Sanjay Rastogi, IIFT,New Delhi
Descriptive Statistics

Sanjay Rastogi, IIFT,New Delhi
Measures of Central Tendency:
Ungrouped Data
•
Measures of central tendency yield information
about “particular places or locations in a group of
numbers.”
•
Common Measures of Location
–
Mode
–
Median
–
Mean
–
Percentiles
–
Quartiles

Sanjay Rastogi, IIFT,New Delhi
Mode
•
The most frequently occurring value in a data
set
•
Applicable to all levels of data measurement
(nominal, ordinal, interval, and ratio)
•
Bimodal -- Data sets that have two modes
•
Multimodal -- Data sets that contain more
than two modes

Sanjay Rastogi, IIFT,New Delhi
•
The mode is 44.
•
There are more 44s
than any other value.
35
37
37
39
40
40
41
41
43
43
43
43
44
44
44
44
44
45
45
46
46
46
46
48
Mode -- Example

Sanjay Rastogi, IIFT,New Delhi
Median
•
Middle value in an ordered array of
numbers.
•
Applicable for ordinal, interval, and ratio
data
•
Not applicable for nominal data
•
Least affected by extremely values.

Sanjay Rastogi, IIFT,New Delhi
Median:
Computational
Procedure
•
First Procedure
–
Arrange the observations in an ordered array.
–
If there is an odd number of terms, the median is the
middle term of
the ordered array.
–
If there is an even number of terms, the median is
the average of the middle two terms.
•
Second Procedure
–
The median’s position in an ordered array is given
by (n+1)/2.

Sanjay Rastogi, IIFT,New Delhi
Median:
Example
with an Odd Number of Terms
Ordered Array
3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 22
•
There are 17 terms in the ordered array.
•
Position of median = (n+1)/2 = (17+1)/2 = 9
•
The median is the 9th term, 15.
•
If the 22 is replaced by 100, the median is 15.
•
If the 3 is replaced by -103, the median is 15.

Sanjay Rastogi, IIFT,New Delhi
Median:
Example
with an Even Number of Terms
Ordered Array
3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21
•
There are 16 terms in the ordered array.
•
Position of median = (n+1)/2 = (16+1)/2 = 8.5
•
The median is between the 8th and 9th terms,
14.5.
•
If the 21 is replaced by 100, the median is
14.5.
•
If the 3 is replaced by -88, the median is 14.5.

Sanjay Rastogi, IIFT,New Delhi
Arithmetic Mean
•
Commonly called ‘the mean’
•
is the average of a group of numbers
•
Applicable for interval and ratio data
•
Not applicable for nominal or ordinal data
•
Affected by each value in the data set,
including extreme values
•
Computed by summing all values in the data
set and dividing the sum by the number of
values in the data set

Sanjay Rastogi, IIFT,New Delhi
Population Mean
X
N
N
X
X
X
X
N
1
2
3
24
13
19
26
11
5
93
5
18 6
...
.

Sanjay Rastogi, IIFT,New Delhi
Sample Mean
X
X
n
n
X
X
X
X
n
1
2
3
57
86
42
38
90
66
6
379
6
63 167
...

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- Spring '17