a measure of central tendency is a way of filemerits and limitations of arithmetic mean limitations
TRANSCRIPT
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