1. Descriptive Statistics
Prepared By
Masood Amjad Khan
GCU, Lahore

2. Index Slide No. Subject Slide No. Subject 1. Index 2 2. Index 3
3. Statistics (Definitions)
4. Descriptive Statistics
Inferential Statistics
Examples of 4 and
7. Data, Level of measurements
8. Variable
9. Discrete variable
10. Continues variable
11. Frequency Distribution
12. Constructing Freq. Distn , 23
13. Example of , 25
14. Displaying the Data
15. Bar Chart, Pie Chart
16. Stem Leaf Plot
17. Graph
18. Histogram , 27
19. Frequency Polygon , 29
20. Cumulative Freq. Polygon , 31
Subject
Slide No.
21. Summary Measures
22. Goals
23. Arithmetic Mean , 40
24. Characteristic of Mean
25. Examples of
26. Weighted Mean
27. Example weighted Mean
28. Geometric Mean
29. Example: Geometric Mean
30. Median
31. Example of Median
32. Properties of Median
33. Mode
34. Examples of Mode
35. Positions of mean, median
and mode
36. Dispersion
37. Range and Mean Deviation
39. Example of Mean Deviation
40. Variance
Subject
Prepared by Masood Amjad Khan

3. Index Subject Subject Slide No. 61. 62. 63. 64. 65. 66. 67. 68. 69.
70.
71.
72.
73.
74.
75.
76.
77.
79.
80.
Slide No.
Subject
41. Examples of variance
42. Moments
43. Examples of Moments
44. Skewness
45. Types of Skewness
46. Coefficient of Skewness
47. Example of skewness
48. Empirical Rule
49. Exercise
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.

4. Examples of Descriptive
STATISTICS
Numerical Facts
(Common Usage)
Field or Discipline of Study
Definition
The Science of Collection, Presentation,
Analyzing and Interpretation of Data to make
Decisions and Forecasts.
1. No. of children born in a hospital
in some specified time.
2. No. of students enrolled in GCU
in 2007.
3. No of road accidents on motor
way.
4. Amount spent on Research
Development in GCU
during
5. No. of shut down of Computer
Network on a particular day.
Probability provides the transition
between Descriptive and
Inferential Statistics
Examples of Descriptive
And Inferential
Statistics
Descriptive
Statistics
Inferential
Statistics 1

5. Descriptive Statistics
Consists of methods for Organizing, Displaying,
and Describing Data by using Tables, Graphs,
and Summary Measures.
Data
A data set is a collection of observations on one
or more variables.
Types of Data 1

6. Frequency Distribution
Organizing the Data 1
Construction of
Frequency Distribution
Tables
Frequency Table
Frequency Distribution
A grouping of quantitative data into
mutually exclusive classes showing
the number of observations in each
class.
A grouping of qualitative data into
mutually exclusive classes showing
the number of observations in each
class.
Selling price of 80 vehicles
Vehicle Selling Number of
Price Vehicles
15000 to
24000 to
33000 to
Preference of four type of beverage
by 100 customers.
Beverage Number
Cola-Plus
Coca-Cola
Pepsi
7-UP

7. Displaying the Data Diagrams/Charts Graph Stem and Leaf Plot 1
Bar Chart
Pie Chart
Histogram
Frequency Polygon
Stem and Leaf Plot 1

8. Go to Descriptive Statistics
Variable
A characteristic under study that assumes different
values for different elements. (e.g Height of persons,
no. of students in GCU )
Qualitative or
Categorical variable
Quantitative
Variable
A variable that can be measured
numerically is called quantitative
variable.
A variable that can not assume
a numerical value but can be
classified into two or more
non numeric categories is
called qualitative or categorical
variable.
Continuous
variable
Discrete
variable
Educational achievements
Marital status
Brand of PC 1
Go to Descriptive Statistics

9. A variable whose observations can assume any
Continuous variable
A variable whose observations can assume any
value within a specific range.
Amount of income tax paid.
Weight of a student.
Yearly rainfall in Murree.
Time elapsed in successive network breakdown. 1
Back

10. Variable that can assume only certain values, and there
Discrete variable
Variable that can assume only certain values, and there
are gaps between the values.
Children in a family
Strokes on a golf hole
TV set owned
Cars arriving at GCU in an hour
Students in each section of statistics course 1
Back

