Graphical Representation of Statistical Data

Steps:

1. On a graph paper, draw two mutually perpendicular lines OX and OY which intersect each other at point O. The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.

2. On the x-axis, starting from O, mark points at equal distances. At these points write the names of the different subjects given. In this question, the names are Hindi, Science, English, Mathematics and Social Studies.

3. Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (percentage of marks obtained). Along the y-axis, take 1 big division (1 cm) to represent 10%.

The required bar graph is as given below:

Steps:

1. On a graph paper, draw two mutually perpendicular lines OX and OY which intersect each other at point O. The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.

2. On the x-axis, starting from O, mark points at equal distances. At these points write the different times given. In this question, the times are 5 a.m., 8 a.m., 11 a.m., 3 p.m. and 6 p.m.

3. Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (temperature in °C). Along the y-axis, take 1 big division (1 cm) to represent 4°C.

The required column graph is as given below:

Steps:

1. On a graph paper, draw two mutually perpendicular lines OX and OY which intersect each other at point O. The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.

2. On the x-axis, starting from O, mark points at equal distances. At these points write the names of the different months given. In this question, the names are January, February, March, April and May.

3. Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (number of pressure cookers produced). Along the y-axis, take 1 big division (1 cm) to represent 200 pressure cookers.

The required bar graph is as given below:

Steps:

1. On a graph paper, draw two mutually perpendicular lines OX and OY which intersect each other at point O. The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.

2. On the x-axis, starting from O, mark points at equal distances. At these points write the different years given. In this question, the years are 2010-11, 2011-12, 2012-13, 2013-14 and 2014-15.

3. Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (number of students). Along the y-axis, take 1 big division (1 cm) to represent 200 students.

The required column graph is as given below:

Steps:

1. On a graph paper, draw two mutually perpendicular lines OX and OY which intersect each other at point O. The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.

2. On the x-axis, starting from O, mark points at equal distances. At these points write the different years of census given. In this question, the years are 1951, 1961, 1971, 1981, 1991 and 2001.

3. Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (population in millions). Along the y-axis, take 1 big division (1 cm) to represent 120 million people.

The required bar graph is as given below:

The subjectwise marks obtained by Varun may be recorded from the given bar graph as under:

Subject	Marks obtained
English	87
Hindi	92
Mathematics	98
Social Science	84
Science	95

(i) From the above table, the least marks obtained by Varun are 84, which are in Social Science.

Hence, Varun's performance was the poorest in Social Science, as the marks obtained in it are the lowest.

(ii) Marks obtained in Mathematics = 98

Marks obtained in Social Science = 84

Required ratio = 98 : 84

= $\frac{98}{84}$

= $\frac{7}{6}$

= 7 : 6

Hence, the required ratio is 7 : 6.

(iii) Total marks obtained by Varun = 87 + 92 + 98 + 84 + 95

= 456

Maximum marks in each subject = 100

Maximum marks for 5 subjects = 5 x 100 = 500

Total percentage score = $\frac{456}{500} \times 100$% = 91.2%

Hence, Varun's total percentage score is 91.2%.

(iv) Arranging the marks obtained in descending order, we get :

98, 95, 92, 87, 84

So, the three highest scores attained by Varun are 98, 95 and 92.

Average of the three highest scores = $\frac{98 + 95 + 92}{3}$

= $\frac{285}{3}$

= 95

Hence, the average of the three highest scores attained by Varun is 95.

In the given histogram, the frequencies are depicted by the heights of the rectangles whose bases are the class intervals.

(i) Clearly, the class intervals are 0 – 5, 5 – 10, 10 – 15, 15 – 20, etc.

So, class size = 5 − 0 = 5.

Hence, the class size is 5.

(ii) Marks less than 5 means marks in the interval 0 – 5 and for this interval the height of the rectangle is 2 units.

Hence, 2 students obtained less than 5 marks.

(iii) Marks 20 or more but less than 25 means marks in the interval 20 – 25 and for this interval the height of the rectangle is 7 units.

Hence, 7 students obtained 20 or more marks but less than 25 marks.

