Business Statistics I
Classification, Tabulation, Frequency Distribution , Diagrams & Graphical Presentation.
In this section we study the following point :
i. Classification and it types.
ii. Tabulation.
iii. Frequency and Frequency Distribution.
iv. Some important concepts.
v. Diagrams & Graphical Presentation
I. Classification and it's types:
Classification:- The process of arranging data into different classes or groups according to their common characteristics is called classification. e.g. we dividing students into age, gender and religion. It is a classification of students into age, gender and religion.
Or
Classification is a method used to categorize data into different groups based on the values of specific variable. The purpose of classification is to condenses the data, simplifies complexities, it useful to comparison and helps to analysis.
The following are some criteria to classify the data into groups.
i. Quantitative Classification :- Quantitative classification deals with data consist of numerical value and it can divided into two types as a) Discrete variable and b) Continuous Variable
a) Discrete Variable :- Discrete Variable take only the specific values, like integer values only. For example the number of children's in a family, the numbers of cars in parking lot etc.
b) Continuous Variable:- Continuous variable takes any value in the range, or it measure on continuous scale. For example age, height, weight, temperature etc.
ii. Qualitative Classification:- it is also known as categorical or nominal classification, this type of classification is used when data is divided into different groups or categories. Without any numerical value. For example we classify the data into gender ( male and female), according to car brand( Toyota, Ford, Suzuki) etc.
iii. Chronological Classification :- The data are arranged according to time is called Chronological Classification, or data based in time order. This type of data are used in time series analysis. In this type data are recorded or collected in regular in time interval. For example daily sale data, daily price of gold, record of daily temperature etc.
iv. Geographical Classification:- The process of dividing data based on their geographical location. Or the data collected based on different locations then it is called geographical data. And this data is divided according to geographical location is called geographical Classification. For example we collecting data of population indifferent states, sales data collected form different city's. etc.
II.TABULATION:- Tabulation is a next step of the Classification. It is defined as the process of arranging data into row and column. The purpose of the Tabulation is same as the Classification.
The following are parts of table:-
1. Table number
2. Title of table, Head-Note
3.Caption
4.Stub
5.Body of the table
6. Source note, foot note.
we see as:
1. Table Number:- Each table should be give number. it is helpful to give reference in any chapter.
2. Title of the table and Head note:- Each table should be give a short and clear title. The purpose of title describe the about data. A Head note is give information about data, and it's units.
3. Caption :- Caption means heading of the column. If more columns in table we give sub-headings.
4. Stub: Stub refers to the heading of the row and they give at the extreme left.
5. Body of the table: Body is the main part of the table, data are given in the numerical form.
6.Source note and foot note:- If data are taken from the other sources it can be Mentioned in this note is Sources note. and foot note provide the additional information or explanation about data presented in table.
this part are shown in following table.
Table
Number:
Title:
(Head
Note if any)
|
|
Caption ( column heading) |
Total (Row) |
|||
|
Sub - Heading |
Sub - Heading |
||||
|
Stub (row heading) |
|
|
|
||
|
Column Total |
|
|
|
||
(Foot Note: if any )
Types of table:
There two types of table based on the number of
characteristics shown in the table.
i. Simple
and ii. Complex tables.
The classification of
the tables are based on the number of
characteristics shown in the table. We consider one variable two divided data into two parts is called simple table or
one-way table. (because here consider one variable or attribute). Other wise
complex table i.e. we table shown the more than two variable mean data divided
into four parts is called complex table.
We see the example of simple
table and complex table.
i.
Simple table: in this case we shown one
variable that divide data into two part, e.g. the age of students in a certain
college there the table contain two column as name of the student and their
age. Here we are interested in the age of the students only.
|
Students Name |
Age |
|
Student1 |
20 |
|
Student2 |
18 |
|
Student3 |
19 |
|
Student4 |
21 |
This
is an example of simple table.
ii.
Complex table:
Two
–way table: collect the data of number of persons arrival at college collecting
data with arrival time and gender.
|
Arrival time |
Number of
persons |
|
|
|
M |
F |
|
07:00 AM TO
08:00 A.M. |
14 |
12 |
|
08:00 A.M. TO
09:00 A.M. |
102 |
105 |
|
09:00A.M. TO
10:00 A.M. |
50 |
45 |
Now
we colleting dada according to arrival time, gender and the number of persons
can divided into students and college staff.
|
Arrival time |
Number of persons |
|||
|
|
Students |
College staff |
||
|
|
M |
F |
M |
F |
|
07:00 AM TO
08:00 A.M. |
17 |
5 |
1 |
4 |
|
08:00 A.M. TO
09:00 A.M. |
34 |
36 |
14 |
5 |
|
09:00A.M. TO
10:00 A.M. |
10 |
7 |
4 |
5 |
It
is an another example of complex table.
