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Descriptive Statistics
Chapter 3
MSIS 111
Prof. Nick Dedeke
1
Objectives
Define measures of central tendency,
variability, shape and association
Define statistical measures
Compute statistical measures for
ungrouped and grouped data
Interpret statistical results
2
Introduction
In most competitive sports, one looks
for the position of the athletes, e.g.
who came in first, second, and so on.
In statistics, one is interested in the
following measures:
- most frequent value in data set
- summary of all values in data set
- midpoint position of data set
- positions of data in data set
- distances to midpoint of data set
3
Exercise: Statistical Measure 1
We want to find out which of the
following students is the better one
using the available data.
Kuli 1st 2nd 1st 2nd 1st 4th 3rd 3rd 2nd 5th 1st
Marti 3rd 2nd 3rd 1st 2nd 1st 1st 1st 3rd 2nd 3rd
4
Using Statistical Measures
Kuli 1st 2nd 1st 2nd 1st 4th 3rd 3rd 2nd 5th 1st
Marti 3rd 2nd 3rd 2nd 2nd 1st 1st 1st 3rd 2nd 1st
Mode: Most frequently occurring value of variable
Mode for Kuli:
Mode for Marti:
Mean: Average of the values of a variable
Sample mean =  Xi
n
Mean or average score for Kuli
Mean or average score for Marti
5
Using Frequency Distributions
Analysis of Kuli’s performance
Mean =  Fi * Xi
 Fi
=
Mode =
Median point =
(11+ 1)/2 = 6th
Median value = 2nd
Using cumul. Freq.
column = 2nd
Xi
1st
2nd
3rd
4th
5th

Frequency Fi * Cum.
(Fi)
Xi
(C
Fi)
4
4
4
3
6
7
2
6
9
1
4
10
1
5
11
11
25
6
Using Frequency Distributions
Analysis of Marti’s performance
Mean =  Fi * Xi
 Fi
=
Mode =
Median point =
(11+ 1)/2 = 6th
Median value = 2nd
Using cumul. Freq.
column = 2nd
Xi
Frequency
(Fi)
1st
2nd
3rd
4th
5th

4
4
3
0
0
11
Fi * Cum.
Xi (C Fi)
4
8
9
0
0
21
4
8
11
0
0
7
New Case: Median measure
Analysis of Katie’s performance
Mean =  Fi * Xi
 Fi
=
Mode =
Median point =
(12+ 1)/2 = 6.5th
Median value
=(2nd+3rd)/2 = 2.5th
Average of the 6th
and 7th positions.
Xi
Frequency
(Fi)
1st
2nd
3rd
4th

4
2
5
1
12
Fi * Cum.
Xi (C Fi)
4
8
15
4
31
4
6
11
12
8
Percentiles
Sometimes we are not analyzing several
values from one person, but one value for
several persons or objects. For example we
have data from the performance of several
fund manager’s for year 2006. We want to
present the data in the form, XX manager is
in the top 10 or tenth percentile or top 25 or
25th percentile.
The method used consists of three steps
- organize data in ascending order
- calculate location of percentile you want
- identify the object in the percentile location
from the data set
9
Interpretation: Percentiles
If manager is in the tenth percentile of
of a group, this means that 90 % of
everyone in the data set scored better
than the manager.
If manager is in the 95th percentile of
of a group, this means that 5 % of
everyone in the data set scored higher
or better than the manager.
10
Exercise: Percentiles for Known Values
First
name
Bill
Fund
performance
106%
Jane
Sven
109%
114%
Larry
Dub
Anna
Cole
Salome
116%
121%
122%
125%
129%
In which percentile is Sven?
11
Response: Percentiles for Known Values
First
name
Fund
performance
Bill
Jane
Sven
106%
109%
114%
Larry
Dub
Anna
Cole
Salome
116%
121%
122%
125%
129%
Fi
Rel.
fi
Cum
fi
Percentiles
1
1
1
1/8
1/8
1/8
1/8
2/8
3/8
12.5th Percentile
1/8
1/8
1/8
1/8
1/8
4/8
5/8
6/8
7/8
1
50th Percentile
1
1
1
1
1
N=8
In which percentile is Sven?
25th Percentile
37.5th Percentile
62.5th Percentile
75th Percentile
87.5th Percentile
100th Percentile
12
Example: Percentiles for UnKnown Values
First
name
Fund
performance
Bill
Jane
Sven
106%
109%
114%
Larry
Dub
Anna
Cole
Salome
116%
121%
122%
125%
129%
Fi
Rel.
fi
Cum
fi
Percentiles
1
1
1
1/8
1/8
1/8
1/8
2/8
3/8
12.5th Percentile
1
1/8 4/8
1
1/8 5/8
1
1/8 6/8
1
1/8 7/8
1
1/8
1
N=8
What is the value of the 90th percentile?
25th Percentile
37.5th Percentile
50th Percentile
62.5th Percentile
75th Percentile
87.5th Percentile
100th Percentile
13
Computing Percentile locations
90th percentile location i = (P/100) * N
= 0.9 * 8 = 7.2th position
90th percentile is 0.2 or 20% between the 7th and 8th
The value for the 90th percentile is computed by
averaging the following values =
7th position’s value + (8th position’s value - 7th
position value)* Fraction got from computing i
125% + (129% - 125%)*0.2 = 125.8%
(~ 126%)
50th percentile location i = (P/100) * N
= 0.5 * 8 = 4th position
14
Computing Central Tend. Measures
Mean=  Fi *Xi
 Fi
= 1655/15
=110.33
Xi
Fi
Fi * Xi
55
60
100
125
2
1
3
5
110
60
300
625
140