(iv) Marks 25 or more means marks in the intervals 25 – 30 and 30 – 35, for which the heights of the rectangles are 9 units and 12 units respectively.

∴ Number of first divisioners = 9 + 12

= 21

Hence, there are 21 first divisioners.

(v) Marks 10 or more but less than 20 means marks in the intervals 10 – 15 and 15 – 20, for which the heights of the rectangles are 2 units and 5 units respectively.

∴ Number of such students = 2 + 5

= 7

Hence, 7 students obtained 10 or more but less than 20 marks.

In the given histogram, the frequencies are depicted by the heights of the rectangles whose bases are the class intervals.

(i) The tallest rectangle in the histogram stands on the class interval 35 – 40 and its height is 11 units.

Hence, the maximum number of teachers are in the age group 35 – 40 and their number is 11.

(ii) Ages 25 years or more but less than 30 means the ages in the interval 25 – 30 and for this interval the height of the rectangle is 4 units.

Hence, 4 teachers have their ages 25 years or more but less than 30.

(iii) Age 40 or more means the ages in the intervals 40 – 45 and 45 – 50, for which the heights of the rectangles are 8 units and 6 units respectively.

∴ Number of such teachers = 8 + 6

= 14

Hence, 14 teachers are of the age 40 or more.

(iv) Clearly, the class intervals are 25 – 30, 30 – 35, 35 – 40, 40 – 45 and 45 – 50.

So, class size = 30 − 25 = 5.

Hence, the class size is 5.

(v) By formula,

Class mark = $\frac{\text{Upper limit + Lower limit}}{2}$

Class mark of 25 – 30 = $\frac{25 + 30}{2} = \frac{55}{2}$ = 27.5

Class mark of 30 – 35 = $\frac{30 + 35}{2} = \frac{65}{2}$ = 32.5

Class mark of 35 – 40 = $\frac{35 + 40}{2} = \frac{75}{2}$ = 37.5

Class mark of 40 – 45 = $\frac{40 + 45}{2} = \frac{85}{2}$ = 42.5

Class mark of 45 – 50 = $\frac{45 + 50}{2} = \frac{95}{2}$ = 47.5

Hence, the class marks of the classes are 27.5, 32.5, 37.5, 42.5 and 47.5.

In the given histogram, the frequencies are depicted by the heights of the rectangles whose bases are the class intervals.

(i) The tallest rectangle in the histogram stands on the class interval 950 – 1000 and its height is 8 units.

Hence, the largest number of workers are kept in the wage-group 950 – 1000 and their number is 8.

(ii) The shortest rectangle in the histogram stands on the class interval 900 – 950 and its height is 2 units.

Hence, the least number of workers are getting wages in the group 900 – 950 and the number of such workers is 2.

(iii) The heights of the rectangles standing on the class intervals 700 – 750, 750 – 800, 800 – 850, 850 – 900, 900 – 950, 950 – 1000, 1000 – 1050 and 1050 – 1100 are 3, 7, 5, 4, 2, 8, 6 and 5 units respectively.

∴ Total number of workers = 3 + 7 + 5 + 4 + 2 + 8 + 6 + 5

= 40

Hence, the total number of workers is 40.

(iv) Clearly, the class intervals are 700 – 750, 750 – 800, 800 – 850, etc.

So, class size = 750 − 700 = 50.

Hence, the class size is 50.

Steps:

1. On a graph paper, draw two mutually perpendicular lines OX and OY which intersect each other at point O. The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.

2. We take the marks obtained along the x-axis and the number of students along the y-axis.

3. Choose a suitable scale. Along the x-axis, take 1 big division (1 cm) to represent 10 marks and along the y-axis, take 1 big division (1 cm) to represent 2 students.

4. Draw rectangles with the class intervals 0 – 10, 10 – 20, 20 – 30, 30 – 40, 40 – 50, 50 – 60, 60 – 70 and 70 – 80 as bases and the corresponding frequencies 2, 3, 14, 10, 6, 5, 8 and 2 as heights. Since the data is in exclusive form, the rectangles are drawn adjacent to each other, leaving no gap between them.

