Numerical Measures of
Variability
Measures of Variability
Measures of Variability are a set of characteristics that
examine the dispersion or spread of the distribution that
the researcher may be interested in.
This group describes the amount of variability
(spread, dispersion, difference) in a set of values.
Range, Interquartile Range, Variance, Standard Deviation,
and Coefficient of Skewness
Range
The range is the difference between the highest value
and the lowest value.
Ex. If the highest value is 20 and the lowest value is 2,
the range is 18.
The range can be reported as 18 or that the “values range
from 2 to 20”.
Interquartile Range
The interquartile range measures the range of the middle 50%
of an ordered data set.
not effected by outliers
still preserves the idea of the range
In finding the interquartile range, we actually locate the 25th,
50th, & 75th quartiles
Interquartile Range
Steps to finding the Interquartile Range
1. Rank order the data, Find the Median, Mark
that value with Q2 (this is the 50th quartile)
2. Find the Median of the Lower 50% w/o including the
Q2 value, Mark that value with Q1 (this is the 25th
quartile)
3. Find the Median of the Upper 50% w/o including the
Q2 value, Mark that value with Q3 (this is the 75th
quartile)
4. IQR = ( Q3 – Q1 )
Variance and Standard Deviation
Most commonly reported and utilized measures of variability
Can be influenced by outliers
Variance and Standard Deviation
If the data is widely scattered,
larger values
Variance
The variance is the average squared amount of deviation
from the mean. Generally used as a step toward the
calculation of other statistics.
Sample
Variance (s2 ) = (x – x )2 / n - 1
x2 = deviation scores squared
Population
Variance (σ2) = (x - µ)2 / N
Standard Deviation
The standard deviation is the average amount of
deviation from the mean. It is the square root of the
variance.
Sample
Standard Deviation (s) =
(x - x)2 / n - 1
(s) = S2
Population
Standard Deviation (σ) =
(σ) = σ2
(x - µ)2 / N
Variance and Standard Deviation
Raw Score
Mean
x
12
11
11
10
9
9
9
8
7
7
6
x
9
9
9
9
9
9
9
9
9
9
9
Deviation Score
(x - x )
Deviation Scores Squared
(x - x )2
∑(x - x )2 =
Variance and Standard Deviation
Raw Score
Mean
x
12
11
11
10
9
9
9
8
7
7
6
x
9
9
9
9
9
9
9
9
9
9
9
Deviation Score
(x - x )
(12-9) = 3
(11-9) = 2
(11-9) = 2
(10-9) = 1
(9-9) = 0
(9-9) = 0
(9-9) = 0
(8-9) = -1
(7-9) = -2
(7-9) = -2
(6-9) = -3
Deviation Scores Squared
(x - x )2
∑(x - x )2 =
Variance and Standard Deviation
Raw Score
Mean
x
12
11
11
10
9
9
9
8
7
7
6
x
9
9
9
9
9
9
9
9
9
9
9
Deviation Score
(x - x )
3
2
2
1
0
0
0
-1
-2
-2
-3
Deviation Scores Squared
(x - x )2
(3)2 = 9
(2)2 = 4
(2)2 = 4
(1)2 = 1
(0)2 = 0
(0)2 = 0
(0)2 = 0
(-1)2 = 1
(-2)2 = 4
(-2)2 = 4
(-3)2 = 9
∑(x - x )2 =
Variance and Standard Deviation
Raw Score
Mean
x
12
11
11
10
9
9
9
8
7
7
6
x
9
9
9
9
9
9
9
9
9
9
9
Deviation Score
(x - x )
3
2
2
1
0
0
0
-1
-2
-2
-3
Deviation Scores Squared
(x - x )2
9
4
4
1
0
0
0
1
4
4
9
∑(x - x )2 = 36
Variance (s2 ) = (x – x )2 / n – 1
(s2 ) = 36 / 11 – 1 = 36 /10 = 3.6
Standard Deviation (s) = S2
(s) =
n = 11
3.6 = 1.8973 = 1.90
Coefficient of Skewness
To measure the skew of the distribution, Pearson’s coefficient
of skewness is often found
based on the relationship between the mean, median,
and standard deviation
Can range in values from -3 to +3
Coefficient of Skewness
Mean = 10
Median = 7
S=4
sk = 3(10 – 7)/4 = 3(3)/4 = 9/4 = 2.25
Data is positively skewed
For our data set
Mean = 9
Median = 9
S = 1.90
sk = 3(9 – 9)/1.90 = 3(0)/1.90 = 0/1.90 = 0
Data is normally distributed
Coefficient of Skewness
Coefficients that are zero or near zero will have data that will
display equal tails
Coefficient of Skewness
Coefficients that are more positive in value will have data that
displays a longer tail to the right
Coefficient of Skewness
Coefficients that are more negative in value will have data that
displays a longer tail to the left