2. A bar graph: A display of information using bars of uniform width, their heights being proportional to the respective values. (i) What is the information given by the bar graph? (ii) In which year is the increase in the number of students maximum? (iii) In which year is the number of students maximum? (iv) State whether true or false: ‘The number of students during 2005-06 is twice that of 2003-04.’ 3. Double Bar Graph: A bar graph showing two sets of data simultaneously. It is useful for the comparison of the data. (i) What is the information given by the double bar graph? (ii) In which subject has the performance improved the most? (iii) In which subject has the performance deteriorated? (iv) In which subject is the performance at par? It is not easy to answer the question looking at the choices written haphazardly. We arrange the data in Table 5.1 using tally marks. Table 5.1 Subject Tally Marks Number of Students Art | | | | | | 7 Mathematics | | | | 5 Science | | | | | 6 English | | | | 4 The number of tallies before each subject gives the number of students who like that particular subject. This is known as the frequency of that subject. Frequency gives the number of times that a particular entry occurs. From Table 5.1, Frequency of students who like English is 4 Frequency of students who like Mathematics is 5 The table made is known as frequency distribution table as it gives the number of times an entry occurs. 5.3 Grouping Data The data regarding choice of subjects showed the occurrence of each of the entries several times. For example, Art is liked by 7 students, Mathematics is liked by 5 students and so on (Table 5.1). This information can be displayed graphically using a pictograph or a bargraph. Sometimes, however, we have to deal with a large data. For example, consider the following marks (out of 50) obtained in Mathematics by 60 students of Class VIII: 21, 10, 30, 22, 33, 5, 37, 12, 25, 42, 15, 39, 26, 32, 18, 27, 28, 19, 29, 35, 31, 24, 36, 18, 20, 38, 22, 44, 16, 24, 10, 27, 39, 28, 49, 29, 32, 23, 31, 21, 34, 22, 23, 36, 24, 36, 33, 47, 48, 50, 39, 20, 7, 16, 36, 45, 47, 30, 22, 17. If we make a frequency distribution table for each observation, then the table would be too long, so, for convenience, we make groups of observations say, 0-10, 10-20 and so on, and obtain a frequency distribution of the number of observations falling in each group. Thus, the frequency distribution table for the above data can be. Table 5.2 Groups Tally Marks Frequency 0-10 | | 2 10-20 | | | | | | | | 10 20-30 | | | | | | | | | | | | | | | | | 21 30-40 | | | | | | | | | | | | | | | | 19 40-50 | | | | | | 7 50-60 | 1 Total 60 Data presented in this manner is said to be grouped and the distribution obtained is called grouped frequency distribution. It helps us to draw meaningful inferences like – (1) Most of the students have scored between 20 and 40. (2) Eight students have scored more than 40 marks out of 50 and so on. Each of the groups 0-10, 10-20, 20-30, etc., is called a Class Interval (or briefly a class). Observe that 10 occurs in both the classes, i.e., 0-10 as well as 10-20. Similarly, 20 occurs in classes 10-20 and 20-30. But it is not possible that an observation (say 10 or 20) can belong simultaneously to two classes. To avoid this, we adopt the convention that the common observation will belong to the higher class, i.e., 10 belongs to the class interval 10-20 (and not to 0-10). Similarly, 20 belongs to 20-30 (and not to 10-20). In the class interval, 10-20, 10 is called the lower class limit and 20 is called the upper class limit. Similarly, in the class interval 20-30, 20 is the lower class limit and 30 is the upper class limit. Observe that the difference between the upper class limit and lower class limit for each of the class intervals 0-10, 10-20, 20-30 etc., is equal, (10 in this case). This difference between the upper class limit and lower class limit is called the width or size of the class interval. 150-175 55 175-200 125 200-225 140 225-250 55 250-275 35 275-300 50 300-325 20 Total 550 (i) What is the size of the class intervals? (ii) Which class has the highest frequency? (iii) Which class has the lowest frequency? (iv) What is the upper limit of the class interval 250-275? (v) Which two classes have the same frequency? 