A measure of Central Tendency is a way of “summarising the data in the form of a typical or representative value .”

DEFINITION

'' A measure of Central Tendency is a typical value around which other figures

congregate''

REQUISITES OF A GOOD AVERAGE

1. It should be easy to understand

2. It should be simple to compute

3. It should be based on all the items

4. It should not be unduly affected by extreme items

5. It should be rigidly defined

6. It should be capable of further algebraic treatment

7. It should have sampling stability

TYPES OF AVERAGES

1. ARITHMETIC MEAN

i) Simple Mean

ii) Combined Mean

iii) Weighted Mean

2. MEDIAN

3. MODE

ARITHMETIC MEAN

(SIMPLE MEAN)

On the basis of the type of data series that has provided to us (ie,

Individual, Discrete, Continuous), it will be convenient if we use appropriate

formula for finding averages in each of these series.

NOTE

There are three methods by which Simple mean can be calculated in each of these three series.They are :

Direct Method Assumed Mean Method Step Deviation Method

Since Direct Method is the simplest method, we discuss only the same.

(Equations of other methods are also given just for reference).

NOTE

SIMPLE MEAN

DISCRETEINDIVIDUAL CONTINUOUS

DIRECT

ASSUMED

STEP DEVIATION

DIRECT

ASSUMED

STEP DEVIATION

DIRECT

ASSUMED

STEP DEVIATION

?

STEPS- INDIVIDUAL SERIES

1. Find the sum of observations ( X)∑

2. Take the number of observations (N)

3. Use the formula

INDIVIDUAL SERIES(DIRECT METHOD)

STEPS- DISCRETE SERIES 1. Multiply the frequency against each observations (fx)

2.Find the sum ( fX)∑

2. Take the number of observations (N)

3. Use the formula

DISCRETE SERIES (DIRECT METHOD)

STEPS- CONTINUOUS SERIES

1. Find the mid value of each class (m)

2.Multiply the frequency against each mid value (fm)

2.Find the sum ( fm)∑

2. Take the number of observations (N)

3. Use the formula

CONTINUOUS SERIES (DIRECT METHOD)

Note : N = ∑f

PROPERTIES OF ARITHMETIC MEAN

The sum of the deviations of items in a series from its Arithmetic Mean is always zero.

Eg. 60, 25, 75, 38, 50, 52 Arithmetic Mean = 50

Sum of deviations (+10, -25, +25, -12, 0, +2) = 0

MERITS AND LIMITATIONS OF ARITHMETIC MEAN

MERITS

It is simple to understand

It is easy to compute

It is amenable to further algebraic treatment

It is relatively reliable It is affected by the value of every items in the series

It is defined by a rigid mathematical formula

MERITS AND LIMITATIONS OF ARITHMETIC MEAN

LIMITATIONS

It is not always a good measure of Central tendency

It can not be calculated for qualitative data

It is not suitable for averaging ratios and percentages

It is a figure which does not exist in the series

If there are extreme items in a series, it will unduly affect the value.

NOTE

Median refers to the middle value in a distribution

It has a middle position in a series.

It is also called positional average

It will not be affected by extreme items

It splits the observations into two halves

INDIVIDUAL SERIES

Steps Individual Series –

Arrange the items in ascending or descending order

Take the number of observations (N)

Use the formula

(It 'N' is an even number, one more step is needed to arrive at answer)

DISCRETE SERIES

Steps Discrete Series –

Arrange the data in ascending or descending order

Take cumulative frequencies

Use the formula

Locate the value through cumulative frequency

CONTINUOUS SERIES

Value of

Steps Continuous Series –

Take cumulative frequencies

Calculate

Find the class where

Find values: L, cf, f and c with respect to median class

Apply the formula

N

2

th item

2

NItem falls using the cumulative frequency

th

Locating Median graphical ly

Convert the data into less than method

Draw less than Ogive

Calculate and mark it on the Y axis

Draw a line parallel to X axis from this point

From the point where it meets the Ogive, draw perpendicular to the

X axis.

The meeting point of this line on the X axis is the median.

