Measure of central tendency (Mean, Median and Mode)

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Measure of Central Tendency
Mean ̅ , Median and Mode
Tutorial-1
Meaning , Properties and Method of calculation
Designed and prepared
By
Narender sharma
A
Quality professional
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Administrator
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www.shakehandwithlife.blogspot.in

Index
S.No. Contents Page No.
1.
Mean
 Meaning and definition
 General Properties
 Mathematical Properties
 Calculation of Mean for Individual Series
 Calculation of Mean for Discrete Series
 Calculation of Mean for Continuous Series
2.
Median
 Properties of median
 Location of median
 Calculation of median for individual series
 Calculation of median for discrete series
 Calculation for median for continuous series
3.
Mode
 Properties of Mode
 Calculation of mode for Individual series
 Calculation of mode for discrete series
 Calculation of mode for continuous series
4. Empirical Relation between Mean, Median and Mode
5. Excel Commands for calculation of Mean, Median and Mode
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Measure of central tendency
An average is single value within the range of the data that is used to represent all of the values in the series.
Such an average is somewhere within the range of the data, it is therefore called measure of central tendency.
Mean
is the sum of observations
divided by the number of
observations in the set.
Median
is a special point , which lies
in the centre of the data so
that half the data lie below
it and half above it.
Mode
The mode of the data set is
the value that occurs most
frequently.
Mean
The mean of a set of observation is their average. It is equal to the sum of all observations divided by the
number of observations in the set.
Let some observations denoted by … … … . Then the sample mean is denoted by
̅
∑
Where is notation of summation . The summation extends over all data points.
Where ̅ is used for sample mean.
For denoting the population mean the symbol used is (mu) and N used as the number of elements. Hence the
population mean is defined as
∑

General Properties Mean
 The mean summarizes all the information in the data.
 It is the average of all the observations.
 The mean is a single point that can be viewed as the point where all the mass –the weight – of the
observation is concentrated.
 It is the center of mass of the data.
 If all the observations in our data set were the same size, then (assuming the total is the same) each would
be equal to the mean.
 The mean is sensitive to extreme observations.
 The mean, however, does have strong advantages as a measure of central tendency. The mean is based on
information contained in all the observations in data set, rather than being an observation lying “in the
middle” of the set.
 The mean also has some desirable mathematical properties that make it useful in many contexts of
statistical inference.
Mathematical Properties
1) The sum of the deviations of the items from arithmetic mean is always zero
Symbolically, ∑ ̅
2) The sum of the squared deviations of the items from arithmetic mean is minimum i.e.
∑ ̅ ∑
3) If each item of a series is increased, decreased , multiplied or divided by some constant , then A.M. also
increases, decreases , multiplied or is divided by the same constant.
4) The product of the arithmetic mean and number of items on which mean is based is equal to the sum of all
given items i.e.
̅
∑
∑ . ̅
5) If each item of the original series is replaced by actual mean, then the sum of these substitutions will be
equal to the sum of the individual items.
̅ ̅ ̅ … ̅
Calculation of Mean for Individual Series
In case of individual series (Series of any Individual numbers), arithmetic mean can be computed by applying
any of the two methods.
Example 1
The pocket allowances of ten students are given below in ₹;
15, 20, 30, 22, 25, 18, 40, 50, 55 and 65
Calculate the arithmetic mean of pocket allowance.
Mean for Individual series
Direct Method
̅
∑
Shortcut Method
̅
∑

Solution :
15 20 30 22 25 18 40 50 55 65
,
̅
Example 2
Solve the above example by shortcut / Assumed mean method
Solution; Take Assumed mean (A) =40, and calculate the deviation (d) of assumed mean from each individual
value, then find the sum of this deviation.
̅
Calculation of Mean for Discrete Series
Discrete Series : The series of complete numbers like Men Cars etc. we can’t take numbers like . men .
cars.
Mean for Discrete series
Direct Method
̅
∑
Shortcut Method
̅
∑
Direct Method
When direct method is used the following formula is used ;
̅
∑

. .
Steps of Calculation
 Multiply the frequency of each item with the values of variable and obtain total
 Find the sum of frequencies i.e.
 Divide the total obtained ( by the number of observations N or . The result would be the required
arithmetic mean.
Example 3
In a small factory the data of wages to the workers given in below table. Calculate the average wage.
Wages (₹) 150 130 120 180 160
No. of Workers 4 5 3 2 5
Solution: Denote wages by (x)and number of workers by (f)
.
̅ .
Thus the average wage is ₹145.78
Shortcut Method/ Assumed Mean Method
When this method is used, the formula for calculation arithmetic mean is;
̅
. . .
Steps for calculation
 Any one of the items in the series is taken as assumed mean A.
 Take the deviations of the items from the assumed mean i.e. and denote these deviations by d
 Multiply these deviations with respective frequency and obtain the total i.e. .
 Divide the total obtained by the total frequency or total number of observations i.e. N
Let us solve the example 3 by shortcut method