11. Inferential Statistics
Consists of methods, that use sample results to help
make decisions or predictions about population. 1

12. Go to Inferential Statistics
Sample
A portion of population selected for study.
2. A sub set of Data selected from a population.
Estimation
Testing of
Hypothesis
Point
Interval
Selecting a Sample
Go to Inferential Statistics 1

13. Go to Inferential Statistics
Population
1. Consists of all-individual items or objects-whose
characteristics are being studied.
2. Collection of Data that describe some phenomenon
of interest.
Examples
Finite Population
Infinite Population
Length of fish in particular lake.
No. of students of Statistics
course in BCS.
No. of traffic violations on some
specific holiday.
Depth of a lake from any conceived
position.
Length of life of certain brand of
light bulb.
Stars on sky. 1
Go to Inferential Statistics

14. Descriptive and Inferential Statistics
Examples
Inferential
Descriptive
At least 5% of all fires reported
last year in Lahore were
deliberately set.
Next to colonial homes, more
residents in specified locality
prefer a contemporary design.
As a result of recent poll, most
Pakistanis are in favor of
independent and powerful parliament.
As a result of recent cutbacks by the
oil-producing nations, we can expect
the price of gasoline to double in the
next year. 1

15. Types of Data 1 Level of measurement
Data can be classified according to level of measurement.
The level of measurement dictates the calculations that can
be done to summarize and present the data.
It also determines the statistical tests that should be performed.
Level of measurement
Nominal
Ordinal
Interval
Ratio
Data are ranked no
meaningful difference
between values
Data may
only be
classified
Meaningful
difference
between values.
Meaningful 0 point
and ratio between
values.
Jersey numbers
of football
player.
Make of car.
Your rank in
class.
Team standings.
Temperature
Dress size
No. of patients seen
No of sales call made
Distance students
travel to class

16. Diagrams/Charts 1 Bar Chart Pie Chart A graph in which the classes
are reported on the horizontal
axis and the class frequencies on
vertical axis. The class frequencies
are proportional to the heights of
the bars.
A chart that shows the proportion or
percent that each class represents
of the total number of frequencies.
360
1300 79
286
Red
126
455
Orange 90
325
Lime 29
104
Black 36
130
White f
Angle
n =
Angle = (f/n)360 1
Back

17. Go to Descriptive Statistics
Graphs
Histogram
Frequency
Polygon
Cumulative Frequency
Go to Descriptive Statistics 1

18. Describing the Data Summary Measures Goals Skewness 1 Measures of
Location
Dispersion
Goals
Moments
Arithmetic Mean
Weighted Arithmetic
Mean
Geometric Mean
Median
Mode
Range, Mean Deviation
Variance, Standard
Deviation
Moments about Origin
Moments about mean
Skewness 1

19. Summary Measures Goals 1 Calculate the arithmetic mean,
weighted mean, median, mode,
and geometric mean.
Explain the characteristics, uses,
advantages, and disadvantages
of each measure of location.
Identify the position of the mean,
median, and mode for both
symmetric and skewed distributions.
Compute and interpret the range,
mean deviation, variance, and
standard deviation.
Understand the characteristics, uses,
advantages, and disadvantages
of each measure of dispersion.
Understand Chebyshev’s theorem and
the Empirical Rule as they relate to a set
of observations. 1

20. Characteristics of the Mean
The arithmetic mean is the most widely used measure of location. It requires the interval scale.
Its major characteristics are:
All values are used.
It is unique.
The sum of the deviations from the mean is 0.
It is calculated by summing the values and dividing by the number of values.
Every set of interval-level and ratio-level data has a mean.
All the values are included in computing the mean.
A set of data has a unique mean.
The mean is affected by unusually large or small data values.
The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero. 1

21. Use of Tables of Random Numbers
Selecting a Sample 1
Use of Tables of Random Numbers
Random numbers are the randomly produced digits from 0 to 9.
Table of random numbers contain rows and columns of these randomly
produced digits.
In using Table, choose:
the starting point at random
read off the digits in groups containing either one, two, three, or more
of the digits in any predetermined direction (rows or columns).
Example
Choose a sample of size 7 from a group of 80 objects.
Label the objects 01, 02, 03, …, 80 in any order.
Arbitrarily enter the Table on any line and read out the pair of digits in any two
consecutive columns.
Ignore numbers which recur and those greater than 80.
Go to Sample

