Introduction To Graphs And Data Handling

1.0Organizing And Grouping Of Data

Some Basic Terms

• Data : The word data means information in the form of numerical figures or a set of given facts.
• Raw data : Data obtained from direct observation is called raw data. The marks obtained by 10 students in a monthly test is an example of raw data or ungrouped data.
• Observation : Each numerical figure in a data is called an observation. Frequency of an observation : The number of times a particular observation occurs is called its frequency.
• Ungrouped frequency distribution : When the given set of observation or data is written in the form of a table in terms of its frequency, this is called ungrouped frequency distribution of raw data.
• Grouped data : To present the data in a more meaningful way, we condense the data into convenient number of classes or groups, generally not exceeding 10 and not less than 5.
• Arrayed data : The data arranged in an order, whether ascending or descending is called an arrayed data.
• Range : The difference between the maximum and the minimum values of the given observations is called the range of the data. Range $=($ Maximum Value $)-($ Minimum Value $)$

Data is a collection of numbers gathered to give some information.

2.0Data Using Tally Marks

If we have our observation as $23,49,50,19,18,17,26,36,38$, $39,12,6,8,5,4,22,40,47,42,43$ This above data is difficult to handle & store correctly. So, we need to reduce the data by dividing it into groups & then get the frequency of observations.

$0-10$ is known as Class Interval. 0 is the lower limit and 10 is the upper limit. The class size is $10-0=10$. The difference between upper limit and lower limit is called the Class Size/Class Width. The mid-point of a class is known as class mark. Eg. Class mark of class $10-20$ is $\frac{10+20}{2}=15$.

3.0Representation Of Data

For data to be useful, it is very important to collect complete, accurate and relevant data. After collection of data, it is necessary to represent data in a precise manner so that it is easily understood.

Representation of data in a visual manner is known as its graphical presentation or simply, graphs. Data can also be presented in the form of a table; however a graphical form is easier to understand.

Consider the family of Mr. Khanna. They are very fond of eating ice-creams. Some like butterscotch, some like vanilla flavour, some like strawberry and the rest of them likes chocolate. Some demand cones & some of them demand cups. A child of VIII class decides to construct the graphs according to their amount of ice-creams consumed. There are 7 ways of representing the data.

4.0Pictograph

Pictorial representation of numerical data, using picture symbol is known as a pictograph. Most business and industrial organizations use this method to represent their data.

• Q. Study the following pictograph & answer the questions of sale of toys.

Monday
Tuesday
Wednesday

(i) How many toys were sold on Tuesday? (ii) On which 2 days the sale of days was same? Explanation: (i) 4 (ii) Monday and Wednesday

5.0Bar Graph

A bar graph represents observations using rectangles of equal width. The height of the rectangles [known as "BARS"] are of length depending upon the value of observation. Remark : (i) The distance between 2 bars is same. (ii) Width of all bars is same.

If we draw the bar graph for the Khanna's family about the amount of ice-creams consumed in 3 months, then it will be shown in the following way.

• Q. Study the bar graph & answer the following : (i) What are the marks of Rajesh and Sita? (ii) Who has got highest marks? (iii) State true or false: "Difference between marks of Ram and Rita is 50".
  
  Explanation: (i) The marks of Rajesh and Sita are $200\&150$ respectively. (ii) Rajesh (The bar of Rajesh is of highest value). (iii) True (marks of Ram is 100 & marks of Rita is $50.\therefore$ Difference is 50)
• You can take 2 different scales on $x$-axis and $y$-axis in the same graph.
• Always write the scale on x -axis and y -axis on the bar graph.

6.0Double Bar Graph

A bar graph having two sets of data together is known as double bar graph. It is mostly used to compare the values.

• Q. Study & answer the questions :-
  
  (i) In which subject, marks of boys are greater than marks of girls? (ii) In which subjects, marks of girls are higher than marks of boys? (iii) What is the total number of marks of girls in all subjects? Explanation: (i) In English, marks of boys are greater than marks of girls. (ii) Marks of girls are higher than marks of boys in Maths & Science. (iii) Total number of marks of girls in all subjects $=30+20+40=90$

7.0Histogram

A histogram is a bar graph that is used to show the class intervals of different activities. It has adjacent bars over the given class intervals.