2. Construct a frequency distribution table for the data on weights (in kg) of 20 students of a class using intervals 30-35, 35-40 and so on. 40, 38, 33, 48, 60, 53, 31, 46, 34, 36, 49, 41, 55, 49, 65, 42, 44, 47, 38, 39. 5.3.1 Bars with a difference Let us again consider the grouped frequency distribution of the marks obtained by 60 students in Mathematics test. (Table 5.4) Table 5.4 Class Interval Frequency 0-10 2 10-20 10 20-30 21 30-40 19 40-50 7 50-60 1 Total 60 This is displayed graphically as in the adjoining graph (Fig 5.1). Is this graph in any way different from the bar graphs which you have drawn in Class VII? Observe that, here we have represented the groups of observations (i.e., class intervals) Fig 5.1 on the horizontal axis. The height of the bars show the frequency of the class-interval. Also, there is no gap between the bars as there is no gap between the class-intervals. The graphical representation of data in this manner is called a histogram. The following graph is another histogram (Fig 5.2). Fig 5.2 From the bars of this histogram, we can answer the following questions: (i) How many teachers are of age 45 years or more but less than 50 years? (ii) How many teachers are of age less than 35 years? 1. Observe the histogram (Fig 5.3) and answer the questions given below. Fig 5.3 (i) What information is being given by the histogram? (ii) Which group contains maximum girls? (iii) How many girls have a height of 145 cms and more? (iv) If we divide the girls into the following three categories, how many would there be in each? 150 cm and more — Group A 140 cm to less than 150 cm — Group B Less than 140 cm — Group C EXERCISE 5.1 For which of these would you use a histogram to show the data? (a) The number of letters for different areas in a postman’s bag. (b) The height of competitors in an athletics meet. (c) The number of cassettes produced by 5 companies. (d) The number of passengers boarding trains from 7:00 a.m. to 7:00 p.m. at a station. Give reasons for each. 2. The shoppers who come to a departmental store are marked as: man (M), woman (W), boy (B) or girl (G). The following list gives the shoppers who came during the first hour in the morning: W W W G B W W M G G M M W W W W G B M W B G G M W W M M W W W M W B W G M W W W W G W M M W W M W G W M G W M M B G G W Make a frequency distribution table using tally marks. Draw a bar graph to illustrate it. 3. The weekly wages (in `) of 30 workers in a factory are. 830, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855, 845, 804, 808, 812, 840, 885, 835, 835, 836, 878, 840, 868, 890, 806, 840Using tally marks make a frequency table with intervals as 800–810, 810–820 and so on. 4. Draw a histogram for the frequency table made for the data in Question 3, and answer the following questions. (i) Which group has the maximum number of workers? (ii) How many workers earn ` 850 and more? (iii) How many workers earn less than ` 850? 5. The number of hours for which students of a particular class watched television during holidays is shown through the given graph. Answer the following. (i) For how many hours did the maximum number of students watch TV? (ii) How many students watched TV for less than 4 hours? (iii) How many students spent more than 5 hours in watching TV? 5.4 Circle Graph or Pie Chart Have you ever come across data represented in circular form as shown (Fig 5.4)? The time spent by a child during a day Age groups of people in a town (i) Fig 5.4 (ii) 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 above graph, the proportion of the sector for hours spent in sleeping number of sleeping hours 8 hours 1 == = whole day24 hours 3 1So, this sector is drawn as rd part of the circle. Similarly, the proportion of the sector 3number of school hours 6 hours 1 for hours spent in school = = = whole day24 hours 4 a fraction of 360°. We make a table to find the central angle of the sectors (Table 5.5). Table 5.5 Flavours Students in per cent preferring the flavours In fractions Fraction of 360° Chocolate 50% 50 1 100 2 = 1 2 of 360° = 180° Vanilla 25% 25 1 100 4 = 1 4 of 360° = 90° Other flavours 25% 25 1 100 4 = 1 4 of 360° = 90° 1. Draw a circle with any convenient radius. Mark its centre (O) and a radius (OA). 