Welcome dear students! Today we are going to learn about Data Handling and Presentation from Class 6 Maths. If you ask your classmates about their favourite colours, you will get a list of colours. This list is an example of data. Similarly, if you measure the weight of each student in your class, you would get a collection of measures of weight, which is again data. Any collection of facts, numbers, measures, observations or other descriptions of things that convey information about those things is called data. We live in an age of information. We constantly see large amounts of data presented to us in new and interesting ways. In this chapter, we will explore some of the ways that data is presented, and how we can use some of those ways to correctly display, interpret and make inferences from such data! Let us begin our journey. [CHECKPOINT]
Section four point one is Collecting and Organising Data. Navya and Naresh are discussing their favourite games. Navya says cricket is her favourite game. Naresh says he plays cricket sometimes but hockey is the game he likes the most. Navya thinks cricket is the most popular game in their class, but Naresh is not sure. They wonder how to find the most popular game in their class. To figure it out, Naresh and Navya decided to go to each student in the class and ask what their favourite game is. Then they prepared a list. Navya shows the list to the class. The list contains names paired with games. Mehnoor likes Kabaddi. Pushkal likes Satoliya, which is also called Pittu. Anaya likes Kabaddi. Jubimon likes Hockey. Densy likes Badminton. Jivisha likes Satoliya. Simran likes Kabaddi. Jivika likes Satoliya. Rajesh likes Football. Nand likes Satoliya. Leela likes Hockey. Thara likes Football. Ankita likes Kabaddi. Afshan likes Hockey. Soumya likes Cricket. Imon likes Hockey. Keerat likes Cricket. Navjot likes Hockey. Yuvraj likes Cricket. Gurpreet likes Hockey. Hemal likes Satoliya. Rehana likes Hockey. Arsh likes Kabaddi. Debabrata likes Football. Aarna likes Badminton. Bhavya likes Cricket. Ananya likes Hockey. Kompal likes Football. Sarah likes Kabaddi. Hardik likes Cricket. Tahira likes Cricket. Navya says happily that she has collected the data and can figure out the most popular game now. But other children look at the list and wonder how to get the answer from it. [CHECKPOINT]
Let us figure this out together. To find the most popular game among Naresh and Navya's classmates, we should count how many times each game appears in the list. We can use tally marks to make counting easier. Let us count them now. Kabaddi appears for Mehnoor, Anaya, Simran, Ankita, Arsh, and Sarah. That is six times. Satoliya appears for Pushkal, Jivisha, Nand, and Hemal. That is four times. Hockey appears for Jubimon, Leela, Afshan, Imon, Navjot, Gurpreet, Rehana, and Ananya. That is eight times. Badminton appears for Densy and Aarna. That is two times. Football appears for Rajesh, Thara, Debabrata, and Kompal. That is four times. Cricket appears for Soumya, Keerat, Yuvraj, Bhavya, Hardik, and Tahira. That is six times. Comparing the counts, Hockey has the highest frequency with eight votes. Therefore, the most popular game in their class is Hockey. Now, try to find out the most popular game among your own classmates by collecting data and counting frequencies. [CHECKPOINT]
Next, Pari wants to respond to some questions. We need to put a tick for questions where she needs to carry out data collection, and a cross for questions where she does not need to collect data. For the question asking the most popular TV show among her classmates, she must collect data from her classmates, so we put a tick. For the question asking when India got independence, this is a known historical fact, so she does not need to collect data. We put a cross. For the question asking how much water is getting wasted in her locality, she must measure or observe the water usage, so she needs to collect data. We put a tick. For the question asking the capital of India, this is a known geographical fact, so she does not need to collect data. We put a cross. Let us discuss these answers in the classroom. [CHECKPOINT]
Shri Nilesh is a teacher. He decided to bring sweets to the class to celebrate the new year. The sweets shop nearby has jalebi, gulab jamun, gujiya, barfi, and rasgulla. He wanted to know the choices of the children. He wrote the names of the sweets on the board and asked each child to tell him their preference. He put a tally mark for each student, and when the count reached five, he put a line through the previous four. The table shows the tally marks and the number of students. For jalebi, the tally shows one group of five and one single mark, making six students. For gulab jamun, the tally shows one group of five and four single marks, making nine students. For gujiya, the tally shows two groups of five and three single marks, making thirteen students. For barfi, the tally shows three single marks, making three students. For rasgulla, the tally shows one group of five and two single marks, making seven students. Now let us complete the table to help Shri Nilesh purchase the correct numbers of sweets. Six students chose jalebi. Barfi was chosen by three students. Thirteen students chose gujiya. Rasgulla was chosen by seven students. Nine students chose gulab jamun. [CHECKPOINT]
Shri Nilesh requested one of the staff members to bring the sweets as given in the table. The table helped him purchase the correct numbers of sweets. Now, is the table sufficient to distribute each type of sweet to the correct student? The table only shows the total count for each sweet. It does not tell us which specific student chose which sweet. Therefore, it is not sufficient for individual distribution. The alternative would be to record each student's name next to their chosen sweet, creating a list that links names to preferences. To organise the data, we can write the name of each sweet in one column and using tally signs, note the number of students who prefer that sweet. The numbers six, nine, thirteen, three, and seven are the frequencies of the sweet preferences for jalebi, gulab jamun, gujiya, barfi, and rasgulla respectively. [CHECKPOINT]
