Welcome dear students! Today we are going to learn about Data Handling from Class 8 Maths.
In your day-to-day life, you might come across information such as runs made by a batsman in the last 10 test matches, number of wickets taken by a bowler in the last 10 ODIs, marks scored by students of your class in a Mathematics unit test, or the number of story books read by each of your friends. The information collected in all such cases is called data. Data is usually collected in the context of a situation that we want to study. For example, a teacher may like to know the average height of students in her class. To find this, she will write the heights of all students, organise the data systematically, and then interpret it. Sometimes, data is represented graphically to give a clear idea of what it represents. Do you remember the different types of graphs which we have learnt in earlier classes?
First is a pictograph, which is a pictorial representation of data using symbols. Consider a table showing car production where one car symbol equals 100 cars. In July, there are two full car symbols and one half car symbol. Since half a symbol denotes one half of 100, the total for July is 250. In August, there are three full car symbols, which equals 300. In September, the table shows four full car symbols followed by a question mark. Let us answer the questions posed by the textbook. Question one: How many cars were produced in July? As calculated, 250 cars. Question two: In which month were the maximum number of cars produced? September has four full symbols, which equals 400 cars, making it the month with maximum production.
Second is a bar graph: a display of information using bars of uniform width, their heights being proportional to the respective values. Bar heights give the quantity for each category. Bars are of equal width with equal gaps in between. Looking at a sample bar graph showing student numbers over different academic years, we can answer key questions. The graph gives information about student enrollment across years. To find the year with the maximum increase, compare the height differences between consecutive years. The year with the tallest bar shows the maximum number of students. You can also verify statements like whether the number of students during 2005-06 is twice that of 2003-04 by comparing the bar heights directly.
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Third is a double bar graph: a bar graph showing two sets of data simultaneously. It is useful for the comparison of the data. For instance, a double bar graph comparing student marks in various subjects across two academic years allows us to answer questions. What information is given by the graph? It compares performance in different subjects over two years. In which subject has the performance improved the most? Look for the largest positive difference between the two bars. In which subject has the performance deteriorated? Look for a decrease in bar height. In which subject is the performance at par? Look for equal bar heights.
Think, discuss and write: If we change the position of any of the bars of a bar graph, would it change the information being conveyed? Why? The answer is no, because the height of the bar represents the value, not its horizontal position. Rearranging bars does not alter the data values themselves.
Try These: Draw an appropriate graph to represent the given information. The first dataset shows monthly watch sales: July 1000, August 1500, September 1500, October 2000, November 2500, December 1500. A bar graph is appropriate here. The second dataset compares walking and cycling preferences across three schools. School A has 40 walking and 45 cycling. School B has 55 walking and 25 cycling. School C has 15 walking and 35 cycling. A double bar graph is appropriate to compare the two modes across schools. The third dataset shows percentage wins in ODI by 8 top cricket teams across two periods. South Africa 75% and 78%, Australia 61% and 40%, Sri Lanka 54% and 38%, New Zealand 47% and 50%, England 46% and 50%, Pakistan 45% and 44%, West Indies 44% and 30%, India 43% and 56%. A double bar graph is appropriate to compare the two time periods for each team.
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4.2 Circle Graph or Pie Chart: Have you ever come across data represented in circular form as shown in Figure 4.1? Figure 4.1 shows two circle graphs. The first shows the time spent by a child during a day, divided into sectors for Sleep 8 hours, School 6 hours, Play 3 hours, Others 3 hours, and Home work 4 hours. The second shows age groups of people in a town: 0 to 14 years is 30 thousand, 60 years and above is 20 thousand, and 15 to 60 years is 50 thousand. These are called circle graphs. A circle graph shows the relationship between a whole and its parts. Here, the whole circle is divided into sectors. The size of each sector is proportional to the activity or information it represents.
For example, in the first graph, the proportion of the sector for hours spent in sleeping equals 8 hours divided by 24 hours, which is 1/3. So, this sector is drawn as 1/3rd part of the circle. Similarly, the proportion for school hours is 6/24, which is 1/4. So this sector is drawn as 1/4th of the circle. Similarly, the size of other sectors can be found. Add up the fractions for all the activities. Do you get the total as one? A circle graph is also called a pie chart.