The required histogram is as given below:

Steps:

1. On a graph paper, draw two mutually perpendicular lines OX and OY which intersect each other at point O. The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.

2. We take the heights along the x-axis and the number of students along the y-axis.

3. Choose a suitable scale. Along the x-axis, take 1 big division (1 cm) to represent 5 cm and along the y-axis, take 1 big division (1 cm) to represent 2 students. Since on the x-axis we have to start from 140, we give a break (called kink) and then start it at 140.

4. Draw rectangles with the class intervals 140 – 145, 145 – 150, 150 – 155, 155 – 160, 160 – 165 and 165 – 170 as bases and the corresponding frequencies 6, 10, 15, 18, 2 and 1 as heights. Since the data is in exclusive form, the rectangles are drawn adjacent to each other, leaving no gap between them.

The required histogram is as given below:

Steps:

1. On a graph paper, draw two mutually perpendicular lines OX and OY which intersect each other at point O. The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.

2. We take the weights along the x-axis and the number of teachers along the y-axis.

3. Choose a suitable scale. Along the x-axis, take 1 big division (1 cm) to represent 5 kg and along the y-axis, take 1 big division (1 cm) to represent 2 teachers. Since on the x-axis we have to start from 45, we give a break (called kink) and then start it at 45.

4. Draw rectangles with the class intervals 45 – 50, 50 – 55, 55 – 60, 60 – 65 and 65 – 70 as bases and the corresponding frequencies 2, 8, 11, 7 and 3 as heights. Since the data is in exclusive form, the rectangles are drawn adjacent to each other, leaving no gap between them.

The required histogram is as given below:

Steps:

1. On a graph paper, draw two mutually perpendicular lines OX and OY which intersect each other at point O. The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.

2. We take the monthly salary along the x-axis and the number of teachers along the y-axis.

3. Choose a suitable scale. Along the x-axis, take 1 big division (1 cm) to represent ₹ 1,000 and along the y-axis, take 1 big division (1 cm) to represent 1 teacher. Since on the x-axis we have to start from 16000, we give a break (called kink) and then start it at 16000.

4. Draw rectangles with the class intervals 16000 – 17000, 17000 – 18000, 18000 – 19000, 19000 – 20000, 20000 – 21000, 21000 – 22000, 22000 – 23000 and 23000 – 24000 as bases and the corresponding frequencies 8, 4, 3, 5, 2, 3, 1 and 1 as heights. Since the data is in exclusive form, the rectangles are drawn adjacent to each other, leaving no gap between them.

The required histogram is as given below:

Steps:

1. Since the angle about the center of a circle is 360°, divide 360° in proportion to the given data. In this question, divide 360° in proportion 120 : 400 : 170 : 135 : 75 to obtain the corresponding central angles.

2. Make a table as given.

Creatures	Number	Central Angles
Beast animals	120	120/900 × 360° = 48°
Other land animals	400	400/900 × 360° = 160°
Birds	170	170/900 × 360° = 68°
Water animals	135	135/900 × 360° = 54°
Reptiles	75	75/900 × 360° = 30°
Total	900	360°

3. Draw a circle of any suitable radius.

4. Draw a horizontal radius.

5. For each central-angle, draw a sector.

Steps:

1. Since the angle about the center of a circle is 360°, divide 360° in proportion to the given data. In this question, divide 360° in proportion 120 : 90 : 75 : 105 : 150 to obtain the corresponding central angles.

2. Make a table as given.

Subject	Marks obtained	Central Angles
English	120	120/540 × 360° = 80°
Hindi	90	90/540 × 360° = 60°
Mathematics	75	75/540 × 360° = 50°
Science	105	105/540 × 360° = 70°
Social Studies	150	150/540 × 360° = 100°
Total	540	360°

3. Draw a circle of any suitable radius.

4. Draw a horizontal radius.

5. For each central-angle, draw a sector.

Steps:

1. Since the angle about the center of a circle is 360°, divide 360° in proportion to the given data. In this question, divide 360° in proportion 25 : 12 : 6 : 8 : 9 to obtain the corresponding central angles.