4
15
560
1655
15
Computing Dispersion Measures
Mean (μ) =  Fi *Xi
 Fi
=1655/15
=110.33
Variance (s 2) =  Fi * (Xi- μ)2
(n –1)
=13573.335/(15 –1)
=969.52
Standard deviation (s) = 31.137
Xi
Fi
Fi * Xi
(Xi- μ)
(Xi- μ)2
Fi * (Xi- μ)2
55
60
100
2
1
3
110
60
300
-55.33
-50.33
-10.33
3061.409
2533.109
106.709
6122.818
2533.109
320.127
125
140

5
4
15
625
560
1655
14.67
29.67
215.209
880.309
1076.045
3521.236
13573.335
16
Computing Dispersion Measures 2
Var (s 2) = Fi* Xi 2 -  (Fi*Xi)2/n
(n –1)
= 196175 – (1655 2/15)/(15 –1)
=(196175 – 182601.66)/14 =
= 969.52
Standard deviation (s) = 31.137
Xi
Fi
Fi * Xi
(Xi)
55
60
100
2
1
3
110
60
300
3025
3600
10000
6050
3600
30000
125
140

5
4
15
625
560
1655
15625
19600
78125
78400
196175
2
Fi*(Xi)2
17
Exercise: Dispersion Measures
Var (s 2) = Fi* Xi 2 -  (Fi*Xi)2/n
(n –1)
Standard deviation (s) =
Xi
Fi
5
6
10
2
1
3
12
14

2
1
Fi * Xi
(Xi)
2
Fi*(Xi)2
18
Excel Examples
19
Grouped Data Examples
Class interval
Freq
(Fi)
M
Fi * M
Fi * M2
[1 – 3) inch
16
2 inches
32 inches
64 inches
[3 – 5) inch
2
4 inches
8 inches
32 inches
[5 – 7) inch
4
6 inches
24 inches
144 inches
[7 – 9) inch
3
8 inches
24 inches
192 inches
[9 – 11) inch
9
10 inches
90 inches
900 inches
[11 – 13) inch
6
12 inches
72 inches
864 inches

40
250
2,196
Var (s 2) = Fi* Mi 2 -  (Fi*Mi)2/n
(n –1)
Standard deviation (s) = 4.03 inches
= 2196 – 1562.5 = 16.24
39
20
Grouped Data Exercise
Class interval
Freq
(Fi)
[1 – 4) inches
4
[4 – 8) inches
4
[8 – 12) inches
6
[12 – 16) inches
12
[16 – 20) inches
8
[20 – 24) inches
6

40
M
Var (s 2) = Fi* Mi 2 -  (Fi*Mi)2/n
(n –1)
Fi * M
Fi * M2
=
Standard deviation (s) =
21