CBSE Class 8 Mathematics Data Handling Worksheet Set 04

Q.1 Fill in the blanks
(a) The collection of numerical facts in a particular context is called data.
(b) The difference between the highest and the lowest values of observations in a data is called range.
(c) The number of times an observation occurs is called frequency.
(d) Tally marks for 13 are given by \(\text{NN NN III}\).
(e) A character which is capable of taking several different values is called a variable.
(f) The possible results of a random experiment are called outcomes.
(g) One or more outcomes of an experiment make an event.
(h) Probability of an impossible event is 0.
(i) Probability of a sure event is 1.
Answer:
(a) data
(b) range
(c) frequency
(d) \(\text{NN NN III}\)
(e) variable
(f) outcomes
(g) event
(h) 0
(i) 1
In simple words: These are basic definitions and fundamental concepts of statistics and probability.
Exam Tip: Learn these definitions thoroughly as they frequently appear as one-mark objective questions in exams.

Q.2 The taste for different flavours of ice creams among the children of a society is depicted in the bar graph in Fig. 16.6. Study it and answer the following questions :
(i) What is the scale along vertical axis?
(ii) Which flavor is the most popuar among the children?
(iii) How many children like to have Vanilla ice cream?
(iv) Write in decreasing order, the number of children preferring different flavours of ice cream.
(v) What percentage of children like to have Mango ice cream?
Answer:
(i) Along the vertical axis, 1 unit length = 20 children.
(ii) Chocolate (height of bar is 130).
(iii) 70 children.
(iv) Chocolate (130), Strawberry (90), Kesar Pista (80), Vanilla (70), Butter Scotch (60), Mango (40).
(v) Total number of children = 70 + 90 + 130 + 60 + 40 + 80 = 470
Percentage of children who like Mango = \(\frac{40}{470} \times 100 = \frac{400}{47} = 8.51\%\)
In simple words: Read values directly from the bar heights on the vertical axis and calculate percentages using total values.

Exam Tip: Always double-check the scale of the axis before reading the heights of the bars.

Q.3 The bar graph given in Fig. 16.8 shows the number of tourists who visited the different hill stations during the vacations in a particular year. Study the bar graph and answer the following questions :
(i) Write the scale along the horizontal axis.
(ii) Which hill station was visited by maximum number of tourists?
(iii) How many tourists visited Darjeeling?
(iv) How many more tourists visited Nainital as compared to Mussoorie?
(v) From the most visited hill station to the least visited one, write the hill stations in order.
Answer:
(i) Along the horizontal axis, 1 unit length = 1000 tourists.
(ii) Darjeeling (bar length extends up to 6000).
(iii) 6000 tourists.
(iv) Tourists who visited Nainital = 3000, Tourists who visited Mussoorie = 4000 (Wait, reading graph: Nainital = 3000, Mussoorie = 4000, Simla = 5000, Manali = 5500, Darjeeling = 6000. Let us re-read Nainital vs Mussoorie: Nainital has 3000, Mussoorie has 4000. Actually, Nainital is at 3000 and Mussoorie is 4000. Difference = 4000 - 3000 = 1000 tourists. If measured other way, let's state standard absolute difference). Absolute difference = 1000 tourists.
(v) Darjeeling (6000), Manali (5500), Simla (5000), Mussoorie (4000), Nainital (3000).
In simple words: Horizontal bar graphs represent data values along the horizontal axis using lengths of bars.

Exam Tip: Pay attention to whether the bar graph is horizontal or vertical when reading axes scales.

Q.4 The marks obtained by 40 students of a class in an examination are given below :
8, 47, 22, 31, 17, 13, 38, 26, 3, 34, 29, 11, 22, 7, 15, 24, 38, 31, 21, 35, 42, 24, 45, 23, 21, 27, 29, 49, 25, 48, 15, 18, 27, 19, 45, 14, 34, 37, 34, 21.
Prepare a frequency distribution table, starting with a class interval 0 - 10 (where 10 is not included), etc.
Answer:
The range of data is from 3 to 49. With class interval of size 10, the table is as follows:

In simple words: Group continuous data into equal intervals and count occurrences using tally marks.

Exam Tip: Always verify that the sum of frequencies equals the total number of given observations (40).

Q.5 Yearly savings (in rupees) of 30 students of class VIII are as under :
38 42 40 35 72 27 57 40
59 80 84 73 65 40 76 44
38 60 58 38 54 39 50
71 83 45 38 80 77 62
Construct a frequency table with 30-35 as one of the class intervals (35 not included).
Answer:
Minimum value = 27, Maximum value = 84. Class interval size = 5. Grouping into intervals starting from 25-30:

In simple words: Organize raw financial data into uniform class intervals using tally marks.

Exam Tip: Ensure that boundary values (e.g., 35) are counted in the upper interval where they are included (35-40), not the lower one.

Q.6 Given below is the frequency distribution of the heights of 50 players of a school :

Draw a histogram representing the above data.
Answer:
To draw the histogram:
1. Take class intervals (heights in cm) along the horizontal X-axis.
2. Take frequencies (number of players) along the vertical Y-axis.
3. Choose a suitable scale on the Y-axis (e.g., 1 unit = 2 players).
4. Construct adjacent rectangular bars with bases equal to the class intervals (140-145, 145-150, etc.) and heights proportional to their frequencies (6, 12, 18, 9, 5 respectively).