III. FREQUENCY AND FREQUENCY DISTRIBUTION
The statistical table which shows the values of the variable arranged in order of magnitude of in group with respective frequencies side by side is called frequency distribution.
Frequency: The number of times the particular value of the variable repeated in data set is called Frequency of value. it is denoted as later f.
The way in which the value of variable is
distributed into different classes is called frequency distribution. And the
table in which the value of variable and their frequency are shown is called
frequency distribution table or simply frequency distribution.
we study the following points as i. Grouped Frequency Distribution,
ii. Un-Grouped Frequency Distribution.
iii. Cumulative Frequency Distribution.
i. Grouped Frequency Distribution (Continuous frequency distribution):- a frequency distribution in which different classes of the variable with corresponding their frequencies are shown in table is called Grouped Frequency Table.
for grouped frequency distribution in which the frequency of the class formed as we count the number of observations fall in particular class. and it write as frequency of that class.
e.g. Construct a suitable frequency distribution,
for the following data.
70, 45, 55, 18, 17, 07, 6, 84, 57, 23, 26, 59, 98,
41, 15, 17, 02, 11, 37, 58, 66, 77, 82, 87, 59, 86, 37, 26, 18, 15, 45, 78, 98,
71, 72, 57, 64, 24, 92, 18, 48, 41, 48, 35, 62, 68.
|
Class |
Frequency |
|
0-10 |
3 |
|
10-20 |
8 |
|
20-30 |
4 |
|
30-40 |
3 |
|
40-50 |
6 |
|
50-60 |
6 |
|
60-70 |
4 |
|
70-80 |
5 |
|
80-90 |
4 |
|
90-100 |
3 |
ii. Un-Grouped Frequency Distribution (Discrete frequency distribution) :- a frequency distribution in which different values of the variable along with their frequency are shown is called Un-Grouped Frequency Distribution.
the formation of the Un-Grouped Frequency Distribution as we count the number of times particular value is repeated in data, that count is frequency of that variable.
e.g. consider a following data to prepare discrete
frequency distribution.
4, 7, 4, 5, 8, 1, 1, 4, 5, 8, 10, 4, 2, 7, 7, 8, 4, 5, 1, 6, 7, 5, 1, 2, 4.
|
Values |
Frequency |
|
1 |
4 |
|
2 |
2 |
|
4 |
6 |
|
5 |
4 |
|
6 |
1 |
|
7 |
4 |
|
8 |
3 |
|
10 |
1 |
iii. Cumulative Frequency Distribution:
the cumulative frequency is obtained as the frequency of first class is added to that of the second class, this sum is added to that of the third and so on then the frequency are obtained are called cumulative frequencies. and they are denoted as c.f.
there are two types of the cumulative frequencies frequencies i. less than and greater than cumulative frequency. for calculating the less than cumulative frequency we add up the frequencies from above to bottom. we get less than cumulative frequency. for calculating Grater Than Cumulative Frequency we add up the frequencies from bottom to top.
we see in the example both cumulative frequencies less than and grater than cumulative frequency.
|
Marks |
Frequency |
Cumulative Frequency |
|
|
Less than |
Greater than |
||
|
0-10 |
1 |
1 (first frequency as it is 1) |
71 (adding below sum 70 and class
frequency 1, 70+1= 71) |
|
10-20 |
7 |
8 (adding first and second frequency
= 1+7=8) |
70 (adding below sum 63, and class
frequency7, 63+7 = 70) |
|
20-30 |
26 |
34 (adding the above sum 8 and class
frequency = 8+26=34) |
63 (adding the last 37 and class
frequency 26=26+37=63) |
|
30-40 |
37 |
71 (adding the above sum 34 and class
frequency 37=34+37=71) |
37 ( last frequency as it is 37) |
IV. Some Important Concepts:
i. Constant:- A measureable characteristic which does not change it's value is called Constant. e.g. area of earth. area of room.
ii. Variable: A measureable characteristics which change it's value is called Variable. e.g. age, height.
iii. Class limit: the lowest and highest value of the class are called Class Limits. e.g. in above example the 0-10 is a class it has to values as 0 and 10 that are the lower and upper limit of that class that are the class limit of that class, 10-20 that class has 10 and 20 are the class limits.
iv. Class Width: Class width is the difference between upper limit and Lower limit of class. and it is denoted as C. it is formulated as C= Upper limit - Lower limit.
v. Frequency: Number of time particular value of the variable is repeated in data is called Frequency. and it is denoted as f.
vi. Mid-Point:- The mid-point of class is defined as average of lower and upper limit.
i.e. mid-point = [lower limit + upper limit]/2.
