Download - Lecture 7-Measure of Dispersion
-
8/10/2019 Lecture 7-Measure of Dispersion
1/31
-
8/10/2019 Lecture 7-Measure of Dispersion
2/31
-
8/10/2019 Lecture 7-Measure of Dispersion
3/31
Another important characteristic of adata set is how it is distributed, or how fareach element is from some measure of
central tendency (spread).
Two variables can have same value inthe measure of central tendencies but
dissimilar in other aspects such asconsistency, performance anddependable.
-
8/10/2019 Lecture 7-Measure of Dispersion
4/31
Measurements of central tendency
(mean, mode and median) locate thedistribution within the range of possiblevalues, measurements of dispersiondescribe the spread of values.
A small dispersion or variability ensures agood representation of the data bymeasures of central tendency.
In other words, mean, median or modethat is used has more credibility (or morebelievable or true) when the variation
about it is small.
-
8/10/2019 Lecture 7-Measure of Dispersion
5/31
Example:
Number of minutes 20clients waited to see a
consultant
ConsultantX Y
05 15 11 12
12 03 10 13
04 19 11 1037 11 09 13
06 34 09 11
Consultant X:
Sees some clients
almostimmediately
Others wait over1/2 hour
Highly inconsistent
Consultant Y:
Clients wait about10 minutes
9 minutes leastwait and 13minutes most
Highly consistent
-
8/10/2019 Lecture 7-Measure of Dispersion
6/31
Measure the total spread in the batch ofdata.
Simple, easily calculated measure of
total variation in the data.
However, it does not take account howthe data are actually distributed
between the smallest and largest value.
-
8/10/2019 Lecture 7-Measure of Dispersion
7/31
Calculation:
1. Find largest and smallest number in data
set
2. Subtract smallest number from largest
number
3. Difference = Range
-
8/10/2019 Lecture 7-Measure of Dispersion
8/31
Ungrouped data:
= highest value lowest value
Grouped data:
= upper limit of last class
lower limit of first class
-
8/10/2019 Lecture 7-Measure of Dispersion
9/31
Example:
Number of minutes 20
clients waited to see a
consultant
Consultant
X Y
05 15 11 12
12 03 10 13
04 19 11 10
37 11 09 13
06 34 09 11
Consultant X:
37 minutes highest
value 3 minutes smallest
value
Range = 37 - 3 = 34
minutes
Consultant Y:
13 minutes highest
value
9 minutes smallest
value
Range 13 - 9 = 4
minutes
-
8/10/2019 Lecture 7-Measure of Dispersion
10/31
http://www.mathgoodies.com/lessons/v
ol8/range.html
http://www.mathgoodies.com/lessons/vol8/range.htmlhttp://www.mathgoodies.com/lessons/vol8/range.htmlhttp://www.mathgoodies.com/lessons/vol8/range.html -
8/10/2019 Lecture 7-Measure of Dispersion
11/31
The interquartile range (IQR) is thedistance between the 75th percentileand the 25th percentile. The IQR is
essentially the range of the middle50% of the data. Because it uses themiddle 50%, the IQR is not affected byoutliers or extreme values.
This range is difference between thethird and first quartile = Q3 - Q1
-
8/10/2019 Lecture 7-Measure of Dispersion
12/31
Advantages over the range:1. Not sensitive to extreme values in a
data set
2. Not sensitive to the sample size Calculation:
1. Put the values in order from low to high
2. Divide the set of values into quarters(1/4s)
3. For the values in the middle 50% --subtract the lower value from the
higher value
-
8/10/2019 Lecture 7-Measure of Dispersion
13/31
Example:
16 sales people were given 12 problemsassociated with on-the-road sales
For keeping automobile expenses, therankings follow:
1 1 1 2 3 4 5 6 7 8 8 9 10 11 11 12
Q3= 10Q1=2
Range = 12 -1 = 11 Interquartile Range = 10 - 2 = 8
50% of respondents lie within 8 rank order points of each other!
Location Q1
= (n +1)
= (16 + 1)
= 4.25 = 4th
observation
Location Q3
= 3/4 (n +1)
= 3/4(16 +1)
= 12.75 = 13rd
observation
-
8/10/2019 Lecture 7-Measure of Dispersion
14/31
It is based on the lower quartile Q1 andthe upper quartile Q3.
The difference Q3 - Q1 is called the inter
quartile range. The difference Q3 - Q1
divided by 2 is called semi-inter-quartilerange or the quartile deviation.
ThusQuartile Deviation (Q.D) = Q3 - Q1
2
-
8/10/2019 Lecture 7-Measure of Dispersion
15/31
The quartile deviation is a slightly bettermeasure of absolute dispersion than therange. But it ignores the observation onthe tails.
