7STASTICS

Tabulation

There are 40 children in class 8B of a school. The grades secured by the students in the mathematics paper during the half yearly examination is as follows.

A E C B E A B C B D A B C A D B D E A D E A B C B E B A A E C B C B A B A C C B

i. How many children secured A grade? ii. How many got B grade ?iii. How many got below C grade?iv. What is the grade scored by majority number

of students ?v. How many children got the least grade ?

To answer the first question , we need to count the grade A only for the second we count the B grade and the third we need to count the grades D & E .What about the fourth one ?We have to add each type separately ,right ?Here it is convenient to record this counting first:

Grade Number of student

A 10

B 12

C 8

D 4

E 6

Total 40

Now can’t you answer all the questions easily?

Here the table shows A grade repeated 10 times , B grade repeated 12 times and C grade 8 times so on.

The number of repetitions (or occurrences) is known as frequency.

Now consider another problem.

The temperatures (in degree Celsius) of a particular town in a month are listed below :

Statistics is a branch of science which deals with data collected for specific purposes .We can make decisions about the data by analyzing & interpreting it

26 29 27 26 29 30 31 30 28 26 27 31 28 31 27 28 29 30 26 28 30 30 29 28 27 30 28 27 27 28

i. How many days have temperature as 26 ̊C ?ii. How many days have temperature below 28 ̊C ? iii. How many days have temperature as 30 ̊C ? iv. How many days have temperature in between 29 &

31 ̊C ? v. Which temperature occurs the most ?

To answer the first question , we need to count the temperature 26 ̊C only, for the second we count the temperatures 26 &27 ̊C . And for the third question ,count 30 ̊C only .

What about the fourth question ?

For convenient first record this counting :

Temperature(in degree Celsius )

Tally Number of repetitions

26 IIII 427 IIII I 628 IIII II 729 IIII 430 IIII I 6

31 III 3Total 30

Lets make a table , we must note how many times each temperature occurs .The lowest temperature is 26 ̊C and the highest is 31 ̊C . Write the numbers from 26 to 31 in a column and check how many times each is repeated . We can use method of tallies .

Now it is easy to answer all questions above , just by looking t the table is n’t it ?

This table shows how many times each temperature occurs ,such as 26 four times ,27 six times ,28 seven times so on .In tables of this kind ,the number of occurrences is generally called frequency.

A table shows counts for a variable is called a frequency table .

1) The height of the 10 th standard students are listed below .(height is in centimeters) .

130 138 130 136 135 134 134 13 1 13 0 138 137 135 133 123 132 136 138 134130 136 137 130 135 134130 131 133 135 132

Make a frequency table and answer these questions : i. How many students have height

of 135 cm ?ii. How many students have height

below 136 cm ? iii. How many students have height

of 137 & 138 cm ? iv. Which height occurs the most ?v. Which height occurs the least ?

The sores obtained by the students in a class are listed below . Make a frequency table and answer the following questions :

2)

“Statistics may rightly called the science of averages” –A.L BOWLEY

10 8 4 5 6 9 7 6 5 9 10 8 8 9 7 6 3 2 10 9 9 7 8 6 5 4 7 7 6 3

i. How many children got above 8 ?ii. How many scored 7 marks ?iii. How many children got 10 ?iv. How many scored in between 5 to 9 ?

3) The daily wages of labors in a factory are listed

below ( in rupees) .

110 115 120 115 110 120 110 120 115 125 125 115 120 110 120

Make a frequency table and answer these questions :

i. How many of them have salary RS 115 ?ii. How many of them have highest salaries ?iii. How many of them got least salary RS 110 ?iv. How many of them have salary RS 115 ?v. How many labors are working in that factory ?

Here the lowest score is 1 & the highest is 69 .

Another form

The total marks of an exam is 100.Following represents the marks obtained by the student.

69 55 1 17 35 56 22 48 67 46 33 45 53 68 32 48 49 38 47 58 52 54 55 47 39 28 5659 46 5 36 57 66 43 55 374 1 27 16 11 31 40 61 21 5155 20 52 48

To make a table as we did so far, we would have to write all numbers from 1 to 100 .But all such numbers are not really needed .More over from such table , we don't get a general idea of the score obtained by the students . so we do it in a slightly different way to find the frequency table .Instead of writing the actual scores in a column , we need to classify them as groups .That is ,Below 10 10 to 20 (20 is not included)20 to 30 (30 is not included)30 to 40 (40 is not included)40 to 50 (50 is not included)

The frequency of a particular data value is the number of times the data value occurs. The frequency of a data value is often denoted by “f ” . A frequency table is constructed by arranging collected data values in ascending order of magnitude with their corresponding frequencies .

s

50 to 60 (60 is not included)60 to 70 (70 is not included)

For convenient we can write

0 - 1010 -20 20 -30 30 -40 40 - 50 ……….. So on

Each group is known as class.

The scores included in the 0 - 10 is the scores Below 10 .What about in the second class. ? The scores 10 & above 10 but below 20 .Similarly what about other classes ?

In the class 0 - 10 , 0 is the lower limit & 10 is the Upper limit . Here we are excluded the upper limit values & lower

limit values are included .Such classes are called exclusive classes .

Then what about the differences ?

