Lecture 7-Measure of Dispersion

8/10/2019 Lecture 7-Measure of Dispersion

1/31


2/31


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.


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.


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


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.


7/31

Calculation:

1. Find largest and smallest number in data

set

2. Subtract smallest number from largest

number

3. Difference = Range


8/31

Ungrouped data:

= highest value lowest value

Grouped data:

= upper limit of last class

lower limit of first class


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


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


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


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


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


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


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.


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.


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


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


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


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


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


22/31

-1 +1

2.29.6

1411

25.812.4

68%

NORMAL DISTRIBUTION CURVE

1 Standard Deviation


23/31

-2 +2

018.2

1411

3713.8

95%


2 Standard Deviations


24/31


3 Standard Deviations

-3 +3

016.8

1411

3715.2

99.7%


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


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


27/31

CV = 100x

s


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.


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.


30/31

PEARSONS MEASURE OF SKEWNESS

3 Mean MedianMean Modeor

S tan dard Deviation S tan dard Deviation


31/31

Date post:	02-Jun-2018
Category:	Documents
Upload:	amirhazieq
View:	228 times
Download:	0 times

Lecture 7-Measure of Dispersion

Documents