22. Construction of Frequency Distribution
Step 1
Step 2
How many no. of groups (classes)?
Just enough classes to reveal the
shape of the distribution.
Let k be the desired no. of classes.
k should be such that 2k > n.
If n = 80 and we choose k = 6,
then 26 = 64 which is < 80, so k = 6
is not desirable. If we take k = 7,
then 27 = 128, which is > 80, so no.
of classes should be 7.
Determine the class interval (width).
the class interval should be the same
for all classes.
The formula to determine class width:
where i is the class width, H is the
highest observed value, L is the
lowest observed value, and k is the
number of classes.
Next 1

23. Construction of Frequency Distribution (continued)
Step 3
Step 4
Set the individual class limits.
Class limits should be very clear.
Class limits should not be
overlapping.
Some time class width is rounded
which may increase the range H-L.
Make the lower limit of the first
class a multiple of class width.
Make tally of observations falling
in each class.
Step 5
Count the number of items in each
class (class frequency)
Back
Example 1

24. Construction of Frequency Distribution ( Example )
Raw Data
( Ungrouped Data )
20445
19251
26613
22817
25449
24571
22374
21740
23613
17266
27443
35925
27896
26285
22845
20962
18890
29237
15546
32277
19331
21722
21442
20356
20633
20642
35851
23657
18263
15794
25799
24052
17968
32492
27453
24533
26661
25783
23765
20203
21981
19766
20818
28670
19688
20155
17357
20004
20895
17399
28337
23169
28034
25277
25251
19873
19889
29076
26651
24609
24324
24285
20047
15935
24296
21639
21558
19587
30872
28683
18021
17891
22442
30655
24220
23591
20454
23372
23197
Continued
Back 1

25. Construction of Frequency Distribution ( Example Continued )
Following Step 1, with n = 80 k should be 7.
Following Step 2 the class width should be 2911.
The width size is usually rounded up to a number multiple of 10 or 100.
The width size is taken as i = 3000.
Following Step 3, with i = 3000 and k = 7, the range is 7×3000=21000.
Where as the actual range is H – L = =
The lower limit of the first class should be a multiple of class width.
Thus the lower limit of starting class is taken as
Total = 80 2
33000 up to 36000 4
30000 up to 33000 8
27000 up to 30000 18
24000 up to 27000 17
21000 up to 24000 23
18000 up to 21000
15000 up to 18000
Frequency
Selling Price
Following Step 4
and Step 5
Back 1

26. Histogram
A graph in which the classes are marked on the horizontal axis and the
class frequencies on the vertical axis. The class frequencies are represented
by the heights of the bars and the bars are drawn adjacent to each other. 2 40
5.1 6 38
4.5 13 32
3.9
3.4 – 4.0 19
3.3 4
2.7
2.2 – 2.8
2.1 f cf H
Group
Histogram (Example 1) 5 10 15 20 25 30 35
1.60
2.20
2.80
3.40
4.00
4.60
5.20
Groups
Example 1 k = 6
Next 1

27. Histogram 1 Example 1 k = 7 Back Histogram (Example 1) 10 20 30 1.540 5 6 38
4.5 8 32 4 15 24
3.5 9 3
2.5 f cf H
Group
Histogram (Example 1) 10 20 30
1.5
2.0
3.0
4.0
5.0
Groups
Percent
Example 1 k = 7 1
Back

28. Frequency Polygon (Example 1)
A graph in which the points formed by the intersections of the class
midpoints and the class frequencies are connected by line segments. 2 40
4.9 6 38
4.3 13 32
3.7
3.4 – 4.0 19
3.1 4
2.5
1.9 f cf
Mid pt
Group
Example 1 k = 6
Frequency Polygon (Example 1)
1.90
2.50
3.10
3.70
4.30
4.90
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0 1 3 5
Raw Data
Percent
Mid point = ( Li +Hi )/2 1
Back

29. Frequency Polygon Continued3 40
4.75
4.5 – 5.0 5 37
4.25 10 32
3.75
3.5 – 4.0 15 22
3.25 4 7
2.75
2.5 – 3.0 1
2.25 2
1.75
1.5 – 2.0 f cf
Mid pt
Group
Example 1 k = 7
Frequency Polygon (Example 1)
1.75
2.25
2.75
3.25
3.75
4.25
4.75
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0 1 2 3 4
Data Example1
Percent
Back 1

