Statistics

• The systematic and scientific treatment of quantitative measurement is precisely known as statistics.

• Statistics may be called as science of counting.• Statistics is concerned with the collection,

classification (or organization), presentation, analysis and interpretation of data which are measurable in numerical terms.

Stages of Statistical Investigation

Collection of Data

Organization of data

Presentation of data

Analysis

Interpretation of Results

Statistics

• It is divided into two major parts: Descriptive and Inferential Statistics.

• Descriptive statistics, is a set of methods to describe data that we have collected. i.e. summarization of data.

• Inferential statistics, is a set of methods used to make a generalization, estimate, prediction or decision. When we want to draw conclusions about a distribution.

Statistics functions & Uses

• It simplifies complex data• It provides techniques for comparison• It studies relationship• It helps in formulating policies• It helps in forecasting• It is helpful for common man• Statistical methods merges with speed of

computer can make wonders; SPSS, STATA MATLAB, MINITAB etc.

Scope of Statistics

• In Business Decision Making• In Medical Sciences• In Actuarial Science• In Economic Planning• In Agricultural Sciences• In Banking & Insurance• In Politics & Social Science

Distrust & Misuse of Statistics

• Statistics is like a clay of which one can make a God or Devil.

• Statistics are the liers of first order.• Statistics can prove or disprove anything.

Measure of Central Tendency

It is a single value represent the entire mass of data. Generally, these are the central part of the distribution.

It facilitates comparison & decision-making There are mainly three type of measure1. Arithmetic mean2. Median3. Mode

Arithmetic Mean

This single representative value can be determined by:

A.M. =Sum/No. of observationsProperties:1. The sum of the deviations from AM is always

zero.2. If every value of the variable increased or

decreased by a constant then new AM will also change in same ratio.

Arithmetic Mean (contd..)

3. If every value of the variable multiplied or divide by a constant then new AM will also change in same ratio.

4. The sum of squares of deviations from AM is minimum.

5. The combined AM of two or more related group is defined as

Median

Mode

• Mode is that value which occurs most often in the series.

• It is the value around which, the items tends to be heavily concentrated.

• It is important average when we talk about “most common size of shoe or shirt”.

Relationship among Mean, Median & Mode

• For a symmetric distribution: Mode = Median = Mean

• The empirical relationship between mean, median and mode for asymmetric distribution is: Mode = 3 Median – 2 Mean

Advantages and disadvantages

Mean More sensitive than the

median, because it makes use of all the values of the data.

It can be misrepresentative if there is an extreme value.

Median It is not affected by extreme scores, so can give a representative value.

It is less sensitive than the mean, as it does not take into account all of the values.

Mode It is useful when the data are in categories, such as the number of babies who are securely attached.

It is not a useful way of describing data when there are several modes.

Same center,different variation

• Ignores the way in which data are

distributed

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

When the value of Arithmetic mean is fraction value(not an integer), Then to compute variance we use the formulae:

22 21 X

Xn n

x 10 11 17 25 7 13 21 10 12 14

Calculate S.D.;-

Formulae for Frequency distribution

By Definition:

For Computation:

222 fx fx

f f

22 1f x x

f

Example

• An analysis of production rejects resulted in the following figures.Calculate mean and variance for number of rejects per operator

No. of rejects per operator

No. of operators

21-25 5

26-30 15

31-35 23

36-40 42

41-45 12

46-50 03

Example

Sale No. of days

10-20 3

20-30 6

30-40 11

40-50 3

50-60 2

• Calculate variance from the following data. (Sale is given in thousand Rs.)

No. of rejects/ operator No. of operators

20-25 5

25-30 15

30-35 28

35-40 42

40-45 15

45-50 12

50-55 3

An Analysis of production rejects resulted in following observations

Calculate the mean and standard deviation.

• Measures relative variation

• Always in percentage (%)

• Shows variation relative to mean

• Is used to compare two or more sets of data measured in different units

Comparing Coefficient of Variation

• Stock A:– Average price last year = $50– Standard deviation = $5

• Stock B:– Average price last year = $100– Standard deviation = $5

$5100% 100% 5%

$100

SCV

X

$5100% 100% 10%

$50

SCV

X

Coefficient of variation:

Stock A:

Stock B:

. An investment ‘A’ has an Expected return of Rs.1,000 and a standard deviation of Rs. 300. Another investment ‘B’ has a standard deviation of its returns as 400 but its expected return is 4,000. Calculate which investment is more risky.

Example

Length of life (in hrs.)

Company A Company B

15-20 16 18

20-25 26 22

25-30 08 08

• 2. A quality control laboratory received samples of electric bulbs for testing their lives, from two companies. The results were as follows:

(a). Which company’s bulbs have the greater length of life?(b). Which company’s bulbs are more uniform with respect to their

lives?

Determine the Mean and standard deviation of prices of shares .In which marketsare the share prices more stable?

The share prices of a company in Mumbai and Kolkata markets during the last 10 months are recorded below:

Month Mumbai Kolkata

Jan 105 108

Feb 120 117

March 115 120

April 118 130

May 130 100

June 127 125

July 109 125

Aug 110 120

Sep 104 110

Oct 112 135

Shape of a Distribution

• Describes how data is distributed• Measures of shape

– Symmetric or skewed

Mean = Median =Mode Mean < Median < Mode Mode < Median < Mean

Right-SkewedLeft-Skewed Symmetric

Skewness

For a positively skewed distribution:Mean>Median>Mode• For a Negatively skewed distribution: Mean<Median<Mode

Measure of Skewness

• Karl Pearson coefficient of Skewness:

Where -3 <= <= 3

k

Mean ModeS

S.D

kS

Calculate the Karl pearson coefficient of skewness for the given data & comment about the result.

7, 9, 15, 16, 17, 22, 25, 27,33,39.

Advantages and disadvantagesAdvantages Disadvantages

Range Quick and easy to calculate Affected by extreme values (outliers)Does not take into account all the values

Standard deviation More precise measure of dispersion because all values are taken into account

Much harder to calculate than the range