2. Make a table as given.

Item	Money spent (in ₹)	Central Angles
Paper	25	25/60 × 360° = 150°
Printing	12	12/60 × 360° = 72°
Binding	6	6/60 × 360° = 36°
Publicity	8	8/60 × 360° = 48°
Royalty	9	9/60 × 360° = 54°
Total	60	360°

3. Draw a circle of any suitable radius.

4. Draw a horizontal radius.

5. For each central-angle, draw a sector.

Given:

Monthly income of Mr. Roy = ₹ 15,000

Total money spent = ₹ (1,875 + 3,250 + 1,125 + 2,250)

= ₹ 8,500

∴ Savings = ₹ (15,000 − 8,500)

= ₹ 6,500

Steps:

1. Since the angle about the center of a circle is 360°, divide 360° in proportion to the given data. In this question, divide 360° in proportion 1,875 : 3,250 : 1,125 : 2,250 : 6,500 to obtain the corresponding central angles.

2. Make a table as given.

Item	Money (in ₹)	Central Angles
Rent	1,875	1,875/15,000 × 360° = 45°
Food	3,250	3,250/15,000 × 360° = 78°
Education	1,125	1,125/15,000 × 360° = 27°
Miscellaneous	2,250	2,250/15,000 × 360° = 54°
Savings	6,500	6,500/15,000 × 360° = 156°
Total	15,000	360°

3. Draw a circle of any suitable radius.

4. Draw a horizontal radius.

5. For each central-angle, draw a sector.

Steps:

1. Since the angle about the center of a circle is 360°, divide 360° in proportion to the given data. In this question, divide 360° in proportion 20 : 35 : 25 : 15 : 5 to obtain the corresponding central angles.

2. Make a table as given.

Brand	Percentage of buyers	Central Angles
A	20%	20/100 × 360° = 72°
B	35%	35/100 × 360° = 126°
C	25%	25/100 × 360° = 90°
D	15%	15/100 × 360° = 54°
E	5%	5/100 × 360° = 18°
Total	100%	360°

3. Draw a circle of any suitable radius.

4. Draw a horizontal radius.

5. For each central-angle, draw a sector.

Steps:

1. Since the angle about the center of a circle is 360°, divide 360° in proportion to the given data. In this question, divide 360° in proportion 40 : 10 : 25 : 20 : 5 to obtain the corresponding central angles.

2. Make a table as given.

Head	Percentage of expenditure	Central Angles
Wages and salaries	40%	40/100 × 360° = 144°
Fuel and Power	10%	10/100 × 360° = 36°
Materials	25%	25/100 × 360° = 90°
Maintenance	20%	20/100 × 360° = 72°
Depreciation	5%	5/100 × 360° = 18°
Total	100%	360°

3. Draw a circle of any suitable radius.

4. Draw a horizontal radius.

5. For each central-angle, draw a sector.

Given:

Monthly income of the officer = ₹ 12,000

The central angles of the various items, as read from the given pie-diagram, are :

Item	Central Angle
Clothing	60°
Food	135°
Education	30°
Miscellaneous	45°
House rent	90°
Total	360°

(i) Money spent on food = $\frac{135^{\circ}}{360^{\circ}} \times 12,000$

= ₹ 4,500

Hence, he spends ₹ 4,500 on food.

(ii) The central angle of the sector representing house rent is 90°.

Percentage of income spent on house rent = $\frac{90^{\circ}}{360^{\circ}} \times 100$% = 25%

Hence, 25% of his income is spent on house rent.

(iii) Money spent on miscellaneous items = $\frac{45^{\circ}}{360^{\circ}} \times 12,000$

= ₹ 1,500

Hence, he spends ₹ 1,500 on miscellaneous items.

(iv) The central angle of the sector representing clothing is 60° and that representing education is 30°.

Required ratio = 60° : 30°

= $\frac{60^{\circ}}{30^{\circ}}$

= $\frac{2}{1}$

= 2 : 1

Hence, the ratio of expenditure on clothing to that on education is 2 : 1.