The histograms are useful for grouped frequency distribution. Consider the example of Khanna's family, if we draw the histogram for the data of icecreams consumed by different age-groups then it will be in the following way.

• Q. Study the given histogram and answer the following questions . (i) What is the class size? (ii) How many students scored less than $20$ marks? Explanation: (i) The class size is 10 [as 20-10 = 10 and $30-20=10$ ] (ii) 5 students scored less than 20 marks.
  

8.0Difference Between Bar Graph And Histogram

Difference between a Bar graph and a Histogram. (i) In a bar graph, the bars are at a distance from one another, while in a histogram, the bars (rectangles) touch one another. (ii) In a bar graph, we generally have a scale for either the x -axis or y -axis (generally y axis as vertical bars are more popular), but in a histogram, we have scales for both xaxis and $y$-axis and both scales need not be the same. (iii) In a bar graph, one axis may not have numerical values but have names, subjects, flavours etc. along it, but in a histogram we display numerical values along both the axes. Sometimes, the class intervals do not start from 0. So, a KINK or a - -sign is given to indicate that the distance from 0 to that class interval is not shown completely.

Remark : Whenever a histogram is given, certain conclusions can be drawn from the graph. This is known as INTERPRETATION of the histogram.

9.0Pie-charts / Pie-diagrams / Circle graph

Pie-charts show the relationship between a whole and its parts. Here, the whole circle is divided into sectors. Its size is proportional to the information it represents. For the Khanna's family, if we draw the pie-chart of flavours of ice-creams, then it will be in the form as shown in figure.

Let us draw a pie-chart for the amount of food ordered in a hotel. There are total 18 dishes. Out of 18, 3 are Mexican, 5 are Chinese & 10 are French (See figure). This way of representation in known as pie-chart as it looks like a pie & the parts look like the slices of the pie.

In the above pie chart, the proportion of French food ordered $=\frac{\text{ Amount of French food ordered }}{\text{ Whole food ordered }}=\frac{10}{18}=\frac{5}{9}$

• Q. Rashmi watches the following TV channels in the given number of hours as shown in the table. Construct a pie chart for the given information.

Solution: Step-1 - Calculate the central angle of each sector.

Step-2 : Draw a circle of any radius with centre 0.

Reading and Interpreting pie-charts

Reading a pie-chart just means to find out what part of the entire chart does each sector represent. To find the central angle, we use Central angle $=\frac{\text{ Particular part }}{\text{ Total of all parts }}\times 360^{\circ }$ and to find the particular sector, we use Particular sector $=\frac{\text{ Central angle }\times \text{ Total of all parts }}{360^{\circ }}$

10.0Line Graph

A line graph is used to display the data that is continuously changing with time. We can take the example of a car moving on a road. It's speed is continuously changing with time.

To plot take time on x -axis & the changing variable i.e. speed on y -axis.

• Q. Ranu and Manu are two runners in a 500 m race. The graphs shows the progress in a race made by them. Read the graph and answer the following questions: (a) At what time were Ranu and Manu at the same level? (b) Did Ranu finish before Manu or Manu before Ranu? (c) Who had more speed in the first 10 seconds? (d) Who was leading after 50 seconds?
  
  Explanation: (a) After 45 secs Ranu and Manu were at the same level. (b) Ranu finished 500 m distance in 56 secs. Manu finished 500 m distance in 58 sec . So, Ranu finished race before Manu. (c) In the first 10 seconds, Ranu covered 100 m and Manu covered 200 m . So, Manu had more speed in the first 10 seconds as she covered more distance. (d) Ranu was leading after 50 seconds as she covered distance of 400 m and Manu covered distance of 358 m .

11.0Linear Graph

Sometimes, it is not necessary that we get a continuous line segment. We may get a whole unbroken line, which we call as Linear Graph. To plot a linear graph, we need to find the location [or co-ordinates of the points].