The difference between upper limit values & lower limit values is known as class interval.

When the set of data value are spread out ,it is difficult to set up a frequency table for every data value as there will be too many rows in the table .So we group the data in to class intervals (or groups) to help us organize , interpret & analyze the data . Each group starts at data value data value that is a multiple of that group. For eg , if the size of the group is 5 ,then the groups should start at 5,10, 15,20, etc.

That is , above class interval is 10 .Now we need to count the scores in each class .

Class Tally frequency

0 -10 II 210-20 III 320 -30 IIII 530-40 IIII IIII 940-50 IIII IIII II 1250-60 IIII IIII IIII 1460-70 IIII 5

Total 50

Now we can easily answer the following questions :

i. Which is the highest frequency ? ii. Which is the lowest frequency ? iii. How many of them got scores less

than 20 ?iv. How many of them got scores less

than 20 ?v. How many of them got scores in

between 40 & 50 ?

Range is the difference between upper limit & lower limit values . That is , R=U-L

The mid point of the class interval is known as the class mark . That is , R=(U+L)/2

Height(in centimeter) of some people are listed below:

Now consider another problem .

150 100 149 160 109 108 112 119 120 115 129 130 142 145 148 156 162 189 185 145 120 129 130 188 189 142 100 160 109 150 145 148 156 129 130 112

Now we can make a frequency table by choosing the class interval as follows :

Any statistical study starts with a process data collection. The second process is the a of the collected data . we can presented the collected data in numerical as well as graphical presentation. collected data can be numerically presented in three ways .they are row data , discrete continuous frequency distribution.

Here in all the class intervals we include the upper limit also .That is , the classes in which both upper limit & lower limit values can be included in the same class are called inclusive classes .

Height Tally Number

100 -109 IIII 5

110-119 IIII 4

120 -129 …………… ……………

130-139 …………… ……………

140-149 …………… ……………

150-159 …………… ……………

160-169 …………… ……………

170-179 …………… ……………

180-189 …………… ……………

190-199 …………… ……………Total 36

1) The Weight (in kilograms) of the members of the school health club are given below

37 40 59 35 2 ⅟26 2 52 36 2 40 2 ⅟ ⅟ ⅟35 2 59 37 2 46⅟ ⅟55 36 2 40 2 35 2 ⅟ ⅟ ⅟ 26 2 52 40 33⅟44 2 40 26 2 53⅟ ⅟55 37 44 2 35 2 ⅟ ⅟

We want to make a frequency table . would classes like 30 -34 , 35- 39 ,40-44, 45-49 & so on do ?In which class would we put 44 2 ⅟Kg ? For eg , we can take classes as 30 -35 , 35 -40 , 40-45 ,so on. On 44 2 can be put in the class ⅟40-45 .That is , here we need exclusive classes . Now we can make a frequency table.

Discrete frequency distribution is a table of 2 columns in which the first is variable values in individual form & second is the frequency. It is known as discrete frequency table .

Continuous frequency distribution is a table of 2 columns in which the first is variable values in class form & second is the frequency.

in

class Tally Frequency

30-35 …………… ……………35-40 …………… ……………40-45 …………… ……………45-50 …………… ……………50-55 …………… ……………55- 60 …………… ……………

Total ……………

1) Given below are the scores of the children in class 9B .Make a frequency table ?

23 25 38 47 40 39 2631 37 32 41 30 25 3833 23 47 40 39 31 4036 25 47 41 40 47 26 33 29 32

We have seen how numerical data can be pictorially represented as bar charts .

A new picture

Now let’s see how the data in a frequency table can be represented by a picture .

The table below gives wages of labors.

wages Number of labors (frequency)

100 -120 5120-140 11140 -160 16160-180 20180-200 9200-220 2Total 63

See how this data is represented by a picture .

Data can be graphically presented in many ways . Commonly used methods are. Histogram, polygon ,pictogram , pie diagram ,bar diagram , ogive curves , etc.

Draw a horizontal & vertical line.

Step 1

The classes are marked on the horizontal line& frequencies on the vertical line .

Step 2

Step 3

Draw a rectangle for each class .

Step 4

Then shade the rectangle.

100-120

120-140

140-160

160-180

180-200

200-220

0

5

10

15

20

25

5

11

16

20

9

2

Histogram

frequencies

classes

freq

uenc

ies

The width of each rectangle shows the length of the class interval & its height shows the frequency .Such a picture is called a Histogram .

1) Details of rain fall in June & July are given in the table below . Draw a Histogram .

Rain fall(mm) Days

10-20 420-3 0 230-40 1440-50 1250-60 1560- 70 1070-80 680-90 1590-100 18

2) Details of weights are given in the table below . Draw a Histogram .

Weight(kg) Number of children

35-37 337-39 6

39-41 941-43 1243-45 1545- 47 847-49 5

2) Details of ages are given in the table below . Draw a Histogram .

Age Number

20-30 230-40 440-50 850-60 760-70 1270- 80 280-90 3

Total 38

LOOKING BACK

Learning outcomes What I can With teachers help

Must improve

Making a frequency distribution of individual entries from given data

Dividing given data in to classes & make a frequency table .

Explaining the need for grouping in to classes in making a frequency table .

Representing a frequency table as a histogram .