Try These: Each of the following pie charts in Figure 4.2 gives you a different piece of information about your class. Find the fraction of the circle representing each. First, Girls or Boys: Girls 50% equals 1/2, Boys 50% equals 1/2. Second, Transport to school: Walk 40% equals 2/5, Cycle 20% equals 1/5, Bus or car 40% equals 2/5. Third, Love or Hate Mathematics: the chart lists Love and Hate, with Hate given as 15%. The fraction for Hate is 15/100, which simplifies to 3/20. The remaining portion for Love is 1 minus 3/20, which equals 17/20.
Next, answer questions based on the pie chart in Figure 4.3, which shows TV viewership: Entertainment 50%, Sports 25%, News 15%, Informative 10%. Which type of programmes are viewed the most? Entertainment at 50%. Which two types of programmes have viewers equal to those watching sports channels? News at 15% and Informative at 10% add to 25%, matching Sports exactly.
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4.2.1 Drawing pie charts: The favourite flavours of ice-creams for students are Chocolate 50%, Vanilla 25%, Other flavours 25%. Let us represent this in a pie chart. The total angle at the centre is 360°. We calculate the central angle for each. Chocolate: 50% is 1/2. 1/2 of 360° is 180°. Vanilla: 25% is 1/4. 1/4 of 360° is 90°. Other flavours: 25% is 1/4. 1/4 of 360° is 90°.
To draw it: First, draw a circle with a convenient radius, mark centre O and radius OA. Second, use a protractor to draw angle AOB = 180° for chocolate. Third, continue marking the remaining sectors from point OB. Label clearly: Chocolate 180°, Vanilla 90°, Other flavours 90°.
Example 1: The pie chart in Figure 4.4 shows family expenditure percentages: Food 25%, Education 15%, House rent 10%, Transport 5%, Others 20%, Savings 15%, Clothes 10%. Expenditure is maximum on food. Expenditure on Education equals savings, both 15%. If monthly savings is ₹3000, what is expenditure on clothes? 15% equals ₹3000. So 10% represents (3000/15) × 10, which is ₹2000.
Example 2: Baker shop sales: ordinary bread ₹320, fruit bread ₹80, cakes and pastries ₹160, biscuits ₹120, others ₹40. Total ₹720. Calculate central angles. Ordinary bread: 320/720 = 4/9. (4/9) × 360° = 160°. Biscuits: 120/720 = 1/6. (1/6) × 360° = 60°. Cakes and pastries: 160/720 = 2/9. (2/9) × 360° = 80°. Fruit bread: 80/720 = 1/9. (1/9) × 360° = 40°. Others: 40/720 = 1/18. (1/18) × 360° = 20°. Draw Figure 4.5 with these sectors labeled accordingly.
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Try These: Draw a pie chart for daily time: Sleep 8 hours, School 6 hours, Home work 4 hours, Play 4 hours, Others 2 hours. Total 24 hours. Sleep: 8/24 = 1/3, angle 120°. School: 6/24 = 1/4, angle 90°. Home work: 4/24 = 1/6, angle 60°. Play: 4/24 = 1/6, angle 60°. Others: 2/24 = 1/12, angle 30°. Draw and label accordingly.
Think, discuss and write: Which form of graph would be appropriate to display the following data? First, production of food grains of a state over years 2001 to 2006: 60, 50, 70, 55, 80, 85 lakh tons. A bar graph is appropriate to show changes over time. Second, choice of food for a group of people: North Indian 30, South Indian 40, Chinese 25, Others 25. A pie chart is appropriate to show parts of a whole. Third, the daily income of a group of factory workers across 7 income brackets. A histogram or bar graph is appropriate to show frequency distribution across ranges.
EXERCISE 4.1 Question 1: Music survey: Semi Classical 20%, Classical 10%, Folk 30%, Light 40%. If 20 people liked classical, total surveyed is 200. Light music is liked most. For 1000 CDs: Semi Classical 200, Classical 100, Folk 300, Light 400. Question 2: 360 people voted: Summer 90, Rainy 120, Winter 150. Winter got most. Central angles: Summer (90/360) × 360° = 90°. Rainy (120/360) × 360° = 120°. Winter (150/360) × 360° = 150°. Draw pie chart. Question 3: Colours: Blue 18, Green 9, Red 6, Yellow 3. Total 36. Proportions: Blue 1/2, Green 1/4, Red 1/6, Yellow 1/12. Angles: Blue 180°, Green 90°, Red 60°, Yellow 30°. Draw. Question 4: Student marks total 540. Angles: Math 90°, S.Science 65°, Science 80°, Hindi 70°, English 55°. For 105 marks, angle = (105/540) × 360° = 70°. Subject is Hindi. Math minus Hindi: 90° - 70° = 20°. Marks = (20/360) × 540 = 30. Sum of S.Science and Math angles = 65° + 90° = 155°. Sum of Science and Hindi = 80° + 70° = 150°. 155° > 150°, so first sum is higher. Question 5: Languages: Hindi 40, English 12, Marathi 9, Tamil 7, Bengali 4. Total 72. Angles: Hindi (40/72) × 360° = 200°. English 60°. Marathi 45°. Tamil 35°. Bengali 20°. Draw pie chart.