Sushri Sandhya asked her students about the sizes of the shoes they wear. She noted the data on the board. The raw data shows shoe sizes like four, five, three, four, three, four, five, five, four, five, five, four, five, six, four, three, five, six, four, six, four, five, seven, five, six, four, five. She then arranged the shoe sizes of the students in ascending order. The ordered list is three, three, three, four, four, four, four, four, four, four, four, five, five, five, five, five, five, five, five, five, five, five, six, six, six, six, seven. Let us help her figure out the following. The largest shoe size in the class is seven. The smallest shoe size in the class is three. There are eleven students who wear shoe size five. There are sixteen students who wear shoe sizes larger than four. This includes the eleven students wearing size five, the four students wearing size six, and the one student wearing size seven. Arranging the data in ascending order helped to quickly identify the smallest and largest values and to easily count the frequency of each size. Other ways to arrange the data include descending order, or grouping them into a frequency table using tally marks. [CHECKPOINT]
Now let us try an activity. Write the names of a few trees you see around you. When you observe a tree on the way from your home to school, record the data and fill in a table. For example, you might find ten Peepal trees, eight Neem trees, five Banyan trees, and three Mango trees. To answer the questions, you would compare the counts. The tree found in the greatest number would be the one with the highest count. The tree found in the smallest number would be the one with the lowest count. You would check if any two trees have the exact same count. Next, take a blank piece of paper and paste any small news item from a newspaper. Each student may use a different article. Prepare a table to count the number of each of the letters c, e, i, r, and x in the words of the news article, and fill in the table. After counting, you will find the letter found the most number of times and the letter found the least number of times. List the five letters c, e, i, r, x in ascending order of frequency. Compare your order with your classmates. Almost everyone is likely to get the order x, c, r, i, e. This happens because in English, the letter e is the most commonly used vowel, followed by i, then r, then c, and x is very rare. Write the process you followed to complete this task, discuss it with friends, and consider how you would approach it with another news item. [CHECKPOINT]
The teacher's note reminds us to provide more opportunities to collect and organise data. Ask students to guess what is the most popular colour, game, toy, or school subject amongst the students in their classroom, and then collect the data for it. It can be a fun activity in which they also learn about their classmates. Discuss how they can organise the data in different ways, each way having its own advantages and limitations. For all these tasks, discuss them with the children, let them understand the tasks, and then let them plan and present their research processes and conclusions in the class. Now we move to section four point two, Pictographs. Pictographs are one visual and suggestive way to represent data without writing any numbers. Look at a picture representing modes of travelling and the number of students. One smiley face symbol equals one student. For private car, there are four smiley faces, meaning four students. For public bus, there are five smiley faces, meaning five students. For school bus, there are eleven smiley faces, meaning eleven students. For cycle, there are three smiley faces, meaning three students. For walking, there are seven smiley faces, meaning seven students. This picture helps you understand at a glance the different modes of travel used by students. The mode of travel used by the most number of students is the school bus. The mode of travel used by the least number of students is the cycle. A pictograph represents data through pictures of objects. It helps answer questions about data with just a quick glance. In this pictograph, one unit or symbol is used to represent one student. There are also other pictographs where one unit or symbol stands for many people or objects. [CHECKPOINT]
Let us look at an example. Nand Kishor collected responses from the children of his middle school in Berasia regarding how often they slept at least nine hours during the night. He prepared a pictograph from the data. The scale is one triangle equals ten children. For the response Always, there are five triangles. For Sometimes, there are two and a half triangles. For Never, there are four triangles. Let us answer the questions. First, what is the number of children who always slept at least nine hours at night? In the table, there are five pictures for Always. Each picture represents ten children. Therefore, five pictures indicate five times ten equals fifty children. Second, how many children sometimes slept at least nine hours at night? There are two complete pictures, which is two times ten equals twenty, and a half picture, which is half of ten equals five. Therefore, the number of children who sleep at least nine hours only sometimes is twenty plus five equals twenty-five. Third, how many children always slept less than nine hours each night? There are four complete pictures for Never. Hence, four times ten equals forty children never sleep at least nine hours in a night, meaning they always sleep less than nine hours. [CHECKPOINT]
Now let us learn about drawing a pictograph. One day, Lakhanpal collected data on how many students were absent in each class. The data is: Class one has three absent, class two has five, class three has four, class four has two, class five has zero, class six has one, class seven has five, and class eight has seven. He created a pictograph to present this data and decided to show one student as one smiley face. The pictograph shows class one with three smileys, class two with five smileys, class three with four smileys, class four with two smileys, class five with zero smileys, class six with one smiley, class seven with five smileys, and class eight with seven smileys. Meanwhile, his friends Jarina and Sangita collected data on how many students were present in each class. The data is: class one has thirty, class two has thirty-five, class three has twenty, class four has twenty-five, class five has thirty, class six has twenty-five, class seven has thirty, and class eight has twenty. If they want to show their data through a pictograph using one symbol for each student, they would face the challenge of drawing too many symbols, which takes too much time and space. Jarina made a plan to address this. Since there were many students, she decided to use one symbol to represent five students. This would save time and space too. Sangita decided to use one symbol to represent ten students instead. Since she used one symbol to show ten students, she had a problem in showing twenty-five students and thirty-five students in the pictograph. Then, she realised she could use half a symbol to show five students. [CHECKPOINT]