In simple words: A histogram consists of contiguous bars where area represents class frequencies.
Exam Tip: Since class intervals are continuous, there are no gaps between the bars in a histogram.

Q.7 The histogram (Fig. 16.13) depicts the marks of 40 students of class VIII obtained in a test of mathematics :
(Look at the histogram and answer the following questions :
(i) What is the class size?
(ii) How many students obtained less than 5 marks?
(iii) How many students obtained 20 or more marks but less than 25?
(iv) If 25 or more marks make first division, how many first divisions are there?
(v) How many students obtained 10 or more but less than 20 marks?
Answer:
(i) Class size = Upper limit - Lower limit = \(5 - 0 = 5\).
(ii) 2 students.
(iii) 8 students (corresponding to interval 20-25).
(iv) Students with marks \(\ge 25\) are in intervals 25-30 (10 students) and 30-35 (12 students). Total = \(10 + 12 = 22\).
(v) Students in intervals 10-15 (4 students) and 15-20 (6 students). Total = \(4 + 6 = 10\).
In simple words: Read required student frequencies directly from the bar heights for the specified mark intervals.

Exam Tip: Be careful to sum the frequencies of all relevant contiguous intervals when a range spans multiple bars.

Q.8 The histogram (Fig. 16.14) show the distribution of ages (in years) of 38 teachers in a school. Look at the histogram and answer the questions that follow :
(i) In which age group maximum number of teachers are there and what is their number?
(ii) What is the number of teachers having their ages 25 years or more but less than 30?
(iii) How many teachers are of the age 40 or more?
(iv) What is class size?
(v) What are the class marks of the classes?
Answer:
(i) Age group 35-40 with a maximum of 11 teachers.
(ii) 4 teachers.
(iii) Teachers aged 40 or more include intervals 40-45 (7 teachers) and 45-50 (6 teachers). Total = \(7 + 6 = 13\).
(iv) Class size = \(30 - 25 = 5\).
(v) Class marks = \(\frac{\text{Lower Limit} + \text{Upper Limit}}{2}\):
- For 25-30: 27.5
- For 30-35: 32.5
- For 35-40: 37.5
- For 40-45: 42.5
- For 45-50: 47.5
In simple words: Class mark is the midpoint of each class interval, calculated by averaging its lower and upper limits.

Exam Tip: Remember the formula for class mark: \(\frac{\text{Lower limit} + \text{Upper limit}}{2}\).

Q.9 Monthly income of a family is Rs. 12,0000. The adjoining pie chart (Fig. 16.18) shows how the family used the money. Study the pie chart and answer the following questions.
(i) How much was spent on food?
(ii) How much was spent on education?
(iii) How much does the family save?
(iv) How much was spent on rent and bills?
(v) Find the ratio of the amount spent on other items to that spent on food.
Answer:
Total monthly income = Rs. 1,20,000 (interpreting Rs. 12,0000 as 1,20,000).
Central angle for Food = \(90^\circ\), Education = \(60^\circ\), Savings = \(30^\circ\), Rent and Bills = \(120^\circ\), Other items = \(360^\circ - (90^\circ + 60^\circ + 30^\circ + 120^\circ) = 360^\circ - 300^\circ = 60^\circ\).
(i) Expenditure on Food = \(\frac{90^\circ}{360^\circ} \times 1,20,000 = \frac{1}{4} \times 1,20,000 = \text{Rs. } 30,000\).
(ii) Expenditure on Education = \(\frac{60^\circ}{360^\circ} \times 1,20,000 = \frac{1}{6} \times 1,20,000 = \text{Rs. } 20,000\).
(iii) Savings = \(\frac{30^\circ}{360^\circ} \times 1,20,000 = \frac{1}{12} \times 1,20,000 = \text{Rs. } 10,000\).
(iv) Expenditure on Rent and Bills = \(\frac{120^\circ}{360^\circ} \times 1,20,000 = \frac{1}{3} \times 1,20,000 = \text{Rs. } 40,000\).
(v) Ratio of amount spent on other items to food = \(\frac{60^\circ}{90^\circ} = \frac{2}{3}\) or \(2:3\).
In simple words: The value of each component in a pie chart is proportional to its central angle out of \(360^\circ\).

Exam Tip: Use the fraction \(\frac{\text{Central Angle}}{360^\circ} \times \text{Total Income}\) to find individual component values easily.

Q.10 180 people donated money in the following organization.

(a) Represent this information using a pie chart.
(b) Do you think it is a good gesture to donate money? If yes, write one incidence when you donate money or something for the poor and needy.
Answer:
(a) Calculation of central angles for each organization:
- NGO: \(\frac{72}{180} \times 360^\circ = 144^\circ\)
- Blind School: \(\frac{40}{180} \times 360^\circ = 80^\circ\)
- Old Age Home: \(\frac{37}{180} \times 360^\circ = 74^\circ\)
- Orphanage Home: \(\frac{31}{180} \times 360^\circ = 62^\circ\)
(Sum of angles = \(144^\circ + 80^\circ + 74^\circ + 62^\circ = 360^\circ\)).
(b) Yes, donating money is a noble gesture that helps uplift society and supports the underprivileged. For example, I donated my old storybooks and warm winter clothes to an orphanage during the festival season.
In simple words: Central angle is found by multiplying the fraction of people by \(360^\circ\).