30. Cumulative Frequency Polygon
A graph in which the points formed by the intersections of the class
midpoints and the class cumulative frequencies are connected by line
segments.
A cumulative frequency polygon portrays the number or percent of
observations below given value. 2 40
4.9 6 38
4.3 13 32
3.7
3.4 – 4.0 19
3.1 4
2.5
1.9 f cf
Mid pt
Group
Example 1 k = 6 1
Next

31. Cumulative Frequency Polygon Continued
Example 1 K = 7
Group
Mid pt cf f
1.5 – 2.0
1.75 2 2
2.25 3 1
2.5 – 3.0
2.75 7 4
3.25 22 15
3.5 – 4.0
3.75 32 10
4.25 37 5
4.5 – 5.0
4.75 40 3
Back 1

32. What is A Stem and Leaf Plot Diagram?
What Are They Used For?
A Stem and Leaf Plot is a type of graph that is similar to a histogram but shows more information.
Summarizes the shape of a set of data.
provides extra detail regarding individual values.
The data is arranged by placed value.
Stem and Leaf Plots are great organizers for large amounts of information.
The digits in the largest place are referred to as the stem.
The digits in the smallest place are referred to as the leaf
The leaves are always displayed to the left of the stem.
Series of scores on sports teams, series of temperatures or rainfall over a period of time, series of classroom test scores are examples of when Stem and Leaf Plots could be used.
Constructing
Stem and Leaf Plot 1

33. Constructing Stem and Leaf Plot
Begin with the lowest temperature.
The lowest temperature of the month was 50.
Enter the 5 in the tens column and a 0 in the ones.
The next lowest is 57.
Enter a 7 in the ones
Next is 59, enter a 9 in the ones.
find all of the temperatures that were in the 60's, 70's and 80's.
Enter the rest of the temperatures sequentially until your Stem and Leaf Plot contains all of the data.
Make Stem and Leaf Plot with
the following temperatures for June.
Stem (Tens) and Leaf (Ones) 8 7 6
0 7 9 5
Leaf (Ones)
Stem (Tens)
Temperature 1
Next

34. Stem and Leaf Example Make a Stem and Leaf Plot for the
following data.
Freq
Stem
Leaf 6 14 1 17 2 8 3 4
3 6 5
3 9 50
1.7
1.2
2.1
2.5
1.9
2.0
0.2
6.3
5.3
5.9
1.1
3.9
1.4
3.5
2.8
0.4
2.7
1.8
2.6
1.3
2.4
4.3
1.5
2.3
3.4
0.9
4.6
0.3
3.1
3.2
3.7
2.9
1.6
0.7 1
Next
Back

35. Stem and Leaf Plot Example
Following are the car battery life
Data.
Make a Stem and Leaf Plot. f S L 2 1
6 9 5 25 3 8 4 40
2.2
4.1
3.5
4.5
3.2
3.7 3
2.6
3.1
1.6
3.3
3.8
4.7
2.5
4.3
3.4
3.6
2.9
3.9
4.4
1.9
4.2 1
Next
Back

36. Stem and Leaf Plot Example
Frequency
Stem
Leaf 2 1
6 9 4 15 3 10 5
5 7 7 40
Go to Stem and Leaf Plot 1
Back

37. Measures of Location 1 Point of Arithmetic Mean Equilibrium Next
Ungrouped Data
Grouped Data
Population
Sample
N observations
X1, X2,…, XN in
the population.
n observations
X1, X2 ,…, Xn in
the sample
Let Xi and fi be the mid
point and frequency
respectively of the ith
group in the population
The mean is defined as
Let Xi and fi be the mid
point and frequency
respectively of the ith
group in the sample
The mean is defined as
Next

38. Numerical Examples Of Arithmetic Mean Ungrouped Data
Example of Sample Mean
Following is a random sample of
12 Clients showing the number of
minutes used by clients in a
particular cell phone last month.
What is the mean number of
Minutes Used?
Example of Population Mean
There are automobile manufacturing
Companies in the U.S.A. Listed below
is the no. of patents granted by the US
Government to each company.
Is this information a sample or population?
Number of
Company
Patent Granted
General Motors
511
Mazda
210
Nissan
385
Chrysler 97
DaimlerChrysler
275
Porsche 50
Toyota
257
Mistubishi 36
Honda
249
Volvo 23
Ford
234
BMW 13 90
110 89
113 91 94
100
112 77 92
119 83 1
Next
Back

