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F.Y. B. Sc. Statistics Notes Paper I Unit III
Notes Prepared by Prof (Mrs) M.J. Gholba
Measures of Central Tendency
According to Prof Bowley “Measures of central tendency (averages) are statistical constants which
enable us to comprehend in a single effort the significance of the whole.”
The main objectives of Measure of Central Tendency are
1) To condense data in a single value.
2) To facilitate comparisons between data.
There are different types of averages, each has its own advantages and disadvantages.
Requisites of a Good Measure of Central Tendency:
1. It should be rigidly defined.
2. It should be simple to understand & easy to calculate.
3. It should be based upon all values of given data.
4. It should be capable of further mathematical treatment.
5. It should have sampling stability.
6. It should be not be unduly affected by extreme values.
Measure of Central Tendency
Locational (positional ) average
Partition values
Mode
Mathematical Average
Arithmetic
Mean
Median
Geometric
Mean
Harmonic
Mean
Quartiles Deciles Percentiles
Partition values: The points which divide the data in to equal parts are called Partition values.
Median: The point or the value which divides the data in to two equal parts., or when the data is
arranged in numerical order
The data must be ranked (sorted in ascending order) first. The median is the number in the middle.
Depending on the data size we define median as:
F.Y. B. Sc. Statistics Notes Paper I Unit III
Notes Prepared by Prof (Mrs) M.J. Gholba
It is the middle value when data size N is odd. It is the mean of the middle two values, when data
size N is even.
Ungrouped Frequency Distribution
Find the cumulative frequencies for the data. The value of the variable corresponding to which a
cumulative frequency is greater than (N+1)/2 for the first time. (Where N is the total number of
observations.)
Example 1: Find the median for the following frequency distribution.
X
1
2
3
4
5
6
7
8
9
Freq
8
10
11
16
20
25
15
9
6
Solution: Calculate cumulative frequencies less than type.
X
1
2
3
4
5
6
7
8
9
Freq
8
10
11
16
20
25
15
9
6
Cum freq
8
18
29
45
65
90
105
114
120
N=120, ( N+1)/2=60.5 this value is first exceeded by cumulative frequency 65 , this value is
corresponding to X-value 5, hence median is 5
Grouped Frequency Distribution First obtain the cumulative frequencies for the data. Then mark
the class corresponding to which a cumulative frequency is greater than (N)/2 for the first time. (N
is the total number of observations.) Then that class is median class. Then median is evaluated by
interpolation formula.