Download Measures of Variability

Measures of Variability
Vernon Enage Reyes
Review
• Measures of central tendency
1. Mean
2. Median
3. Mode
However, measures of central tendency shows
an incomplete picture
Example
• Honolulu, Hawaii
Mean daily temperature = 75 degrees
• Phoenix, Arizona
Mean daily temperature = 75 degrees
However:
Honolulu’s temperature range between 70 – 80
degrees
Arizona’s temperature range between 40 degrees
(January) to 100 degrees (July and August)
Measures of Variability
An index of how the scores are scattered around
the center of the distribution (measure of
central tendency – usually the mean)
Variability is also called: spread, width or
dispersion
Well know measures of variability: the range,
mean deviation, the variance and standard
deviation
1. The RANGE
• A quick and easy way to get a rough measure
of variability. The Range (R) is the difference
between the highest and lowest score of
distribution
• Honolulu hottest temp = 89 degrees
• Honolulu coldest temp = 65 degrees
Range = 24 degrees
Formula is... R = H - L
Disadvantage
• Although the range is quick-and-easy, it is
DEPENDENT only on two scores (the highest
and lowest), thus it shows inaccurate data.
Example:
Prof A: 18, 18, 19, 19, 20, 20, 22 and 23
Prof B: 18, 18, 19, 19, 20, 21 and 43
Thus scores are only affected only by ONE
student.
2. The Mean Deviation
• The mean deviation was discussed last time.
• Remember: the deviation is the distance of
any given raw score from the mean. (X – X)
• Remember also, if we add the deviations it will
equal to zero
Example
-----------------------------X
X–X
----------------------------9
+3
+5
8
+2
6
0
5
-1
-5
2
-4
X=6
• Notice that if we add all
the deviations it will
always equal to zero!
• (+)5 + (-)5 = 0
2. The Mean Deviation
• The mean deviation was discussed last time.
• Remember: the deviation is the distance of
any given raw score from the mean. (X – X)
• Remember also, if we add the deviations it will
equal to zero
• So what we do is convert the deviations into
“absolute deviations”(we just ignore the plus
and minus) then add them.
Mean deviation
• Formula
MD = Σ |X – X|
N
MD = mean deviation
Σ |X – X| = sum of the absolute deviation
(ignoring +/- signs) This is important to not get
a Zero
N = total number of scores
Example
--------------------------------------------------------------------X
X–X
|X – X|
--------------------------------------------------------------------9
+4
4
8
+3
3
6
+1
1
4
-1
1
2
-3
3
1
-4
4
Σ X = 30
0
16
X (Mean) = 30/6 = 5
To get the mean deviation
• Formula
MD = Σ |X – X|
N
= 16
6
MD = 2.67
The Variance and Standard Devition
• The Mean deviation is NOT anymore widely
used because it can not be manipulated
algebraically
• Also it can not be used to advanced statistical
procedures (inferential statistics)
• Thus what we do is just square the deviation
instead of converting it into absolute values
 This also removes the negative and positive
sign but can be algebraically manipulated
Variance
• Formula
s2 = Σ (X – X)2
N
Example
--------------------------------------------------------------------X
X–X
|X – X|
(X – X)
--------------------------------------------------------------------9
+4
4
16
8
+3
3
9
6
+1
1
1
4
-1
1
1
2
-3
3
9
1
-4
4
16
Σ X = 30
0
16
52
X (Mean) = 30/6 = 5
What for?
Distribution A
---------------------------------X
|X – X | (X – X)2
3
2
4
5
0
0
5
0
0
7
2
4
X=5
MD = 1
S2 = 2
Distribution B
---------------------------------X
|X – X | (X – X)2
4
1
1
4
1
1
6
1
1
6
1
1
X=5
MD = 1
S2 = 1
What for? Graph form
Distribution A
Distribution B
_____________________
2 3 4 5 6 7 8
_____________________
2 3 4 5 6 7 8
Standard deviation
• However, since the values has been squared, we
need to turn it back to its original unit of
measurement, that is, we need to take the square
root of the variance
• Formula
Variance
s2 = Σ (X – X)2
N
Standard deviation s =
Σ (X – X)2
N
√
Standard deviation an easier formula
Standard deviation s =
√
Σ (X – X)2
N
s 2 = Σ X2 / N – X2
s=
√
Σ X2 / N - X2
X2 = sum of squared raw
scores ( each raw score is
squared FIRST, then summed)
N = total number of score
X2 = mean squared
In the end
• We shall discuss further
how the standard
deviation is useful next
week.
• Note however that
deviations, are just like
mean, median and
mode
• Mean/median/mode
gives a single score to
describe a group
• Variance/standard
deviations gives a single
score to describe (how
close or how far are
from the mean) the
scores of a group.