Construction of frequency distributions and their analysis in the form of measures of central tendency and variations; types of measures, their relative merits, limitations and characteristics; skew ness; meaning and co-efficient of skew ness.

A. S. Raheja

asraheja@gmail.com

9717871155

MBA – 2104/50704: QUANTITATIVE ANALYSIS

Classification of Data

• Rules of classification– Exhaustive – each data should belong to one

of the class– Mutually Exclusive – fit to one class only– Suitability – confirm to object of inquiry– Stability – Basic principle should be retained– Homogeneity – data should be homogeneous

Types of Data Classification

• Geographical – area wise

• Chronological – basis of time

• Qualitative – according to attribute

• Quantitative – in terms of magnitude– Discrete– Continuous

Frequency Distribution

• Categorical frequency distribution : can be used for data that can be placed in specific categories, such as nominal- or ordinal-level data. Ex - political affiliation, religious affiliation, blood type etc.

• Ungrouped frequency distribution : can be used for data that can be enumerated and when the range of values in the data set is not large. Ex - number of girls in a 4-child family etc

• Grouped frequency distribution: can be used when the range of values in the data set is very large. The data must be grouped into classes that are more than one unit in width. Ex – Marks of students in a class

Categorical frequency distribution

• Blood Type Frequency Blood Type Frequency

DistributionDistributionClass Frequency Percent

A 5 20

B 7 28

O 9 36

AB 4 16

Exercise 1 : Construct a ungrouped Frequency Distribution Chart

Ungrouped Frequency Distribution (Discrete Series)

Grouped Frequency Distribution (Continuous Series)

• Rules for Choosing Classes– Clearly defined– All values should be included– Class should be Mutually Exclusive– Equal Width– Between 5 – 20 Classes– Class intervals are considered after taking in to

account :• Minimum and Maximum values• Number of Classes to be formed

Important Terms• Class Limits – Lowest and Highest values that can be included in

a class ex- (20-40) means lowest value is 20 and highest value is 40.

• Class Intervals – Difference in the upper and lower limit of a class.– H = (L- S) / K

• L = Largest Value• S = Smallest value• K = Number of Classes

Types of Class intervalsExclusive Method: When upper limit of one class is lower limit of the next class, it is known as exclusive method of classification

Inclusive Method: Both Lower and Upper limits of the class are included in the class itself. Overlapping is avoided

More of Important Terms

• Frequency Density– Class Frequency/Width of the Class

• Relative Frequency– Class Frequency/Overall total Frequency

• Percentage Frequency– Relative Frequency X 100

Exercise 2 : Same example as group frequency

Example 2

Arrange the data in 10 class intervals

Obtain the percentage frequency in each class

Find the class boundaries and cumulative frequencies

Solution

Solution

Measure of Central Tendency

• Averages– Mathematical Average

• Arithmetic mean • Geometric mean• Harmonic mean

– Positional Average• Median• Mode• Quartiles• Deciles• Percentiles

Arithmetic MeanSimple Arithmetic MeanFor Ungrouped data

Direct Method

Shortcut Method

Mean for Grouped Data

Direct Method

Shortcut Method

Exercise

And repeat the same exercise with Shortcut method

Arithmetic Mean Ctd.

• Step Deviation Method for Grouped Frequency Distribution:

Exercise

Solution

Merits/Limitation of Mean

• Merits– Easy to understand– Easy to Calculate– Takes all values in to account– Used in further Mathematical treatment– Sum of deviation from mean is always zero– Sum of square of deviation from the mean is minimum

• Limitations– Affected very much by extreme values– In open end class distribution, need to make assumptions– It may lead to false conclusion if details of data is not given

Median• Divides the distribution in to two equal part• 50% observations are above median and 50% are below median

– Median = Md = ½(N+1)

Median – Ungrouped Frequency Distribution

Median – Grouped Frequency Distribution

Exercise

Merits/Limitation

• Merits– Easy to calculate and understand– Not affected by extremes– Computable for open end classes– Useful with qualitative data

• Demerits– Need to rearrange data– Even number: need to estimate as A. Mean of two

middle values– Not Capable for algebraic treatment– Insensitive : Not dependent on all the data in the

series

Mode

• Value in the series that occurs with greatest frequency– Ungroup Data

• 2,4,13,6,8,10,12,10,17,10,21,18,27– Mode = 10

• 10,15,7,6,4,2,8,9– No Mode

– Ungrouped/Grouped Frequency Distribution Data

• Grouping Method is used

Grouping Table

Mode for Discrete Series

Merits/Limitation of Mode

• Merits– Most typical or representative value of distribution– Not affected by extreme values– Value can be determined by open end distributions

without ascertaining class limit

• Limitation– Indeterminate value in case of group data– In case of bi modal , empirical method– Not capable for algebraic manipulation– Not based on all the observation of the series

Mode for Grouped Data (Continuous Series)

Geometric Mean

It is the Nth root of the product of N items

If X1,X2 ….Xn refer to items in series GM = n√X1.X2.X3….Xn

In case of Discrete Seriesf1, f2,…fn are frequency of value X1,X2 ….Xn GM = N√(X1

f1.X2 f2.….Xn

fn) N = ∑f

In case of Continuous SeriesGM = N√(m1

f1.m2 f2.….mn

fn)Where m = middle values of the ranges

Harmonic Mean

Reciprocal of Arithmetic Mean of the reciprocal of individual observation

If X1,X2 ….Xn refer to items in series HM = N/(1/X1 + 1/X2+.….1/Xn)

In case of Discrete Seriesf1, f2,…fn are frequency of value X1,X2 ….Xn HM = N / ∑ (fi / Xi)

In case of Grouped DataHM = N / ∑ (f / m.values)