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4.3 Chance and Probability: Sometimes you carry a raincoat daily and it does not rain, but the day you forget, it rains. Sometimes a student prepares 4 out of 5 chapters, but the test focuses on the unprepared one. These are situations where chances are not equal. We focus on experiments with equal chances.
4.3.1 Getting a result: Coin toss outcomes are Head or Tail. You cannot control it. This is a random experiment. Try These: Starting a scooter yields 2 outcomes: starts or does not start. Throwing a die yields 6 outcomes: 1, 2, 3, 4, 5, 6. Spinning the wheel in Figure 4.6 yields landing on A, B, C, or D. Figure 4.6 shows a wheel with 5 sectors: A, A, B, C, D. Drawing a ball from a bag of 5 different coloured balls in Figure 4.7 yields 5 colour outcomes.
Think, discuss and write: In throwing a die, does the first player have a greater chance of getting a six? Does the next player have a lesser chance? If the second player gets a six, does the third lose the chance? The answer is no. Each throw is independent. Every player has an equal 1/6 chance of getting a six, regardless of previous outcomes.
4.3.2 Equally likely outcomes: Coin toss results table shows: 50 tosses give 27 heads, 23 tails. 60 tosses give 28 heads, 32 tails. 70 tosses give 33 heads, 37 tails. 80 tosses give 38 heads, 42 tails. 90 tosses give 44 heads, 46 tails. 100 tosses give 48 heads, 52 tails. As tosses increase, counts converge. Outcomes with same chance are equally likely.
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4.3.3 Linking chances to probability: Coin toss: P(Head) = 1/2. P(Tail) = 1/2. Die throw: 6 outcomes. P(2) = 1/6. P(5) = 1/6. P(7) = 0. P(1 to 6) = 1. 4.3.4 Outcomes as events: Getting Head is an event. Getting Tail is an event. Getting even number on die is an event (2, 4, 6). P(even) = 3/6 = 1/2. Example 3: Bag has 4 red, 2 yellow balls. Total 6. P(red) = 4/6 = 2/3. P(yellow) = 2/6 = 1/3. Red is more likely. Try These: Spin wheel in Figure 4.8. Figure 4.8 shows 8 sectors: R, G, R, G, G, G, R, G. Green sectors = 5. Red sectors = 3. P(green) = 5/8. P(not green) = 3/8. 4.3.5 Chance and probability in real life: Chance of rain on forgotten coat day might be 1/10. P(no rain) = 9/10. Used in exit polls and weather forecasting.
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EXERCISE 4.2 Question 1: Outcomes: Spinning wheel with A, A, B, C, D gives A, B, C, D. Tossing two coins gives HH, HT, TH, TT. Question 2: Die throw: Prime: 2, 3, 5. Not prime: 1, 4, 6. Greater than 5: 6. Not greater than 5: 1, 2, 3, 4, 5. Question 3: Probabilities: Pointer on D: 1/5. Ace from 52 cards: 4/52 = 1/13. Red apple from 7 apples (4 red, 3 green) as shown in the figure: 4/7. Question 4: Slips 1 to 10. P(6) = 1/10. P(<6) = 5/10 = 1/2. P(>6) = 4/10 = 2/5. P(1-digit) = 9/10. Question 5: Wheel: 3 green, 1 blue, 1 red. Total 5. P(green) = 3/5. P(non-blue) = 4/5. Question 6: Probabilities for Q2: Prime 1/2. Not prime 1/2. >5 is 1/6. ≤5 is 5/6.
WHAT HAVE WE DISCUSSED? 1. To draw meaningful inferences from data, organise it systematically. 2. Data can be presented using circle graph or pie chart, showing relationship between whole and part. 3. Certain experiments have outcomes with equal chances. 4. A random experiment cannot be predicted exactly in advance. 5. Outcomes are equally likely if each has same chance. 6. Probability of an event = (Number of outcomes that make event) / (Total number of outcomes), when equally likely. 7. One or more outcomes make an event. 8. Chances and probability are related to real life.
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]