vii. Frequency density of class interval :- frequency density of class interval is defined as the ratio of class frequency to the class width. i.e. Frequency density = f/c, where f- frequency and c- class width.
viii. methods of classification: a) Inclusive method, and b) Exclusive Method
a) Inclusive Method : In this method the classes are formed that both limits, upper and lower limit included in same class.
b) Exclusive Method: In this method the class are formed as the upper limit of class is lower limit of next class. therefore the upper limit is not included in that class. e.g. if the classes are 0-10, 10-20 then that value 10 in data is not included in first class we include the value 10 in second class 10-20.
ix. Principles of frequency distribution:
a) Number of classes (k) : the number of classes are formed depends on the values in data set it is obtained as k= 1+3.322 x log(N)
where k is the number of classes and N be the population size.
b) Class width (c): the size of class depends on number of classes and range of the data. it is calculated as class width = c= (L-S) / K
where L - S = R = Range =(largest - smallest observation in data set.)
Note that : if the class which has no lower limit of first class or no upper limit of last class, is called Open end classes.
x) Relative Frequency : Relative Frequency is defined as ratio of class frequency and total frequency, Relative Frequency = f / N.
V. Diagrams & Graphical Presentation.
We have observed the classification and tabulation method. We use this method to take a lot of information and make it fit into a small table. The reason we do this is to make the information more organized and easier to understand.
Tabulation helps us arrange data neatly so that it's not messy and confusing. tabulation is a way to make big files of information look neat and tidy in a table. but better and beautiful way to represent data using diagrams and graphs.
the diagram and graph have some advantages because that used to visualise the data. that helps to understand and give information easily to any common man or any one, following are the some advantages of diagram and graph.
I. Advantages
i. Data Representation: Diagrams and graphs are excellent for presenting data visually, making trends, comparisons, and statistical information easier to understand.
ii. Simplification: Diagrams take complex ideas and simplify them into visual representations that are easier to grasp. the both diagrams and graphs are so simple that even an ordinary man also will understand properly.
iii. Clarity: Visualizing information through diagrams can make it clearer and more understandable. Diagrams show relationships, connections, and patterns that might be hard to explain with words alone.
iv. Memory Aid: Visuals are often easier to remember than written or spoken words. hence diagram and graph are remember quickly. they remember longer.
v. Graphs help in analysis: using graph the data can be analysed with minimum calculations. from the graph we can find median, mode, quartiles etc.
II. Guidelines for drawing diagrams.
i. Title: For each chart give clear and concise title, the title gives exact idea about the diagram or chart.
ii. Simplicity: The diagrams should be simple, and that are understand anyone. create each chart must be simple to understand.
iii. Scale: for diagram like bar diagram choose suitable scale. e.g. the scale must be multiple of 5 or 10. i.e. 5, 10, 15, 20. .....or 10, 20, 30, 40,.........etc.
iv. Attractive: The chart should be clean and attractive.
Types of Diagram.
i. One- Dimensional Diagram - Bar Diagram
ii. Two- Dimensional Diagram - Pie -chart
i. Bar Diagram:
the bar diagram is the most common type of diagram, and it is simple diagram. In bar diagram the variable represented by thick bars with uniform width. and the height of the bar is proportional to the value of that variable. the bars are separated by uniform distance. means that the bar are differ by only height. bars are draw vertically as well as horizontally. but vertical bars are more attractive and vertical bar are popular. this bar diagrams are easily understood to every one.
In a Simple bar diagram we represent only one
variable. e.g. to use bar diagram representing the class-wise student strength:
|
Class |
Student Strength |
|
Class 1 |
27 |
|
Class 2 |
34 |
|
Class 3 |
25 |
|
Class 4 |
45 |
|
Class 5 |
26 |
bar diagram is
.png)
for drawing bar diagram simply put or select the value in below
Enter Value Tab to get bar diagram, this is used to check your bar diagram is correct or not.
2. Pie-Diagram:
Pie-Diagram is a circular chart that can be used to
represent data as slice of a Pie. When we are interested to represent more that
three or four variable the bar-diagram is more complex and it doesn’t give
proper visualization of data to understand. That case we use Pie-chart is a
circle that divided into sections or slices. And the area on each section is
proportional to the size or the value of the variable.
For constructing the Pie-Chart the circle is divided
into section is proportional to the
angle at the center of circle. We draw a angle at the center that proportional to
the value of variable or data. The angle is calculated as
e.g. to Draw a Pie-Diagram to represent the
following data. For Family expenditure in percentage.
|
Items |
Family expenditure % |
|
Cloths |
27 |
|
Food |
35 |
|
Rent |
18 |
|
Other |
20 |
First we find the angles for Items.