In a pie chart, the various observations or components are represented by the sectors of a circle and the whole circle represents the sum of the values of all the components. So, in a pie chart we show the relationship between a whole and its parts.

So, Reason (R) is true.

The total angle at the centre of a circle is 360°.

If two central angles are of 180° each, then

Sum of the two central angles = 180° + 180°

= 360°

Since the sum is exactly 360°, the two sectors together make up the whole circle. This happens when the pie chart has only two components, each of which is half of the total value.

So, Assertion (A) is true.

But Reason (R) only states what a pie chart shows and does not explain why two central angles can be of 180° each.

∴ Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).

Hence, option 2 is the correct option.

The graph of grouped frequency distribution, consisting of rectangles with class intervals as bases and the corresponding frequencies as heights, is known as a histogram.

So, Reason (R) is true.

Players above the age of 20 years are the players in the class intervals 20 – 30, 30 – 40 and 40 – 50, for which the heights of the rectangles are 600, 800 and 500 respectively.

∴ Number of players above the age of 20 years = 600 + 800 + 500

= 1900

Since the number of players above the age of 20 years is 1900 and not 1500, Assertion (A) is false.

∴ Assertion (A) is false but Reason (R) is true.

Hence, option 4 is the correct option.

It is given that spectacles were recommended for 80% of the people who got their eyes checked, which means 80% of them had vision problems.

Since 80% is more than half of the people who got their eyes checked, most of the people who had their eyes checked in the camp had vision problems.

The given pie graph shows only the break-up of the types of spectacles recommended. It gives no information about the age groups of the people, about the people of the village who did not get their eyes checked, or about the number of males and females.

∴ Most of the people who had their eyes checked in the camp had vision problems.

Hence, option 3 is the correct option.

In a pie chart, the whole circle represents the sum of the values of all the components.

Pie chart I shows the time spent on different activities in a day.

Total time shown = (8 + 7 + 1 + 2 + 4 + 3) hours

= 25 hours

But a day has only 24 hours. So, pie chart I does not represent the data accurately.

Pie chart II shows the percentage of income spent on different things.

Total percentage shown = (50 + 10 + 32 + 8)%

= 100%

Since the parts add up to the whole, pie chart II represents the data accurately.

∴ Only pie chart II represents the data accurately.

Hence, option 2 is the correct option.

From the given pie chart,

Percentage of students with S. St = 35%

Percentage of students with Science = 25%

Percentage of students with English = 30%

Since the whole circle represents 100% of the students,

Percentage of students with Math = 100% − (35% + 25% + 30%)

= 100% − 90%

= 10%

By formula,

Central angle for a component =

$\Big(\frac{\text{Percentage of component}}{100} \times 360^{\circ}\Big)$

Substituting the values, we get :

Angle representing Math = $\frac{10}{100} \times 360^{\circ}$

= 36°

Hence, option 3 is the correct option.

Given, the central angle of the sector representing Maza = 72°

The total angle at the centre of the circle is 360°, which represents the total cold drink sale.

Percentage of the total cold drink sale that is Maza = $\frac{72^{\circ}}{360^{\circ}} \times 100$% = 20%

Hence, option 3 is the correct option.

Given, Momos account for 60% of the sales.

By formula,

Central angle for a component =

$\Big(\frac{\text{Percentage of component}}{100} \times 360^{\circ}\Big)$

Substituting the values, we get :

Angle of the sector of Momos = $\frac{60}{100} \times 360^{\circ}$

= 216°

Hence, option 1 is the correct option.

Given, total marks obtained by the student = 540

By formula,

Central angle for a component = $\Big(\frac{\text{Value of the component}}{\text{Total value}} \times 360^{\circ}\Big)$

Substituting the values, we get :

Central angle for 120 marks = $\frac{120}{540} \times 360^{\circ}$

= 80°

From the given pie chart, the sector of 80° represents Hindi.

∴ The subject in which he scored 120 marks is Hindi.

Hence, option 3 is the correct option.