If we take difference samples from apopulation and calculate their quartile
deviations, their values are quite likely tobe sufficiently different. This is calledsampling fluctuation. It is not a popular
measure of dispersion. The quartile deviation calculated from
the sample data does not help us todraw any conclusion (inference) aboutthe quartile deviation in the population.
-
8/10/2019 Lecture 7-Measure of Dispersion
16/31
These are the most familiar
measurements of dispersion.
Variance is the arithmetic mean (average)
of the square of the difference betweenthe value of an observation and thearithmetic mean of the value of all
observations.
-
8/10/2019 Lecture 7-Measure of Dispersion
17/31
http://www.mathsisfun.com/standard-
deviation.html
http://www.mathsisfun.com/standard-deviation.htmlhttp://www.mathsisfun.com/standard-deviation.htmlhttp://www.mathsisfun.com/standard-deviation.html -
8/10/2019 Lecture 7-Measure of Dispersion
18/31
Standard deviation is the square root ofthe variance.
Most frequently used measure of
dispersion
It is the average of the distances ofthe observed values from the mean
value for a set of data
Basic rule -- more spread will yield alarger SD
-
8/10/2019 Lecture 7-Measure of Dispersion
19/31
Calculation:
1. Calculate the arithmetic mean (AM)
2. Subtract each individual value from the AM3. Square each value -- multiply it times itself
4. Sum (total) the squared values
5. Divide the total by the number of values (N)
6. Calculate the square root of the value
21s (x x)n 1
2
2x1
xn 1 n
21s f (x x)n 1
2
2f x1
f xn 1 n
OR
OR
-
8/10/2019 Lecture 7-Measure of Dispersion
20/31
SD =Sum of squares of individual deviations from arithmetic mean
Number of items
Example: Scores
Deviations From
Mean
Squares of
Deviations
143
-13
-11
-09
-08
-03
-02
+01
+05
+20
+23
169
121
81
64
9
4
1
25
400
529
140
3
M = 143/10 = 14
No. of scores =
10
SD =1403
1
0
= 11.8
-
8/10/2019 Lecture 7-Measure of Dispersion
21/31
SD =Sum of squares of individual deviations from arithmetic mean
Number of items
Example: Scores
Deviations From
Mean
Squares of
Deviations
09
09
10
10
11
11
11
12
13
13
109
-02
-02
-01
-01
00
00
00
+01
+02
+02
4
4
1
1
0
0
0
1
4
4
19
M = 109/10 = 11
No. of scores =
10
SD =19
1
0
= 1.4
-
8/10/2019 Lecture 7-Measure of Dispersion
22/31
-1 +1
2.29.6
1411
25.812.4
68%
NORMAL DISTRIBUTION CURVE
1 Standard Deviation
-
8/10/2019 Lecture 7-Measure of Dispersion
23/31
-2 +2
018.2
1411
3713.8
95%
NORMAL DISTRIBUTION CURVE
2 Standard Deviations
-
8/10/2019 Lecture 7-Measure of Dispersion
24/31
NORMAL DISTRIBUTION CURVE
3 Standard Deviations
-3 +3
016.8
1411
3715.2
99.7%
-
8/10/2019 Lecture 7-Measure of Dispersion
25/31
Range
InterquartileRange
Use the range sparingly asthe measure of dispersion
Median is measure ofcentral tendency -- usethe interquartile range
Mean is measure ofcentral tendency -- usethe standard deviation
Standard
Deviation
-
8/10/2019 Lecture 7-Measure of Dispersion
26/31
The coefficient of variation (CV), alsoknown as relative variability, equals the
standard deviation divided by the mean. It
can be expressed either as a fraction or a
percent.
It only makes sense to report CV for a
variable, such as mass or enzyme activity,
where 0.0 is defined to really mean zero.A weight of zero means no weight. So it
would be meaningless to report a CV of
values expressed
-
8/10/2019 Lecture 7-Measure of Dispersion
27/31
CV = 100x
s
-
8/10/2019 Lecture 7-Measure of Dispersion
28/31
The terms skewed and askew are used torefer to something that is out of line or distorted
on one side. When referring to the shape offrequency or probability distributions,skewness refers to asymmetry of thedistribution.
-
8/10/2019 Lecture 7-Measure of Dispersion
29/31
A distribution with an asymmetric tail extendingout to the right is referred to as positivelyskewed or skewed to the right, while adistribution with an asymmetric tail extending
out to the left is referred to as negativelyskewed or skewed to the left. Skewness canrange from minus infinity to positive infinity.
-
8/10/2019 Lecture 7-Measure of Dispersion
30/31
PEARSONS MEASURE OF SKEWNESS
3 Mean MedianMean Modeor
S tan dard Deviation S tan dard Deviation
-
8/10/2019 Lecture 7-Measure of Dispersion
31/31