Frequency Distributions Q U A N T T E C H I N T E U Q I A S E V I T 1 0 S

Contents Basics of Data Samples and Populations Data Array Frequency Distributions Relative Frequency Distributions Classes Qualitative versus Quantitative Discreet versus Continuous Illustrating Data Histograms Polygons

Basics of Data

Samples and Populations

Data Array

Frequency Distributions

Relative Frequency Distributions

Classes

Qualitative versus Quantitative

Discreet versus Continuous

Illustrating Data

Histograms

Polygons

Data Basics Data are collections of any number or related observations Number of telephones installed by all workers in one day Number of telephones installed by one worker in one day Number of tourists in Finland on every Diwali day ?? Data is useful when they Reveal some kind of pattern Temperature in December is less than that in June Lead to some logical conclusion Senior citizens avoid investing in equity markets

Data are collections of any number or related observations

Number of telephones installed by all workers in one day

Number of telephones installed by one worker in one day

Number of tourists in Finland on every Diwali day ??

Data is useful when they

Reveal some kind of pattern

Temperature in December is less than that in June

Lead to some logical conclusion

Senior citizens avoid investing in equity markets

Data Collection & Sanity Check Source of Data Actual observation in the field Physical records available with source organisation Third party data sources Commercial data sellers Free data sources available in the web Basic Sanity Check Is the source trustworthy ? Is there something missing in the data ? Do we have enough observations ? Is the conclusion logical ? Garbage In Garbage Out Is there double counting ?

Source of Data

Actual observation in the field

Physical records available with source organisation

Third party data sources

Commercial data sellers

Free data sources available in the web

Basic Sanity Check

Is the source trustworthy ?

Is there something missing in the data ?

Do we have enough observations ?

Is the conclusion logical ? Garbage In Garbage Out

Is there double counting ?

Samples and Populations Population : is a collection of all elements about whom we are trying to draw conclusions Women in Calcutta with age > 18 Sample : is a collection of some, not all, elements of the population about whom we are in a position to gather data Statisticians gather data from a sample and then use this data to draw inferences about the population Representative Sample : it should reflect the characteristics of the underlying population Selecting a sample from of women from Calcutta Club may not be representative of all women in Calcutta !

Population : is a collection of all elements about whom we are trying to draw conclusions

Women in Calcutta with age > 18

Sample : is a collection of some, not all, elements of the population about whom we are in a position to gather data

Statisticians gather data from a sample and then use this data to draw inferences about the population

Representative Sample : it should reflect the characteristics of the underlying population

Selecting a sample from of women from Calcutta Club may not be representative of all women in Calcutta !

Organising Data Organising data enables us to quickly spot some of the characteristics of the data Range : Highest Value ? Lowest Value ? Clustering : Are the values grouped around a specific value ? Popularity : Which value occurs most frequently Ways of organising data Simple ascending or descending order Group by certain characteristic Age ? Income ? Education Level ? Colour ? Material ?

Organising data enables us to quickly spot some of the characteristics of the data

Range : Highest Value ? Lowest Value ?

Clustering : Are the values grouped around a specific value ?

Popularity : Which value occurs most frequently

Ways of organising data

Simple ascending or descending order

Group by certain characteristic

Age ? Income ? Education Level ?

Colour ? Material ?

Examples of Raw Data Retail Sales Figures

Examples of Raw Data Forbes500 Company Data

Examples of Raw Data US Cereals Data

Examples of Raw Data Stockmarket Price Data

Examples of Raw Data Examination Marks

Examples of Raw Data Engine Pollution Data

Data Array The Data Array arranges values in ascending or descending order

The Data Array arranges values in ascending or descending order

Why Create a Data Array ? We can quickly get the highest and lowest value In Hydrocarbon : 0.34 .. 1.1 We can divide the data into sections First 1/3 : Between 0.34 and 0.46 Second 1/3 : Between 0.47 and 0.56 Last 1/3 : Between 0.56 and 1.1 We can see whether some value appears multiple times We can observe difference between successive values of the data

We can quickly get the highest and lowest value

In Hydrocarbon : 0.34 .. 1.1

We can divide the data into sections

First 1/3 : Between 0.34 and 0.46

Second 1/3 : Between 0.47 and 0.56

Last 1/3 : Between 0.56 and 1.1

We can see whether some value appears multiple times

We can observe difference between successive values of the data

Limitations of Data Array Cumbersome to use when the volume of data is very large Utility goes down as human mind cannot comprehend so much data in one shot There is a need to compress this data and make it more accessible

Cumbersome to use when the volume of data is very large

Utility goes down as human mind cannot comprehend so much data in one shot

There is a need to compress this data and make it more accessible

Frequency Distribution A frequency distribution is a table that organises data into classes A class is a group of values describing ONE characteristic of the data It shows the number of observations from the data that fall into each class Frequency distribution can be constructed by determining how often ('with what frequency') values occur inside each class of a data set Fewer classes mean more data compression

A frequency distribution is a table that organises data into classes

A class is a group of values describing ONE characteristic of the data

It shows the number of observations from the data that fall into each class

Frequency distribution can be constructed by determining how often ('with what frequency') values occur inside each class of a data set

Fewer classes mean more data compression

Frequency Distribution

Relative Frequency Distribution Frequency of each value can be expressed as a fraction or percentage of the total number of observations This could help us compare data from samples that are of different sizes

Frequency of each value can be expressed as a fraction or percentage of the total number of observations

This could help us compare data from samples that are of different sizes

Discrete & Continuous Classes DISCRETE : In this case, the data in a class can take ONE discrete value : 0, 1, 2, ... CONTINUOUS : In this case, the data in a class can take any value in a range > 0; <= 1 > 1; <= 2 > 2; <= 3 And so on

DISCRETE : In this case, the data in a class can take ONE discrete value :

0, 1, 2, ...