2

Nitemth

Y

o XMEDIAN VALUE

N

2

MERITS AND LIMITATIONS OF MEDIAN

MERITS

It is easy to compute

It is easy to understand

It is more useful in skewed distributions

It is not very much affected by extreme values

It can be computed graphically

It is especially useful in the case of open – end class

LIMITATIONS

It may not be representative of series in many cases

Its value is not determined by each and every observation

It is not capable of algebraic treatment

The value of Median is affected more by Sampling fluctuations

than the value of Arithmetic Mean

It is tedious to arrange data when the number of items is large

In case of continuous series, the median value can only be an

approximate value.

MERITS AND LIMITATIONS OF MEDIAN

QUARTILES

Median is a value which divides the series into two equal parts.

Quartiles are those values which divides a series into four equal parts – Q1, Q2 and Q3 (Q2 is same as the Median)

Deciles are those values which divides a series into ten equal parts – D1, D2, D3, D4, ....., D9.

Percentiles are those values which divides a series into hundred equal parts – P1, P2, P3, P4, P5, P6, ......, P99.

INDIVIDUAL SERIES

Q1 =

DISCRETE SERIES

Q1 =

CONTINUOUS SERIES

Value of

QUARTILES

INDIVIDUAL SERIES

DISCRETE SERIES

CONTINUOUS SERIES

Value of Value of

DECILES

INDIVIDUAL SERIES

DISCRETE SERIES

CONTINUOUS SERIES

Value of Value of

DECILES

INDIVIDUAL SERIES

DISCRETE SERIES

CONTINUOUS SERIES

Value of Value of

DECILES

INDIVIDUAL SERIES

DISCRETE SERIES

CONTINUOUS SERIES

Value of Value of

PERCENTILES

INDIVIDUAL SERIES

DISCRETE SERIES

CONTINUOUS SERIES

Value of

PERCENTILES

INDIVIDUAL SERIES

DISCRETE SERIES

CONTINUOUS SERIES

Value of Value of

PERCENTILES

INDIVIDUAL SERIES

DISCRETE SERIES

CONTINUOUS SERIES

Value of Value of

MODE

Mode represents the most typical value of a series.

● It is the value which occurs the largest number of

times in a series.

● Mode is the value around which there is the

greatest concentration of values

● It is the item having the largest frequency.

● If there is one value occurs more frequently, it is

called Uni Modal and if there are more than one

value, it is called Bi modal or Multi modal

● If no value repeats, there can be no mode at all.

INDIVIDUAL AND DISCRETE SERIES

Inspect which value repeats highest number of times. Take it as Mode.

If no repetition found or more than one value has samenumber of repetition, use the empirical formula :

MODE = 3 MEDIAN – 2 MEAN

CONTINUOUS SERIES

or

CONTINUOUS SERIES

1. Find the Modal Class (the class having highest frequency)

2. Take the lower limit of the Modal class (L)3. Find the difference between the frequencies of the modal class and the class preceding it (D1 or ) (ignore signs)

4. Find the difference between the frequencies of the modal class and the class succeeding it (D2 or ) (ignore signs)

5. Take the class interval of the Modal class (c or i or h)6. Use the formula

STEPS

or

∆1

∆2

MERITS AND LIMITATIONS OF MODE

MERITS➢ It gives the most typical value of a series

➢ It is not affected by the extreme values

➢ It can be determined graphically

➢ It is commonly understood

➢ It is easy to calculate

➢ Open end class do not pose any problem in finding mode

➢ It is not necessary to know the values of each item of the series

➢ It is used in qualitative data

➢ It is very much useful in the fields of Business and Commerce

MERITS AND LIMITATIONS OF MODE

LIMITATIONS

➢ In the case of Bi Modal series, Mode can not be determined

➢ It is not capable of algebraic treatment

➢ It is not based on each and and every item of the series

➢ It is not a rigidly defined one

➢ It is ill – defined, indeterminate and indefinite

Locating Mode graphical ly

STEPS

➢ Draw a Histogram of the given data

➢ Draw two lines diagonally in the inside of the Modal class bar

(To be started from each corner of the bar to the upper corner of

the adjacent bar)

➢ Then draw a perpendicular line from the point of intersection to

the X axis

➢ That will be the value of Mode

Y

o XMODAL VALUE

SPECIMEN