39. Numerical Examples Of Arithmetic Mean Grouped Data
Following is the frequency distribution of Selling Prices of Vehicles at
Whitner Autoplex Last month.
Find arithmetic mean.
So the mean vehicle selling price is $23100.
1845.0 80
Total
69.0
34.5 2
126.0
31.5 4
228.0
28.5 8
459.0
25.5 18
382.5
22.5 17
448.5
19.5 23
132.0
16.5 fX X f
($ thousands)
Midpoint
Frequency
Selling Price
Go to
Summary measures
Back 1

40. Point of Equilibrium
An object is balanced at when
Back 1

41. Summary Measures EXAMPLE Weighted Mean 1 Weighted Mean
A special case of arithmetic mean.
Case when values of variable are
associated with certain quality, e.g price of medium, large, and big
The weight mean of a set of numbers
X1, X2, ..., Xn, with corresponding
weights w1, w2, ...,wn, is computed
from the following formula: 3
$1.50
Big 4
$1.25
Large
$0.90
Medium
Weights
Price
Soft Drink
EXAMPLE
Weighted Mean 1

42. EXAMPLE Weighted Mean Go to Back Summary measures 1
The Carter Construction Company pays its hourly employees
$16.50, $19.00, or $25.00 per hour. There are 26 hourly employees,
14 of which are paid at the $16.50 rate, 10 at the $19.00 rate, and 2 at the
$25.00 rate. What is the mean hourly rate paid the 26 employees?
Go to
Summary measures
Back 1

43. Summary Measures The geometric mean of a set of n
positive numbers is defined as the
nth root of the product of n values.
The formula for the geometric mean
is written:
The geometric mean used as the
average percent increase over time
n is calculated as:
Useful in finding the average change of percentages, ratios, indexes, or growth rates over time.
It has a wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the GDP, which compound or build on each other.
The geometric mean will always be less than or equal to the arithmetic mean.
Example 1

44. Example of Geometric Mean
The return on investment by certain
Company for four successive years
was 30%, 20%, -40%, and 200%.
Find the geometric mean rate of
return on investment.
Solution:
The 1.3 represents the 30 percent
return on investment, i.e original
Investment of 1.0 plus the return of
0.3. So
Which shows that the average return is
29.4 percent.
If you earned $30000 in 1997 and
$50000 in 2007, what is your annual rate of
increase over the period?
The annual rate of increase is 5.24 percent.
Back
Summary Measures 1

45. Median Example 1 Median is the midpoint of the values
If number of observations n is odd,
the median is( n+1)/2th observation.
If n is even the median is the
average of n/2th and (n/2+1)th
observations
Example:
Determine the median for each set of
data.
Arrange the set of data
n=7 median is 4th observation
that is 33.
Median is the midpoint of the values
after they have been ordered from
the smallest to the largest, or the
largest to the smallest
2) n=6, median is average of 3rd and
4th observation, that is (27+28)/2
= 27.5.
Median for Grouped Data
The median is obtained by using the
formula:
Where m is the group of n/2th obs.
Lm, Im, fm, and cfm-1 are the lowest
value, class width, frequency, and
cumulative frequency respectively of
the mth group.
Example 1

46. Back Go to Summary Measures 1
Example (Median)
n/2 = 20, so median group is
Lm = 3.40, Im = 0.6, fm = 13, cfm-1 = 19
Find the Median for the following
data.
Example 1 L H f cf
1.60
<
2.20 2
2.80 4 6
3.40 13 19
4.00 32
4.60 38
5.20 40
Back
Go to Summary Measures 1

47. Properties of the Median
There is a unique median for each data set.
It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.
It can be computed for ratio-level, interval-level, and ordinal-level data.
It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.
Go to Summary Measures 1

48. Mode 1 Next The mode is the value of the observation that appears most
frequently.
Region
No. of Seniors
New England
524
Middle Atlantic
818
E.N.Central
815
W.N.Central
367
S.Atlantic
679
E.S.Central
196
W.S.Central
436
Mountain
346
Pacific
783
Mode 1
Next

