VARIABILITY
PREVIEW
Figure 4.1 the statistical mode for defining abnormal
behavior. The distribution of behavior scores for the
entire population is divided into three sections.
Extreme
abnormal
Average
Normal behavior
Extreme
abnormal
Overview
Definition
Variability provides a quantitative measure
of the degree to which scores in a
distribution are spread out or clustered
together.
Overview
In general a good measure of variability
will serve two purposes.
Variability describes the distribution.
Specifically, it tells whether the scores are
clustered close together or are spread out
over a large distance.
Variability measure how well an individual
score ( or group of scores ) represents the
entire distribution
The Range
Range = URL X max – LRL X min.
The range is the difference between the
upper real limit of the largest ( maximum )
X value and the lower real limit of the
smallest ( minimum ) x value .
The Interquartile Range And SemiInterquartile Range
The interquartile range is the distance
between the first quartile and the third
quartile :
The interquartile range Q3 – Q 1
The Semi- Interquartile Range
The semi- interquartile range is one-half of
the interquartile range: Q3 –Q12
Q3 – Q1
semi- interquartile range =
2
Standard Deviation And variance A
Population
Deviation is distance from the mean :
Deviation score = X - μ
Population variance = mean squared deviation.
Variance is the mean of the squared deviation
scores.
Standard deviation = variance
Sum Of Squared Deviations ( ss )
Variance = mean squared deviation =
Sum of squared deviations
Member of scores
SS, or sum of squares, is the sum of squared
deviation scores.
Definitional formula : SS = Σ ( x – μ )
Formulas For Population Standard
Deviation And Variance
Variance =
ss
N
Standard deviation is the square root of
variance, so the equation for standard
deviation is
SS
N
SUMMARY OF COMPUTATION FOR
VARIANCE AND STANDARD DEVIATION
1. Find the distance From the mean For each individual
2. Square each distance
3. 2- Find the sum of the squared distance . this value is
called ss or sum of squares. ( note : ss can also be
obtained using the computational formula instead of
steps 1 – 3 )
4. Find the mean of the squared distance. This value
called variance and measure the average squared
distance from the mean.
5. Take the square root of the variance. This value is
called standard deviation and provides a measure of
the standard distance from the mean.
FIGURE 4.4
Graphic representation Of The
Mean And Standard Deviation
Figure 4.5 The graphic representation of a
population with a mean of μ = 40 and
standard deviation of σ = 4 .
Standard Deviation And Variance
For Samples
Figure 4.6 the population of adult heights
form a normal distribution.
Standard Deviation And Variance
For Samples
Find the deviation for each score: deviation = x – x
2. Square of each deviation: squared deviation = ( x – x)2
3. Sum of the squared deviation: SS =Σ ( x – x ) 2
These three steps can be summarized in a definitional
formula SS :
Definitional formula: SS = Σ ( x – x ) 2
The value of ss can also be obtained using the
1.
computational formula using
this formula is:
sample notation,
Computational formula : ss = ΣX2
-
( ΣX )2
n
Standard Deviation And Variance
For Samples
Sample variance = S2 =
SS
n- 1
Sample standard deviation ( identified by
the symbols ) is simply the square root of
the variance .
SS
sample standard deviation = s =
n- 1
Biased And Unbiased Statistics
Definitions a sample statistic is unbiased if
the sample statistic. Obtained over many
different samples is equal to the
population parameter. On the other hand if
the average value for a sample statistic
consistently underestimates or
consistently overestimates the
corresponding population parameter, then
the statistic is biased.
Biased And Unbiased Statistics
Degrees of freedom or df for a
sample are defined as df = n – 1
Where n is the number of scores in
the sample.
Transformations Of Scale
1. Adding a constant to each score will not
change the standard deviation.
2. Multiplying each score by a constant
causes the standard deviation to be
multiplied by the same constant.
Factors That Affect Variability
1. Extreme scores
2. Sample size
3. Stability under sampling
4. Open – ended distributions.