DBSE | Class 9Th | Maths | Statistics | Graphical representation of data | Exercise 03 (Part 05)

DBSE | Class 9Th | Maths | Statistics | Graphical representation of data | Exercise 03 (Part 05)

Graphical representation of data

Task 9 : Readiness task:

1. Calculate the monthly income of your family. Write the expenditure per month from various

heads like rent, grocery, fuel, electricity, dairy product, medicine, entertainment, etc. Every

pair will share their data with each other. They have to present their data in such a way that

every person can easily understand and compare their family expenditure.

2. To find the reason for student dropout in the schools, a survey was conducted by an

organization. The result of the survey is given below:

Causes                                           % of students

Classes were not interesting               20

Drug addiction                                       15

Family carelessness                              25

Child labour                                             8

Engaged in antisocial work                   12

Other reasons                                          20

(i) Represent the data graphically.

(ii) Which is the major reason for student dropout?

(iii) Suggest ways to deal with causes of dropout.

3. Suppose you received a cash award of Rs 10000 (Ten thousand) in the singing competition.

Make a budget plan for spending this amount.

Show your budget plan graphically.

4. The given below are the data showing the career plan of two sections of grade 9.

(i) Draw a double bar graph to represent the above data.

(ii) What you infer from the above data.

5. In figure, there is a histogram showing the daily wages of workers in a factory.

Construct the frequency distribution table.

6. In a year, the number of deaths due to habit of smoking for different age group is given below :

(i) Represent the given information with the help of a histogram.

(ii) What lesson do you learn from this

information?

Task 10 : Formation of Histogram when class size of the interval is unequal.

The weekly pocket expenses of 125 students of a school given below:

Observe the given table.

Think, pair and share:

Do peer observation of the histogram prepared by your pair.

Discuss about how your pair handled the various class sizes in the histogram.

Now, observe the given histograms prepared by two students Raja and Rashi.

Raja’s histogram

Rashi’s histogram

Share your observations and reflections with your pair and then large group.

● Which histogram correctly represents the data? Why or why not?

● In which histogram, the areas of the rectangles are proportional to the frequencies in a histogram?

NOTE

The Raja’s histogram is giving us a misleading picture. As we have mentioned earlier, the areas of the rectangles are proportional to the frequencies in a histogram. Earlier this problem did not arise, because the widths of all the rectangles were equal. But here, since the widths of the rectangles are varying, the histogram above does not give a correct picture. For example, it shows a greater frequency in the interval 80 - 100, that is 80, which is not the case. So, we need to make certain modifications in the lengths of the rectangles so that the areas are again proportional to the frequencies.

The steps to be followed are as given below:

1. Select a class interval with the minimum class size. In the example above, the minimum

class-size is 10.

2. The class size is then modified to be proportionate to the class-size 10. For instance, when the class-size is 30, the frequency is 30, the adjusted class size will be (30/30)×10 = 10.

Here, the class sizes are different, so calculate the adjusted frequency for each class by using the formula.

Adjusted frequency for a class with respect to class size 10 = (𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 / 𝐶𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒) ×10 

Similarly, proceeding in this manner, we get the following table:

Let us represent weekly pocket money along x-axis and corresponding adjusted frequencies

along y-axis on a suitable scale, the required histogram is as given below :

Check your understanding

1. A random survey of the number of children of various age groups playing in a park was found as follows:

i) Draw a histogram to represent the data above.

ii) What do you infer from the above data?

iii) Do the children in your neighbourhood have

enough space to play? If not then how do they

manage?

2. The given below data showing the daily earning of a

Chemist shop in Yamuna Vihar area:

★ Draw a histogram to represent the data above.

★ What do you infer from the above data?

Task 11 : Frequency polygon

Observe the given graphical representation:

Students will share their observations in groups. Every group represents their understanding in the large group. Students will reflect their understanding and ask questions related to their doubts or inquiry.

NOTE

INFORMATION :

A frequency polygon is a graph obtained by joining the midpoints of the tops of the rectangles in the histogram. The height of the point represents frequencies.

Frequency polygons can also be drawn independently without drawing histograms. For this, we require the mid-points of the class-intervals used in the data. These mid-points of the class-intervals are called class-marks. To find the class-mark of a class interval, we find the sum of the upper limit and lower limit of a class and divide it by 2.

Constructing Frequency Polygons

1. The following histogram represents the marks made by 60 students in a math test.

Use the histogram to construct a frequency polygon to represent the data.

2. The following distribution table represents the number of miles run by 20 randomly selected runners during a race:

Using this table, construct a frequency polygon.

3. The frequency polygon below represents the heights, in inches, of a group of professional

basketball players. Use the frequency polygon to answer the following questions:

Construct a frequency distribution table for the

data

Answer the following questions:-

(i) How many player’s heights were

measured?

Sol. To find the number of players whose

heights were measured, just add up all of

the frequencies. This can be done as

follows: 6+9+24+27+18+21+3=108

This means that 108 player heights were

measured.

(ii) What was the scale of the histogram on which the frequency polygon is based?

Sol. The class scale of the histogram on which the frequency polygon is based is 3.

(iii) What range of heights was most common among the basketball players?

Sol. Remember that the x-coordinate of each of the points that were connected to create the

frequency polygon is the midpoint of one of the class interval of the corresponding

histogram. It's obvious from the frequency polygon that 78 inches has the greatest

frequency, but this doesn't necessarily mean that height most common among the

basketball players was 78 inches. All it means is that the range of heights that was most

common among the basketball players was 76.5 inches to 79.5 inches.

(iv) What range of heights was least common among the basketball players?

Sol. The range of heights that was least common among the basketball players was 85.5 inches to 88.5 inches. Remember that the points at 66 inches and 90 inches along the

horizontal axis were just added to give the frequency polygon the appearance of having a

starting point and an ending point.

(v) What percentage of the basketball players measured had a height of less than 76.5?

Sol. The point at 75 inches along the horizontal axis represents the class interval [73.5, 76.5).

Therefore, to find the number of basketball players measured who had a height of less

than 76.5 inches, add the frequencies of the first 3 class interval as follows:

6+9+24=39

This means that 39 players had a height of less than 76.5 inches, so the percentage of the

basketball players measured who had a height of less than 76.5 inches is 39/100=0.39=39%.

4. The following two tables gives the distribution of students of two sections according to the marks obtained by them:

★ Represent the marks of the students of both the sections on the same graph by using

histogram and by using frequency polygons.

★ Which method is easier to compare the performance of the two sections?

NOTE :

Advantages of frequency polygon over histogram:

The main advantage of a frequency polygon over a histogram is that you can draw two or more frequency polygons of several distributions on the same axis, which makes it possible to compare. But a histogram cannot be utilised in the same way, we construct histograms on  eparated graphs to compare them. Hence, frequency polygons are preferred for providing a graphic comparison of frequency distributions due to this limitation.

Check your understanding:

1. What is the midpoint of the class interval 14.5-19.5?

a. 19          b. 17          c. 18.5          d. 34

2. The following frequency polygon represents the weights of players who all participated

in the same sport. Use the polygon to answer the following questions:

A. How many players played the sport?

B. What was the most common weight

for the players?

C. What sport do you think the players

may have been playing?

D.What do the weights of 55 kg and 105

kg represent?

E. What 2 weights have no recorded

players weighing those amounts?

3. In a school marks obtained by 80 students

are given in the table.

Draw a histogram. Also, make frequency polygon.

4. The runs scored by two teams A and B on the first 60 balls in a cricket match are given below:

★ Represent the data of both the teams on the same graph by using frequency polygons.

★ What do you infer from the above data?