Data:
It is a collection of facts such as numbers, words, measurements, alphabets, symbols, observations or even just description of things.
Examples of data include marks of students, present-absent report of students, name of students, runs made by batsman, etc.
Different types of graphs:
1. Pictograph:
It is simply the pictorial representation of data using symbols.
Example: The pictograph below shows the number of fruits collected by a student.
2. Bar Graph:
In this type of graph, bars of uniform widths are used for representing different quantities, but, the heights of these bars are proportional to the respective values of the quantities.
Example: A bar graph shown below represents various data collected for groups A, B, C, D & E.
3. Double Bar Graph:
In this type of graph, a bar graph represents two sets of data simultaneously. This type of graph is mainly used for comparing data.
Example: A double graph shown below represents literacy rate of various states in India for 2001 and 2011. It can be deduced from the graph that there is an increase of literacy rate in all the states for 2011 when compared with 2001.
Data Organization:
(i) The data available in an unorganized form is called as raw data.
(ii) The extraction of the information from these raw data to give meaning to these data is known as data organization.
(a) Frequency of data:
The number of times a particular quantity repeats itself in the given data is known as its frequency.
Example: Table below represents number of cars possessed by different families in a society.
Number of cars |
No of families |
0 |
4 |
1 |
8 |
2 |
2 |
Â Here, the frequency of families who have one car is 8.
(b) Frequency Distribution Table:
The table which represents the number of times a particular quantity is repeated is known as the frequency distribution table.
Example 1: Table below represents number of cars possessed by different families in a society.
Number of cars |
Frequency |
0 |
4 |
1 |
8 |
2 |
2 |
3 |
3 |
4 |
2 |
5 |
1 |
Example 2: A shop has 4 variety of fruits available. The following list shows quantity of these individual fruits.
B C C A A B D D D A B B A D C C B C D A A A D C B B C D D C A D A D C B C B D A D C A A B D D D A B B A D C C B C D A A A D C B B C D D C A D A D C B C B D A D B
Prepare a frequency distribution table and draw a bar graph.
Solution:Â The table below represents the quantity of different fruits:
Fruit |
Quantity |
A |
20 |
B |
18 |
C |
19 |
D |
24 |
Â The figure below shows the bar graph for given table:(c) Data Grouping:
When the amount of data is huge, then the frequency distribution table for individual observation will result into a large table. In such case, we form group of data and then prepare a table. This type of table is called as grouped frequency distribution.
Example: Suppose, we need to prepare a table for Science marks obtained by 60 students in a class. Then preparing table for individual marks will result into a big table, so we will group the data as shown in the table below:
Range of Marks |
No of students |
0 - 10 |
2 |
10-20 |
9 |
20-30 |
22 |
30-40 |
20 |
40-50 |
6 |
50-60 |
1 |
Total |
60 |
Class Interval or Class: It represents the range in which the data are grouped. For the above example, groups 0-10, 10-20, 20-30, etc. represents class interval.
Lower class limit: The lowest number occurring in a particular class interval is known as its lower class limit. For the above example, if we consider the class interval 10-20 then 10 is called the lower class limit of that interval.
Upper class limit: The highest number occurring in a particular class interval is known as its upper class limit. For the above example, if we consider the class interval 10-20 then 20 is called the upper class limit of that interval.
Width or size of class interval: The difference between the upper class limit and the lower class limit is called as the width or size of class interval. For the above example, if we consider the class interval 10-20, then width or size of this class interval will be 10.
Histogram:
If a bar graph is plotted for grouped frequency distributed type of data or data which has continuous class intervals, then such a graph is known as histogram.
Example 1: Consider the table given below:
Class Interval |
Frequency |
0 - 10 |
2 |
10-20 |
10 |
20-30 |
21 |
30-40 |
19 |
40-50 |
7 |
50-60 |
1 |
Total |
60 |
Here, the height of the bars represents the frequency of the class interval.
Note that there is no gap between the bars as the class intervals are continuous.
Example 2: Draw a histogram for the weekly wages of 30 workers in a factory. The wages are as follows: 830, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855, 845, 804, 808, 812, 840, 885, 835, 835, 836, 878, 840, 868, 890, 806, 840.
Solution:
Class Interval Temp (F) |
Frequency |
800 â€“ 810 |
3 |
810 â€“ 820 |
2 |
820 â€“ 830 |
1 |
830 â€“ 840 |
9 |
840 â€“ 850 |
5 |
850 â€“ 860 |
1 |
860 â€“ 870 |
3 |
870 â€“ 880 |
1 |
880 â€“ 890 |
1 |
890 â€“ 900 |
4 |
Following is the histogram for the given data:
Circle Graph or Pie Chart:
When the given data are represented in circular form, then such type of graph is known as circular graph or pie chart.