CONTINUOUS : In this case, the data in a class can take any value in a range

> 0; <= 1

> 1; <= 2

> 2; <= 3

And so on

Qualitative & Continuous Classes Discrete Classes can also be used to model Qualitative Classes Where the data does not take specific numerical values but falls into certain qualitative that is non-numeric categories Continuous classes cannot have qualitative data Unless you want to prove a point !!

Discrete Classes can also be used to model Qualitative Classes

Where the data does not take specific numerical values but falls into certain qualitative that is non-numeric categories

Continuous classes cannot have qualitative data

Unless you want to prove a point !!

Characteristics of Classes All Inclusive All the data must fall into or other class Sum of relative frequencies must add up to 1 Mutually Exclusive Greater Than ( > ) Lower Class Boundary Less Than OR Equal to ( <=) Upper Class Boundary First and Last Class open ended 0 count for ratings <= 10 1 count for ratings > 90 ratings <= 100 0 count for ratings > 100

All Inclusive

All the data must fall into or other class

Sum of relative frequencies must add up to 1

Mutually Exclusive

Greater Than ( > ) Lower Class Boundary

Less Than OR Equal to ( <=) Upper Class Boundary

First and Last Class open ended

Constructing a Frequency Distribution Decide on Type of Class Quantitative or Qualitative measure ? Decide on Number of Classes More classes : give more information Fewer classes : easier to interpret Rule of Thumb : Between 6 and 15 classes Determine width of class interval [Largest Value] – [Unit Value before Smallest Value] Total Number of Class Intervals Determine the number of points in each class Illustrate the data in a chart

Decide on Type of Class

Quantitative or Qualitative measure ?

Decide on Number of Classes

More classes : give more information

Fewer classes : easier to interpret

Rule of Thumb : Between 6 and 15 classes

Determine width of class interval

[Largest Value] – [Unit Value before Smallest Value]

Total Number of Class Intervals

Determine the number of points in each class

Illustrate the data in a chart

Using a spreadsheet to Construct a Frequency Distribution Functions used Max Min Roundup Rounddown Sum Frequency

Functions used

Max

Min

Roundup

Rounddown

Sum

Frequency

Handling Qualitative Data Lookup Converts Text Data to Numeric data Which is then used to create frequency distributions

Lookup

Converts Text Data to Numeric data

Which is then used to create frequency distributions

Creating Histograms Histogram A series of rectangles, each proportional in width to the range of values in a class and proportional in height to the frequency of that class In a spreadsheet, care must be taken to choose the values in the X-axis correctly. Could be Class Number Midpoint of class

Histogram

A series of rectangles, each proportional in width to the range of values in a class and proportional in height to the frequency of that class

In a spreadsheet, care must be taken to choose the values in the X-axis correctly. Could be

Class Number

Midpoint of class

Creating a Frequency Polygon Frequency Polygon Another way to show the frequency of each class X-axis value is midpoint of class

Frequency Polygon

Another way to show the frequency of each class

X-axis value is midpoint of class

Histograms & Polygons Histograms The rectangle clearly shows each separate class in the distribution The area of each rectangle, relative to all other rectangles, shows the proportion of the total number of observations that occur in the class Polygons The polygon is simpler than its histogram counterpart It sketches an outline of the data pattern more clearly The polygon becomes increasingly smooth and curvelike as we increase the number of classes and the number of observations Will lead to more significant kinds of graphs

Histograms

The rectangle clearly shows each separate class in the distribution

The area of each rectangle, relative to all other rectangles, shows the proportion of the total number of observations that occur in the class

Polygons

The polygon is simpler than its histogram counterpart

It sketches an outline of the data pattern more clearly

The polygon becomes increasingly smooth and curvelike as we increase the number of classes and the number of observations

Will lead to more significant kinds of graphs

Charting Cumulative Frequencies using “Less than” Ogives

Exotic Charts

What have we learnt ? Data Array Advantages and Disadvantages Frequency Distribution Relative Frequency Distribution Cumulative Frequency Distribution Classes Qualitative Quantitative Discreet Continuous Charts Histograms Polygons Ogives

Data Array

Advantages and Disadvantages

Frequency Distribution

Relative Frequency Distribution

Cumulative Frequency Distribution

Classes

Qualitative

Quantitative

Discreet

Continuous

Charts

Histograms

Polygons

Ogives