• Q. Draw a graph to represent the relationship between the perimeter and the side x of a square given as "Perimeter $(P)=4\times$ side, i.e., $P=4x$ " using the following table:

Explanation: To sketch the graph, take the side of square x (in cm ) on the X -axis and the perimeter P (in cm ) on the Y -axis as per the following scale: Scale: X-axis: 10 small divisions $=1\mathrm{cm}$ Y-axis: 5 small divisions $=2\mathrm{cm}$

Co-ordinate Axes: The horizontal number scale is called the $\mathbf{x}$-axis and the vertical number scale the $\mathbf{y}$-axis. The point where the two scales cross each other is called the origin. The distance $OM$ of the point $P$ from the $y$-axis is called $x$-coordinate (or abscissa) and the distance PM of the point $P$ from the x -axis is called the y -coordinate (or ordinate).

• Q. A point $(3,2)$ is located 3 points from the left edge and 2 points from the bottom edge. ' 3 ' is known as x-coordinate (or abscissa) of the point and ' 2 ' is known as $y$-coordinate of the point. $\therefore (3,2)$ is known as coordinates of the point. Explanation
  
• Q. Plot the following points on a graph : $\mathrm{A}(1,2),\mathrm{B}(3,3),\mathrm{C}(0,2)$ Solution:
  
• Q. Plot the following points on a graph paper. (a) $A(-4,2)$ (b) $\mathrm{B}(2,5)$ (c) $\mathrm{C}(0,3)$ (d) $\mathrm{D}(-2,-1)$ (e) $\mathrm{E}(-3,0)$ (f) $F(1,-2)$ Explanation: (a) Take a graph paper and draw the X -axis and the Y -axis. (b) Mark the origin 0. (c) Using a proper scale, mark number on X and Y -axis. (d) Plot the above points keeping in mind the quadrant in which it lies.
  

Correct

Incorrect

12.0Chances And Probability

There are many situations in life where we are not sure of the results of some events, like when a coin is tossed, will it be head or tail?

We don't know what the outcome will be. But there is certainty that either it will be head or tail. So, the chance of happening of an event ranges from being certain to happen.

We define probability as the measure of the chance of happening or non-happening of an event.

Probability $P(E)$ of an event is defined as $P(E)=\frac{\text{ Number of favourable outcomes }}{\text{ Number of total outcomes }}$

• The word probability derives from the Latin Probabilities, which can also mean probity, a measure of the authority of a witness in a legal case in Europe.

• Probability of an outcome is represented by a number that lies from 0 to 1 .
• If the probability of an outcome is 0 , it means it is an Impossible event & if it is 1 , we say it is a certain event.
• In an experiment, the sum of probabilities for all possible outcomes is equal to one.
• Q. Study the following events : (i) If we try to start a bike, then what will happen? (ii) A die is thrown, what will be the outcome? Also find the probability of getting a number 5. Explanation: (i) If we try to start a bike, there are 2 possibilities either it will start or it will not start so, there are 2 outcomes. Likelihood of getting started is one out of 2 i.e. $\frac{1}{2}$. So, we say, the probability of getting started is $\frac{1}{2}$ [one out of two] and the probability of getting not started is $\frac{1}{2}$ [one out of two]. (ii) If an unbiased die is thrown, we can get any one of the six faces i.e. 6 number (i.e., 1 , $2,3,4,5,6$ ). So, the likelihood of getting a number is one out of 6 i.e., $\frac{1}{6}$. So, we say the probability of getting a number 5 is $\frac{1}{6}$.
• When a coin is tossed either we get head or tail.

Outcomes As Events

Each outcome of an experiment or a collection of outcomes makes an event. For example, on tossing a coin, getting a head or a tail is an event.

• Q. A bag contains 4 red marbles and 2 green marbles. A marble is drawn from the bag. What is the probability of getting a red marble? Solution: The total number of marbles in the bag $=4+2=6$. We define probability as the ratio of number of outcomes making an event and total number of outcomes. So, total number of red marbles $=4$. $\therefore$ Probability of getting a red marble $=\frac{\text{ Number of outcomes getting a red marble }}{\text{ Total number of outcomes }}=\frac{4}{6}=\frac{2}{3}$

13.0Mind Map