statistics

IT IS THE AGGREGATE OF FACTS WHICH ARE AFFECTED BY MULTIPLICITY OF CAUSES, THEY ARE EXPRESSED NUMERICALLY & ARE COLLECTED IN A SYSTEMATIC MANNER.

STA

TIS

TIC

S

BASICSTYPES OF SERIES - INDIVIDUAL DISCRET

CONTINUOUS S.no

Marks

1 30

2 40

3 50

4 60

5 70

Marks Freq.

20 8

30 12

40 20

50 10

60 6

Marks Freq.

10 -20 8

20 -30 12

30 -40 20

40 -50 10

50 -60 6

CUMMULATIVE FREQUENCY

marks Freq.(f)

20 8

30 12

40 20

50 10

60 6

Cumm.freq.(cf)

8

20

40

50

56

CENTRAL TENDENCY Measures of Central Tendency are used

to describe the set of data.

Mean Median

Mode

MEA

N1.Arithmetic Mean

2.Shortcut Method to Mean

3.Direct Method to mean

4.Shortcut Method for Direct Method

5.Calculation of Mean:-

*Individual series *Discret series *continuous series

3, 4, 4, 10, 15, 16, 18

70 7 =10

Mean

The mean is also known as the average of the set. Add all of the number is in the set and divide this sum by the number of items in the set.

Just

add and

divide

CHARACTERISTICS OF MEAN1.Easy to understand.2.Simple to compute.3. Based on all items.4.Affected by extreme observation.5.Capable of further calculation.

1.Arithmetic Mean:

N =No. of observations/items in a series

N

XX

XnXXXX .....321

x20

30

10

2. Shortcut method to Mean Ex:-

Assumed mean

AXd

AN

dAX

X d= X-A

20

30

40

50

3.Direct Method:-

Nf

f

xfX

Marks(X)

f

20 4

30 2

40 3

50 1

f x

4.Shortcut Method for Direct Method:-

Axd

f

dfAX

Marks

f

20 8

30 12

40 20

50 10

d = x-A

f.d

Calculation of Mean in Continuous series:-

mid point of class interval

m

f

mfX

)(

Marks (x)

f

10 – 20 8

20 – 30 12

30 – 40 20

40 – 50 10

50 - 60 6

m f (m)

ADVANTAGES OF MEAN Since its definition is precise and clear,

calculation is easy and its value always determinate.

Its chief merit consists in the fact that it take onto consideration every item in the data.

NO special arrangement of data is necessary while calculating it, unlike median and mode where we have to arrange and group the data in a certain manner.

It forms a good basis of comparison while comparing different groups of numerical data.

DISADVANTAGES OF MEAN

Omission of even a single item of the data gives an incorrect value of the A.M., unlike median and mode where extreme items can simply be discarded.

It may not identical with any one of the items of the data. That is, it may not be one of the figures that comprise the data.

The fact that it gives a large weight to extreme items is its handicap since it then fails to be a good representatives. That is, the value of A.M. of the data consisting of very and very small items may lead to wrong conclusions.

It cannot be used in qualitative studies, unlike the median.

MED

IAN 1.Definition

2.Characteristics3.Calculating median:- *Individual series *Discret series *Continuous series4.Advantages & Disadvantages of median.

DEFINITION & CHARACTERISTICS The value of the middle item of a series

that is ordered either in the ascending or descending order of magnitude is called the median.

1.Measure of central tendency

2.Middle value in ordered sequence

If n is odd, middle value of sequence

If n is even, average of 2 middle values. (n= no of items in a series)

marks

40

20

50

10

30

order marks

30

60

10

40

20

50

order

MEDIAN IN INDIVIDUAL SERIES Odd series:- middle value of sequence.

(series must be in order)

Even series:- median= item

No. of observation

N

thN

2

1

MEDIAN IN DISCRET SERIES Arrange the data in ascending or descending order Find c.f. Apply

Look for the value in cf. corresponding figure to cf is the median.

Income

No. of person(f)

800 16

1000 24

1500 26

1800 30

2000 20

2500 6

thN

2

1

cf

MEDIAN IN CONTINUOUS SERIES

Method:- same as discrete series but instead of apply to determine the median class.Then apply,

Median = L= lower limit of the median class

cf= cf of the class preciding the median class

f= frequency of median class

i=class interval

2

1N

2

N

if

cfN

L *2

Marks No.of student(f)

0 – 5 4

5 – 10 5

10- 15 10

15 – 20

10

cf

if

cfN

L *2

2

N

ADVANTAGES OF MEDIAN Most of the conditions of an ideal average

are satisfied by it.

The median is of immense use while estimating qualities such as honestly, intelligence, virtue, morality etc. and proves to be a good representative.

It’s possible to find the median by knowing only the values of central items and the no. of items. That is, the values of extreme items are not necessary for finding the median.

DISADVANTAGES OF MEDIAN An irregular series is characterized by extreme

variations between the items, in the case of such a series the median fails to be representative.

Arrangement of the data either in the ascending order of the magnitude or descending order is the chief requisite. This process is tedious when the data is vast.

Sometimes the medians may exist between two values, thus involving the work of estimation.

Unlike the A.M., it is not possible to find the total value of all items if we know the value of the median and the no. of items.

RELATIONAL POSITIONAL MEASURE QUARTILE:- It divides the entire series

into 4 equal parts.Quartiles in:- # Individual & discrete

series # Continuous series

INDIVIDUAL & DISCRETE SERIES

thN

Q

thN

Q

4

133

4

11

CONTINUOUS SERIES

&

if

cfN

LQ

thN

Q

*4

1

41

if

cfN

LQ

thN

Q

*4

33

433

Find Q1 & Q3

class f

0 - 5 7

5 – 10 18

10 -15 25

15 – 20 30

20 – 25 20

cf

MO

D E The value of the item in a variable that is repeated the greatest number of times is called the mode.

MODEThe number that repeats the most

3, 4, 4, 10, 15, 16, 184 is the

modeThe number that appears

the most

MODEthere may be more than one

mode3, 3, 4, 4, 4, 7, 9, 9,

94 & 9 are the

modes

MODE IN CONTINUOUS SERIESCASE: 1

when there is only single value(f) which is highest.

L= lower limit of model class

f1= frequency of model class

fo= frequency of class preciding the model class

f2= frequency of class successiding the model class

ifff

ffLM

212

1

Calculate mode

marks No. of student

0 – 10 10

10 – 20 12

20 – 30 15

30 – 40 8

40 - 50 5

ifff

ffLM

212

1

CASE:-2

When there are two similar frequencies/values which are highest.

MODE = 3 MEDIAN – 2MEAN

AD

VA

NTA

GES

OF

MO

DE

1.It is the most predominant item in a discrete series.

2.If a continuous series is a regular one i.e. with the maximum frequency in the centre the mode can be calculated easily without knowing the frequencies at the two extremities of the series.

3.It is very useful average in studying business relating to sales, profits etc.

DIS

AD

VA

NTA

GES

OF

MO

DE

1.It is quite possible that in certain types of data it may not be properly defined & hence it may be indeterminate & indefinite.

2.The value of the mode is not the value obtained by considering the value of every item of the data.

3.In the case of an irregular series it is not very easy to determine the mode.

4.It is quite possible that in certain instances there may be two or more values for the mode.

TH

AN

K Y

OU

BOOK:- Quantitative Techniques in Business by : Dr. Aditham B.Rao

REFERENCE