₹ .
̅ . .
Calculation of mean for Continuous series
Continuous Series : The series of continuous numbers like 6.5 Kg , 4.5 meters etc. This implies if we choose an
interval of 2 to 3 this means we can take any number between 2 and 3 i.e. 2.1, 2.2, 2.32, 2.70, 2.95, and so on .
Hence in an interval of a continuous series there exist an infinitely large numbers.
Direct Method
Formula for calculation of mean for continuous series is
̅
∑
Where m= mid point of various classes; f = frequency of each class , N = the total frequency.
Steps of calculation
I) Obtain the mid value of each class and denote it by m.
II) Multiply each mid-value by the corresponding frequency and obtain the total ∑
III) Divide the total obtained ∑ by the sum of frequencies i.e. N
Example 4
Calculate the arithmetic mean from the following data
Weight 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50
No. of Pcs 20 24 40 36 20
Solution :
Mean for Continuous series
Direct Method
̅
∑
Shortcut Method
̅
∑
Step deviation Method
̅
∑

Weight
0 – 10 20 100
10 – 20 24 360
20 – 30 40 1000
30 – 40 36 1200
40 – 50 20 900
̅
∑
.
Shortcut Method
Formula for calculation mean by this method is
̅
∑
Where A= assumed mean; d= deviations of mid value from assumed mean i.e. m – A; N =Total number of
observations i.e. .
Steps for calculations
I) Find the mid values of each value of each class and denote it by m.
II) Take any mid value as assumed mean A
III) Take deviations of the mid value(m) from the assumed mean (m – A) and denote it by d.
IV) Multiply the respective frequencies of each class by these deviations and obtain the total ∑ .
V) Divide the total obtained ∑ by the total frequency ∑ or total number of observations.
Now we solve the Example No. 4 by Shortcut method
–
–
–
–
–
̅
∑
. .
Thus , mean weight =25.85
Step Deviation Method
In case of continuous series the formula uses for calculation of mean is

̅
∑
Where m= mid value of the class; i= class interval; A = assumed mean
Note : Step deviation method is most commonly used in case of continuous series.
Steps for Calculation:
1. Find the mid value of each class and denote it by m.
2. Take any mid value as assumed mean A.
3. Take deviations of the mid value (m) from the assumed mean and denote it by d.
4. Compute step deviations . These are obtained by dividing the deviations by the magnitude of class
intervals i.e. ⁄
5. Multiply the respective frequencies of each class by these deviations and obtain the total .
6. Divide the total obtained by the total frequency or N and then multiply by (i) in the formula for
getting arithmetic mean.
Now we solve the example no. 4 by step deviation
̅
∑
. .
Calculation of Mean for different series using different Methods
Method →
Series↓
Direct Method Shortcut Method Step Deviation Method
Individual Series ̅
∑
̅
∑
-------------
Discrete Series ̅
∑
̅
∑
-------------
Continuous Series
̅
∑ ̅
∑
̅
∑

Median
Median is a special point and it lies in the center of the data so the half the data lie below it and half the data lie
above it. It is a positional average. To find the exact position of the median, arrange the data either in ascending
order or descending order. “Thus median is a measure of the location of the observations. It is a value which
divides the arranged series into two equal parts in such a way that the number of observations smaller than the
median is equal to the number of observations greater than it.”
Median is thus a positional average. Median is denoted by symbol ‘M’
Properties of Median
1. It is easy to compute or locate.
2. It is the most suitable average in dealing with qualitative facts such as beauty, intelligence, honesty etc.
3. It is most suitable average in case of open ended classes.
4. It is not affected by extreme items. Unlike the mean which is sensitive to the extreme value i.e. the value
of mean will change if the value of any extreme or outliers are changed Median is not affected by the
outliers i.e. median will not change if the extreme values are changed.
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22
Median for the given series is 12, Now if the value of 2 is changed to 5 and 22 is changed to 19 then the new
series is
4, 5, 6, 8, 10, 12, 14, 16, 18, 19, 20
The median of this new series is 12.
5. Sum of the absolute deviations of the items from the median is less than the sum of deviations of the
items from any other value or average. i.e.
⌈ ⌉ [ ̅]

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