Here total expenditure of family is = 100
the pie-chart is
Example2: Draw a Pie-Diagram to represent the
following data. Data of year wise products of certain company.
|
Year |
Products |
|
1998 |
15 |
|
1999 |
78 |
|
2000 |
89 |
|
2001 |
125 |
|
2002 |
87 |
First we find the angles for each year.
Here total product is 15+78+89+125+87 = 394
the Pie-Chart is
Graphical Representation:
The representation of numerical data into graphs is called
graphical representation of data. following are the graphs to represent a data
i.
Histogram
ii.
Ogive
Curves
iii.
Frequency
Polygon
ii. Histogram:
Histogram is one of the
simplest method to representing the grouped (continuous) frequency distribution.
And histogram is defined as A pictorial representation of grouped (or
continuous frequency distribution ) to
drawing a adjacent rectangles, the area of rectangle is proportional to the
frequency of that class.
For constructing the
histogram we plot the classes on x-axis and the corresponding frequencies on
y-axis. The height of the rectangle is proportional to the frequency of that
class and the width of the rectangle is equal to the length of the class. If
the class width is equal then width of all rectangles are equal but in some
cases the width is not equal that case we adjust the height of the rectangle
and area of the rectangle is proportional to the frequency of that class.
e.g. plot the histogram
for the following data.
|
Marks |
No. of Students |
|
0-10 |
1 |
|
10-20 |
14 |
|
20-30 |
19 |
|
30-40 |
20 |
|
40-50 |
36 |
|
50-60 |
40 |
|
60-70 |
30 |
|
70-80 |
16 |
|
80-90 |
5 |
|
90-100 |
4 |
Histogram for equal class length
e.g. plot the histogram
for the following data.
|
Marks |
No. of Students |
|
0-10 |
1 |
|
10-20 |
14 |
|
20-30 |
19 |
|
30-40 |
20 |
|
40-50 |
36 |
|
50-90 |
40 |
Histogram for unequal class length.
ii. Ogive Curve:
In frequency curve we plot the frequencies
against the value of the variable but in An ogive curve is obtained by plotting
the cumulative frequency against the
upper limit of the class. There are two type of cumulative frequencies
i.
Less
than cumulative frequency
ii.
Greater
than cumulative frequency
So
we get two type of the ogive curve or cumulative frequency curves as
i.
Less
than ogive or less cumulative frequency curve
ii.
Greater
than ogive or cumulative frequency curve
ii. Less
than ogive or less cumulative frequency curve:
For Less than ogive curve is we first
add up the frequencies from top to bottom, (i.e. less than cumulative
frequencies) and then plotting the less than cumulative frequencies against the
upper limit of the corresponding class.
Note that the less than ogive curve is
started from the zero value on y-axis.
ii.Greater than ogive or greater than
cumulative frequency curve:
For
Greater than ogive curve is we first add up the frequencies from top bottom to
top, (i.e. greater than cumulative frequencies) and then plotting the greater
than cumulative frequencies against the lower limit of the corresponding class.
Note that the greater than ogive curve is end at the zero value on y-axis.
e.g draw the ogive curve for following
data.
|
Marks |
No. of Students |
|
0-10 |
1 |
|
10-20 |
14 |
|
20-30 |
19 |
|
30-40 |
20 |
|
40-50 |
36 |
|
50-60 |
40 |
|
60-70 |
30 |
|
70-80 |
16 |
|
80-90 |
5 |
|
90-100 |
4 |
here we find the cumulative frequencies.
|
Marks |
No. of Students |
Less than cumulative frequency |
Greater than cumulative frequency |
|
0-10 |
1 |
1 |
185 |
|
10-20 |
14 |
15 |
184 |
|
20-30 |
19 |
34 |
170 |
|
30-40 |
20 |
54 |
151 |
|
40-50 |
36 |
90 |
131 |
|
50-60 |
40 |
130 |
95 |
|
60-70 |
30 |
160 |
55 |
|
70-80 |
16 |
176 |
25 |
|
80-90 |
5 |
181 |
9 |
|
90-100 |
4 |
185 |
4 |
*Note
that: here in ogive curve we draw a dotted line.*
iii.
Frequency Polygon:
For
constructing a polygon the midpoints of the class are taken along the x-axis
and the corresponding frequencies on the y-axis, then this point are plotted and
joining these points by straight line.
e.g.
To Draw a Frequency polygon for the following data.
|
Marks |
No. of Students |
|
0-10 |
2 |
|
10-20 |
14 |
|
20-30 |
19 |
|
30-40 |
20 |
|
40-50 |
36 |
|
50-60 |
40 |
|
60-70 |
30 |
|
70-80 |
16 |
|
80-90 |
5 |
|
90-100 |
4 |
the frequency polygon is
these term are helpful to understand other terms in next part. if this information help in learning, please share with friends. and join the telegram, to make statistics simple to everyone! https://t.me/gsstats
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