49. Mode (Example)
Back
Next 1

50. Mode Grouped Data Back 1 Calculating Mode for Grouped Data.
Calculate the mode of the following
Distribution.
Solution:
Modal Group is
fm = 14, fm-1 = 4, fm+1 = 12 and Im= 0.6
Group f 2 4 14 12 6
Back
Go to Summary Measures 1

51. The Relative Positions of the Mean, Median and the Mode
Go to Summary Measures 1

52. Dispersion Next 1 Why Study Dispersion?
A measure of location, such as the mean or the median, only describes the center of the data. It is valuable from that standpoint, but it does not tell us anything about the spread of the data.
For example, if your nature guide told you that the river ahead averaged 3 feet in depth, would you want to wade across on foot without additional information? Probably not. You would want to know something about the variation in the depth.
A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions.
Studying dispersion through display.
Next 1

53. Range and Mean Deviation
Example
The number of cappuccinos sold at
the Starbucks location in the Orange
Country Airport between 4 and 7p.m.
for a sample of 5 days last year were
20, 40, 50, 60, and 80. Determine the
mean deviation for the number of
cappuccinos sold.
Range = Largest value – Smallest value
Range = Largest – Smallest value
= 80 – 20 = 60
Next
Back 1

54. Mean Deviation Example
Solution
Number of Cappuccinos
Absolute Deviation
Sold Daily ( X ) 20
= -30 30 40
= -10 10 50
= 0 60
= 10 80
= 30
Total
Example
The number of cappuccinos sold
at he Starbucks location in the
Orange Country Airport between
4 and 7 p.m. for a sample of 5
days last year were 20, 40, 50,60,
and 80.
Determine the mean deviation for
the number of cappuccinos sold.
Next
Back 1

55. Mean Deviation (Grouped Data)
Mean Deviation for Grouped Data
288.6
Total
22.8
11.4
34.5 2
33.6
8.4
31.5 4
43.2
5.4
28.5 8
2.4
25.5 18
10.2
-0.6
22.5 17
82.8
-3.6
19.5 23
52.8
-6.6
16.5 X f 80
Total
34.5 2
31.5 4
28.5 8
25.5 18
22.5 17
19.5 23
16.5 X f
($ thousands)
Frequency
Selling Price
Back 1
Go to Summary Measures

56. Variance and Standard Deviation
Population variance and standard
deviation.
Let X1, X2,…, XN be N observations
in the population.
The variance is defined as:
The standard deviation is defined as:
The sample variance and
Standard deviation.
Let X1, X2,…, Xn be n observations
in the sample.
The variance is defined as:
The standard deviation is defined as:
Next 1

57. Example Variance and standard deviation
The number of traffic citations issued
during the last five months in
Beaufort County, South Carolina, is
38, 26, 13, 41, and 22. What is the
population variance?
The hourly wages for a sample of
part-time employees at Home Depot
are: $12, $20, $16, $18, and $19.
What is the sample variance?
Hourly Wage
$ ( X ) 12 -5 25 20 3 9 16 -1 1 18 19 2 4 85 40
Next 2
Back