2. The angle of the sector for chocolate is 180°. Use the protractor to draw ∠AOB = 180°. 3. Continue marking the remaining sectors. Example 1: Adjoining pie chart (Fig 5.7) gives the expenditure (in percentage) on various items and savings of a family during a month. (i) On which item, the expenditure was maximum? (ii) Expenditure on which item is equal to the total savings of the family? (iii) If the monthly savings of the family is ` 3000, what is the monthly expenditure on clothes? Solution: (i) Expenditure is maximum on food. (ii) Expenditure on Education of children is the same (i.e., 15%) as the savings of the family. MATHEMATICS (iii) 15% represents ` 3000 3000Therefore, 10% represents ` ×10 = ` 200015 Example 2: On a particular day, the sales (in rupees) of different items of a baker’s shop are given below. ordinary bread : 320 fruit bread : 80 cakes and pastries : 160 Draw a pie chart for this data. biscuits : 120 others : 40 Total : 720 Solution: We find the central angle of each sector. Here the total sale = ` 720. We thus have this table. Item Sales (in `) In Fraction Central Angle Ordinary Bread 320 320 4 720 9 = 4 360 1609 × ° = ° Biscuits 120 120 1 720 6 = 1 360 606 × ° = ° Cakes and pastries 160 160 2 720 9 = 2 360 809 × ° = ° Fruit Bread 80 80 1 720 9 = 1 360 409 × ° = ° Others 40 40 1 720 18 = 1 360 2018 × ° = ° Fig 5.8 Draw a pie chart of the data given below. The time spent by a child during a day. Sleep — 8 hours School — 6 hours Home work — 4 hours Play — 4 hours Others — 2 hours THINK, DISCUSS AND WRITE Which form of graph would be appropriate to display the following data. 1. Production of food grains of a state. 2. Choice of food for a group of people. 3. The daily income of a group of a factory workers. Year 2001 2002 2003 2004 2005 2006 Production (in lakh tons) 60 50 70 55 80 85 Favourite food Number of people North Indian 30 South Indian 40 Chinese 25 Others 25 Total 120 Daily Income Number of workers (in Rupees) (in a factory) 75-100 45 100-125 35 125-150 55 150-175 30 175-200 50 200-225 125 225-250 140 Total 480 3. When you spin the wheel shown, what are the possible outcomes? (Fig 5.9) List them. (Outcome here means the sector at which the pointer stops). Fig 5.9 Fig 5.10 4. You have a bag with five identical balls of different colours and you are to pull out (draw) a ball without looking at it; list the outcomes you would get (Fig 5.10). THINK, DISCUSS AND WRITE In throwing a die: • Does the first player have a greater chance of getting a six? • Would the player who played after him have a lesser chance of getting a six? • Suppose the second player got a six. Does it mean that the third player would not have a chance of getting a six? 5.5.2 Equally likely outcomes: A coin is tossed several times and the number of times we get head or tail is noted. Let us look at the result sheet where we keep on increasing the tosses: Number of tosses Tally marks (H) Number of heads Tally mark (T) Number of tails 50 | | | | | | | | | | | | | | | | | | | | | | 27 | | | | | | | | | | | | | | | | | | | 23 60 | | | | | | | | | | | | | | | | | | | | | | | 28 | | | | | | | | | | | | | | | | | | | | | | | | | | 32 70 ... 33 ... 37 80 ... 38 ... 42 90 ... 44 ... 46 100 ... 48 ... 52 Observe that as you increase the number of tosses more and more, the number of heads and the number of tails come closer and closer to each other. This could also be done with a die, when tossed a large number of times. Number of each of the six outcomes become almost equal to each other. In such cases, we may say that the different outcomes of the experiment are equally likely. This means that each of the outcomes has the same chance of occurring. 