Example: Pie-chart below shows distribution of river water into five regions A, B, C, D, and E.A circle graph shows the relationship between a whole and its parts. And the size of each sector is proportional to its contribution of the whole.
Steps to draw pie charts:
Let us understand the process of drawing pie chart with an example.
Example 1: Consider the table below and draw its equivalent pie chart.
Items |
Selling in % |
A |
25 % |
B |
25 % |
C |
50 % |
Â Solution:Â We know that the total angle at center of circle is 360^{o}. So, we will need to divide the circle on basis of the selling percentage.
Consider the table below:
Items |
Selling in % | In fractions | Fraction of 360^{o} |
A |
25 % |
25/100 = 1/4 |
1/4 x 360^{o }= 90^{ o} |
B | 25 % | 25/100 = 1/4 |
1/4 x 360^{o }= 90^{ o} |
C | 50 % | 50/100 = 1/2 |
1/2 x 360^{o }= 180^{ o} |
Step 1: Draw a circle of any radius.Step 2: Divide the circle into two equal parts.
Step 3: Divide any one of the two divided parts into half again. At last, you will obtain the required pie chart.Example 2: Draw a pie chart for the following information:
Colors |
Number of people |
Blue |
18 |
Green |
9 |
Red |
6 |
Yellow |
3 |
Total |
36 |
Solution:Â The angle for each color can be calculated as shown in table below:
Colors |
Number of people | In fractions | Fraction of 360^{o} |
Blue |
18 |
18 / 36 |
18 / 36 x 360^{o }= 180^{ o} |
Green | 9 | 9 / 36 |
9 / 36 x 360^{o }= 180^{ o} |
Red |
6 |
6 / 36 |
6 / 36 x 360^{o }= 180^{ o} |
Yellow | 3 | 3 / 36 |
3 / 36 x 360^{o }= 180^{ o} |
Chance and Probability:
1. Random Experiment:
Any experiment whose result is unknown is termed as a random experiment.
For example, if a coin is tossed then the result could either be head or tail. So, tossing of coin is a random experiment.
2. Outcome:
The possible results of random experiment are known as its outcome.
For example, head and tail can be the two outcomes possible in case of tossing a coin.
3. Equally likely outcome:
When each of the outcomes have same chances of occuring, then such outcomes are termed as equally likely outcomes.
For example, when a dice is tossed there are equal chances of getting any number from one to six. As theÂ number of tosses will increase, the chances of getting any of the six outcomes will almost become equal to each other.
4. Linking chances to probability:
The probability and chances can be related to each other as follows:
Probability = (Number of chances for particular outcome ) / (Total number of outcomes)
Example 1: Let us consider probability of getting Head in case of tossing a coin.
We know that, there is one chance of getting Head out of the two possible outcomes.
Hence, Probability of Head = 1/2.
Example 2: Let us consider probability of getting â€˜twoâ€™ in case of dice tossing.
We know that, there is one chance of getting â€˜twoâ€™ out of the six possible outcomes.
Â Hence, Probability of two = 1/6.
5. Event:
For any experiment, each outcome or a collection of outcomes is known as event.
Example: On tossing a coin, getting a Head is one event and getting a Tail is second event.
Some Examples:
Example 1: A bag has 4 red balls and 2 yellow balls. (The balls are identical in all respects other than color). A ball is drawn from the bag without looking into the bag. What is probability of getting a red ball? Is it more or less than getting a yellow ball?
Solution: From the data, total number of balls = 4 (red) + 2 (yellow) = 6 balls. Hence, 6 outcomes are possible for the given event.
Now, total balls are 6 and number of red balls are 4.
Thus, the probability of getting a red ball is 4/6 = 2/3.
Similarly, total balls are 6 and number of yellow balls are 2.
Thus, the probability of getting a yellow ball is 2/6 = 1/3.
On comparing the probabilities of both the balls, the probability of getting red ball is more than the yellow ball.
Example 2: If you have a spinning wheel with 3 green sectors, 1 blue sector and 1 red sector, what is the probability of getting a green sector? What is the probability of getting a non blue sector?
Solution:Â From the data, the total sectors = 3 (green) + 1 (blue) + 1 (red) = 5 sectors.
Now, total sectors are 5 and the probability of getting green sector is 3.
Thus, probability of getting a green sector = 3/5.
Now, total sectors are 5 and the probability of getting non-blue sector i.e. either green or red is 4.
Thus, probability of getting a non-blue sector = 4/5.