measures of central tendency describe the
distribution more accurately
– Less distances between the scores
• Larger variability indicates
– Scores are inconsistent
– Measures of central tendency describe the
distribution less accurately
– Greater distances between the scores
The Range
The Range
• The range indicates the distance between the
two most extreme scores in a distribution
• Range = Highest score – Lowest score
The Sample Variance and
Standard Deviation
Variance and Standard Deviation
The variance and standard deviation indicate
how much the scores are spread out around the
mean.
Sample Variance
The sample variance is the average of the
squared deviations of scores around the
sample mean.
2
SX
( X  X )

N
2
Sample Standard Deviation
The sample standard deviation is the square
root of the average squared deviation of scores
around the sample mean.
( X  X )
SX 
N
2
The Standard Deviation
• The standard deviation indicates something
like the “average deviation” from the mean,
the consistency in the scores, and how far
scores are spread out around the mean
• The larger the value of SX, the more the scores
are spread out around the mean, and the
wider the distribution
Normal Distribution and
the Standard Deviation
Normal Distribution and the
Standard Deviation
Approximately 34% of the scores in any normal
distribution are between the mean and the
score located one standard deviation from the
mean.
The Population Variance and
Standard Deviation
Population Variance
The population variance is the true or actual
variance of the population of scores.

2
X
( X   )

N
2
Population Standard Deviation
The population standard deviation is the true or
actual standard deviation of the population of
scores.
X
( X   )

N
2
Estimating the Population
Variance and Standard Deviation
2
X
• The sample variance ( S ) is a biased
2
estimator of the population variance ( X )
• The sample standard deviation ( S X ) is a
biased estimator of the population standard
deviation ( X )
Estimated Population Variance
By dividing by N – 1 instead of N, we have an
unbiased estimator of the population variance
called the estimated population variance.
( X  X )
s 
N 1
2
X
2
Estimated Population
Standard Deviation
Taking the square root of the estimated
population variance results in the estimated
population standard deviation.
( X  X )
sX 
N 1
2
Unbiased Estimators
2
X
• The estimated population variance ( s ) is an
unbiased estimator of the population variance
( )
2
X
• The estimated population standard deviation
( s X ) is an unbiased estimator of the
population standard deviation ( X )
2
X
2
X
Uses of S , S X , s , and s X
2
X
• Use the sample variance ( S ) and the
sample standard deviation ( S X ) to describe
the variability of a sample
2
X
• Use the estimated population variance ( s )
and the estimated population standard
deviation ( s X ) for inferential purposes when
you need to estimate the variability in the
population
Organizational Chart of Descriptive and
Inferential Measures of Variability
New Symbols
• X
2
The Sum of Squared Xs
First square each raw score and then sum
the squared Xs
• ( X )
2
The Squared Sum of X
First sum the raw scores and then square
that sum
Computing Formulas
The computing formula for the sample
variance is
(X )
X 
2
N
SX 
N
2
2
Computing Formulas
The computing formula for the sample standard
deviation is
(X )
X 
N
SX 
N
2
2
Computing Formulas
The computing formula for the estimated
population variance is
(X )
X 
2
N
sX 
N 1
2
2
Computing Formulas
The computing formula for the estimated
population standard deviation is
(X )
X 
N
sX 
N 1
2
2
Example
Using the following data set, find
• The range
• The sample variance and standard deviation
• The estimated population variance and standard deviation
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
Example—Range
The range is the largest value minus the smallest
value.
17  10  7
Example
Sample Variance
2
(
X
)
X 2 
2
N
SX 
N
(246) 2
3406 
3406  3362
2
18
SX 

 2.44
18
18
Example
Sample Standard Deviation
2
(
X
)
X 2 
N
SX 
N
(246) 2
3406 
18
SX 
 2.44  1.56
18
Example
Estimated Population Variance
( X )
X 
N
s X2 
N 1
2
2
(246) 2
3406 
3406  3362
2
18
sX 

 2.59
17
17
Example—Estimated Population
Standard Deviation
2
(

X
)
2
X 
N
sX 
N 1
(246) 2
3406 
18
sX 
 2.59  1.61
17