Exam Tip: Always verify that the sum of all calculated central angles equals exactly \(360^\circ\).

Q.11 Akash rolls a die. What are the chances of getting a number :
(i) less than 7
(ii) less than 1
(iii) which is even?
Answer:
Total possible outcomes when rolling a die = \(\{1, 2, 3, 4, 5, 6\}\) (Total = 6).
(i) Numbers less than 7 are \(\{1, 2, 3, 4, 5, 6\}\) (6 outcomes). Probability = \(\frac{6}{6} = 1\) (Sure event).
(ii) Numbers less than 1: none (0 outcomes). Probability = \(\frac{0}{6} = 0\) (Impossible event).
(iii) Even numbers are \(\{2, 4, 6\}\) (3 outcomes). Probability = \(\frac{3}{6} = \frac{1}{2}\).
In simple words: Probability is the ratio of favorable outcomes to the total number of possible outcomes.

Exam Tip: The probability of any event always lies between 0 and 1 inclusive.

Q.12 Toss two coins together. Write all the possible outcomes and then find the probability of getting tail on both the coins.
Answer:
When two coins are tossed together, the sample space (possible outcomes) is:
\(S = \{(H, H), (H, T), (T, H), (T, T)\}\) (Total = 4 outcomes).
Favorable outcome of getting tail on both coins = \(\{(T, T)\}\) (1 outcome).
Probability of getting tail on both coins = \(\frac{1}{4}\).
In simple words: List all combinations of heads and tails for two coins, then count how many match the target condition.

Exam Tip: For \(n\) coins tossed, total possible outcomes are \(2^n\). For 2 coins, \(2^2 = 4\).

Q.13 Using the probability scale, write the probability of each of the following :
(i) Next year, the Christmas Day will be on \(25^{\text{th}}\) December.
(ii) There will be 29 days in February in the year 2006.
(iii) Students of class VIII will go for a picnic.
(iv) India will win the one day cricket match played between India and Australia.
(v) You get an odd number on throwing a dice.
(vi) Independence Day will be celebrated on \(15^{\text{th}}\) August.
(vii) You will score above \(80\%\) marks in the annual exams, if you have worked hard and the level of paper is easy.
(viii) Today Earth will not rotate on its axis.
Answer:
(i) 1 (Sure event)
(ii) 0 (Impossible event, 2006 was not a leap year)
(iii) Can happen but not certain (Probability lies between 0 and 1, or indeterminate without context, typically denoted as likely/chance)
(iv) 0.5 or chance (Equally likely depending on performance)
(v) \(\frac{1}{2}\) or 0.5
(vi) 1 (Sure event)
(vii) Highly likely (close to 1)
(viii) 0 (Impossible event)
In simple words: Sure events have probability 1, impossible events have probability 0, and uncertain events have probabilities between 0 and 1.
Exam Tip: Understand the difference between definite facts (probability 1 or 0) and uncertain events.

Q.14 The following data relates to the expenditure of the families A and B per month.

Represent the expenditure of the families A and B by Pie Chart.
Answer:
First, find the total expenditure and calculate central angles for each family.
Family A (Total = 9060):
- Food: \(\frac{4000}{9060} \times 360^\circ \approx 159^\circ\)
- Rent: \(\frac{500}{9060} \times 360^\circ \approx 20^\circ\)
- Clothing: \(\frac{800}{9060} \times 360^\circ \approx 32^\circ\)
- Education: \(\frac{2000}{9060} \times 360^\circ \approx 79^\circ\)
- Saving: \(\frac{1260}{9060} \times 360^\circ \approx 50^\circ\)
- Miscellaneous: \(\frac{500}{9060} \times 360^\circ \approx 20^\circ\)

Family B (Total = 10080):
- Food: \(\frac{4536}{10080} \times 360^\circ = 162^\circ\)
- Rent: \(\frac{532}{10080} \times 360^\circ = 19^\circ\)
- Clothing: \(\frac{868}{10080} \times 360^\circ = 31^\circ\)
- Education: \(\frac{2380}{10080} \times 360^\circ = 85^\circ\)
- Saving: \(\frac{504}{10080} \times 360^\circ = 18^\circ\)
- Miscellaneous: \(\frac{1260}{10080} \times 360^\circ = 45^\circ\)
Using these calculated angles, two separate pie charts can be drawn with a compass and protractor.
In simple words: Calculate the central angle for each expenditure item by finding its fraction of the total expenditure and multiplying by \(360^\circ\).

Exam Tip: Always double check that the sum of all central angles for a pie chart equals exactly \(360^\circ\).

Students must free download and practice these worksheets to gain more marks in exams. CBSE Class 8 Mathematics Worksheet - Data Handling