58. Example Grouped Data Next Back 2
The sample standard deviation is defined as:
Example:
For the following frequency distribution of prices of vehicle, compute the
standard deviation of the prices.
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59. Go to Measures of Dispersion
Example (continued)
Alternate method of computing variance is:
Example
Group
Mid pt (X) f fX
fX2
1.75 2
3.5
6.125
2.25
4.5
10.13
2.75 5
13.75
37.81
3.25 15
48.75
158.4
3.75 8 30
112.5
4.25 6
25.5
108.4
4.75
9.5
45.13
Total 40
135.5
478.5
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60. Moments 2 Next Moments about Origin The rth moment about origin ‘a’ is
defined as:
Moments about Mean
The rth moment about mean is
First moment about mean is Zero.
Moments of Grouped Data
The rth moment about origin ‘a’ is
defined as:
The rth moment about mean is
First moment about mean is Zero. 2
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61. Example of Moments Next Back 2 Moments about Mean. Group Mid pt (X) ffX
1.75 2
3.5
2.25
4.5
2.75 5
13.75
3.25 15
48.75
3.75 8 30
4.25 6
25.5
4.75
9.5
Total 40
135.5
Moments about Mean.
5.445
2.645
2.1125
0.3375
0.98
0.343
4.335
3.645
19.5
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62. Example of Moments (Continued)
-1.97
-9.84
19.37
-38.11
75.00
-1.17
-10.51
12.28
-14.34
16.75
-0.37
-5.52
2.03
-0.75
0.28
0.43
4.32
1.87
0.81
0.35
1.23
7.39
9.11
11.22
13.82
4.06
8.26
16.78
34.10
2.83
8.02
22.71
64.32
3.63
7.26
26.38
95.82
348.03
87.31
94.14
552.65
Class f X fX 5
0.4 2 9
1.2
10.8 15 30 10
2.8 28 6
3.6
21.6
4.4
8.8 1
5.2 12
Total 50
118.4 2
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63. Skewness
Mean, median and mode are measures of central location for a set of observations and measures of data dispersion are range and the standard deviation.
Another characteristic of a set of data is the shape.
There are four shapes commonly observed:
symmetric,
positively skewed,
negatively skewed,
Bimodal
The coefficient of skewness can range from -3 up to 3.
A value near -3, such as -2.57, indicates considerable negative skewness.
A value such as 1.63 indicates moderate positive skewness.
A value of 0, which will occur when the mean and median are equal, indicates the distribution is symmetrical and that there is no skewness present.
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64. Types of Skewness
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65. Coefficient of Skewness
The Pearson coefficient of skewness is
defined as:
Example
Following are the earnings per share for a sample of 15 software companies for the year The earnings per share are arranged from smallest to largest.
Compute the mean, median, and standard deviation. Find the coefficient of skewness using Pearson’s estimate. What is your conclusion regarding the shape of the distribution?
Solution
The shape is moderately positively skewed.
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66. Example of Skewness (Continued)
Class 5 2
20.65 8 13
9.6
12.14 14 27 28
2.61 11 38
30.8
1.49 7 45
25.2
9.55 47
8.8
7.75 1 48
5.2
7.66 50 12
25.46
Total
118.4
87.31
The skewness can also be measured
with moments as:
m2 = 1.75, m3 = 62
b = 0.492
The shape is slightly positively skewed 2
Go to Skewness
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67. Go to Skewness Back 2 Next
Example Skewness
Mode
Median
Mean
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68. Empirical Rule Next Go to Skewness Back 2 Empirical Rule
For a symmetrical, bell-shaped frequency distribution:
Approximately 68% of the observations will lie within plus and minus one standard deviations of the mean. ( mean ±s.d )
About 95% of the observations will lie within plus and minus two standard deviations of the mean. ( mean ± 2s.d )
Practically all (99.7%) wiill lie within plus and minus three standard deviations of the mean. ( mean ± 3s.d )
Let the mean of a symmetric distribution be 100 and standard deviation be 10, then the empirical rule is as follows:
68%
95%
99.7%
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69. Example Empirical Rule
Mean = 3.25 sd = 0.77
Mean ± sd = ( 2.48 – 4.05) ( 67.5%)
Mean ± 2sd = ( 1.71 – 4.79 ) ( 97.5%)
Mean ± 3sd = ( 0.94 – 5.56 ) ( 100%)
Consider the following distribution:
Check the empirical rule.
Mean = s.d = 0.75
Mean ± sd = ( 2.45 – 3.95 ) ( 67.5%)
Mean ± 2sd = ( 1.7 – 4.7 ) ( 97.5%)
Mean ± 3sd = ( 0.89 – 5.45 ) (100%)
Group f X fX
fX^2 2
1.75
3.5
6.13 5
2.25
11.3
25.3 8
2.75 22
60.5 10
3.25
32.5
106
3.75 30
113
4.25
21.3
90.3
4.75
9.5
45.1 40
130
446
1.6
2.5 3
3.4
3.8
1.8
2.6
3.2
3.5
4.1 2
3.6
2.3
4.2
2.8
3.3
4.3
3.7
2.4
2.9
4.5
4.6
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70. Exercise Back 3 The following is the distribution of
Wages per thousand employees in a
Certain factory.
For the following data of examination
marks find the Mean, Median, Mode,
Mean Deviation and variance. Also
find the Skewness.
No. of Employees
Daily Wages
Marks
30 – 39
40 – 49
50 – 59
60 – 69
70 – 79
80 – 89
No. of students 8 87
190
304
211 85 20 22 24 26 28 30 32 34 36 38 40 42 44 3 13 43
102
175
220
204
139 69 25 6 1
Calculate the
Modal
and Median
wages. Why is
difference b/w
the two.
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