5.5.3 Linking chances to probability Consider the experiment of tossing a coin once. What are the outcomes? There are only two outcomes – Head or Tail. Both the outcomes are equally likely. Likelihood of getting 1a head is one out of two outcomes, i.e., . In other words, we say that the probability of21 getting a head = . What is the probability of getting a tail?2 Now take the example of throwing a die marked with 1, 2, 3, 4, 5, 6 on its faces (one number on one face). If you throw it once, what are the outcomes? The outcomes are: 1, 2, 3, 4, 5, 6. Thus, there are six equally likely outcomes. What is the probability of getting the outcome ‘2’? 1 ← Number of outcomes giving 2It is 6 ← Number of equally likely outcomes. What is the probability of getting the number 5? What is the probability of getting the number 7? What is the probability of getting a number 1 through 6? 5.5.4 Outcomes as events Each outcome of an experiment or a collection of outcomes make an event. For example in the experiment of tossing a coin, getting a Head is an event and getting a Tail is also an event. In case of throwing a die, getting each of the outcomes 1, 2, 3, 4, 5 or 6 is an event. Is getting an even number an event? Since an even number could be 2, 4 or 6, getting an even number is also an event. What will be the probability of getting an even number? 3 ← Number of outcomes that make the eventIt is 6 ← Total number of outcomes of the experiment. Example 3: A bag has 4 red balls and 2 yellow balls. (The balls are identical in all respects other than colour). A ball is drawn from the bag without looking into the bag. What is probability of getting a red ball? Is it more or less than getting a yellow ball? Solution: There are in all (4 + 2 =) 6 outcomes of the event. Getting a red ball consists of 4 outcomes. (Why?) 42 Therefore, the probability of getting a red ball is 6 = . In the same way the probability3 21of getting a yellow ball = = (Why?). Therefore, the probability of getting a red ball is63 more than that of getting a yellow ball. Suppose you spin the wheel 1. (i) List the number of outcomes of getting a green sector and not getting a green sector on this wheel (Fig 5.11). (ii) Find the probability of getting a green sector. (iii) Find the probability of not getting a green sector. Fig 5.11 5.5.5 Chance and probability related to real life We talked about the chance that it rains just on the day when we do not carry a rain coat. What could you say about the chance in terms of probability? Could it be one in 10 1 days during a rainy season? The probability that it rains is then . The probability that it10 9does not rain = . (Assuming raining or not raining on a day are equally likely)10 The use of probability is made in various cases in real life. 1. To find characteristics of a large group by using a small part of the group. For example, during elections ‘an exit poll’ is taken. This involves asking the people whom they have voted for, when they come out after voting at the centres which are chosen off hand and distributed over the whole area. This gives an idea of chance of winning of each candidate and predictions are made based on it accordingly. 2. Metrological Department predicts weather by observing trends from the data over many years in the past. EXERCISE 5.3 1. List the outcomes you can see in these experiments. (a) Spinning a wheel (b) Tossing two coins together 2. When a die is thrown, list the outcomes of an event of getting (i) (a) a prime number (b) not a prime number. (ii) (a) a number greater than 5 (b) a number not greater than 5. 3. Find the. (a) Probability of the pointer stopping on D in (Question 1-(a))? (b) Probability of getting an ace from a well shuffled deck of 52 playing cards? (c) Probability of getting a red apple. (See figure below) 4. Numbers 1 to 10 are written on ten separate slips (one number on one slip), kept in a box and mixed well. One slip is chosen from the box without looking into it. What is the probability of . (i) getting a number 6? (ii) getting a number less than 6? (iii) getting a number greater than 6? (iv) getting a 1-digit number? 5. If you have a spinning wheel with 3 green sectors, 1 blue sector and 1 red sector, what is the probability of getting a green sector? What is the probability of getting a non blue sector? 6. Find the probabilities of the events given in Question 2. WHAT HAVE WE DISCUSSED?