What could be the problems faced in preparing such a pictograph if the total number of students present in a class is thirty-three or twenty-seven? The problem is that the scale of ten does not divide evenly into thirty-three or twenty-seven. You would need to draw three full symbols and a fraction of a symbol for thirty-three, and two full symbols plus seven-tenths of a symbol for twenty-seven. Drawing exact fractions like seven-tenths is difficult and can lead to inaccurate visual representation. Let us review the math talk points. Pictographs are a nice visual and suggestive way to represent data. They represent data through pictures of objects. Pictographs can help answer questions and make inferences about data with just a quick glance. By reading a pictograph, we can quickly understand the frequencies of the different categories and the comparisons of these frequencies. In a pictograph, the categories can be arranged horizontally or vertically. For each category, simple pictures and symbols are then drawn in the designated columns or rows according to the frequency of that category. A scale or key is added to show what each symbol or picture represents. Each symbol or picture can represent one unit or multiple units. It can be more challenging to prepare a pictograph when the amount of data is large or when the frequencies are not exact multiples of the scale or key. [CHECKPOINT]
Let us solve the figure it out questions for pictographs. First, a pictograph shows the number of books borrowed by students in a week from the library of Middle School, Ginnori. The scale is one book symbol equals one book. Monday has four symbols. Tuesday has six symbols. Wednesday has three symbols. Thursday has five symbols. Friday has eight symbols. Saturday has two symbols. On which day were the minimum number of books borrowed? Saturday has only two symbols, so Saturday is the answer. What was the total number of books borrowed during the week? We add the symbols: four plus six plus three plus five plus eight plus two equals twenty-eight books. On which day were the maximum number of books borrowed? Friday has eight symbols, so Friday is the answer. The possible reason could be that students prepare for weekend reading or have more free time on weekends. Second, Magan Bhai sells kites at Jamnagar. Six shopkeepers purchase kites. Chaman bought two hundred fifty. Rani bought three hundred. Rukhsana bought one hundred. Jasmeet bought four hundred fifty. Jetha Lal bought two hundred fifty. Poonam Ben bought seven hundred. We prepare a pictograph using one symbol to represent one hundred kites. For Chaman, we use two and a half symbols. For Rani, we use three symbols. For Rukhsana, we use one symbol. For Jasmeet, we use four and a half symbols. For Jetha Lal, we use two and a half symbols. For Poonam Ben, we use seven symbols. Now we answer the questions. How many symbols represent the kites that Rani purchased? Three symbols. Who purchased the maximum number of kites? Poonam Ben purchased seven hundred, which is the maximum. Who purchased more kites, Jasmeet or Chaman? Jasmeet purchased four hundred fifty, while Chaman purchased two hundred fifty, so Jasmeet purchased more. Rukhsana says Poonam Ben purchased more than double the number of kites that Rani purchased. Is she correct? Rani purchased three hundred. Double that is six hundred. Poonam Ben purchased seven hundred, which is indeed more than six hundred. So yes, she is correct. [CHECKPOINT]
Now we move to section four point three, Bar Graphs. Have you seen graphs like this on TV or in a newspaper? Like pictographs, such bar graphs can help us to quickly understand and interpret information, such as the highest value, the comparison of values of different categories, and so on. However, when the amount of data is large, presenting it by a pictograph is not only time consuming but at times difficult too. Let us see how data can be presented instead using a bar graph. Let us take the data collected by Lakhanpal earlier regarding the number of students absent on one day in each class. Class one has three absent, class two has five, class three has four, class four has two, class five has zero, class six has one, class seven has five, and class eight has seven. He presented the same data using a bar graph. The scale is one unit length equals one student. The horizontal axis shows classes one through eight. The vertical axis shows the number of students from zero to eight. The bar for class one reaches three units. The bar for class two reaches five units. The bar for class three reaches four units. The bar for class four reaches two units. Class five has no bar, showing zero. The bar for class six reaches one unit. The bar for class seven reaches five units. The bar for class eight reaches seven units. Notice the equally spaced horizontal lines. This means each pair of consecutive numbers on the left has the same gap. [CHECKPOINT]
Let us answer questions using this bar graph. In class two, five students were absent that day. In which class were the maximum number of students absent? Class eight has the tallest bar at seven units, so class eight. Which class had full attendance that day? Class five has a height of zero, meaning no students were absent, so class five had full attendance. When making bar graphs, bars of uniform width can be drawn horizontally or vertically with equal spacing between them. Then the length or height of each bar represents the given number. As we saw in pictographs, we can use a scale or key when the frequencies are larger. Let us look at an example of vehicular traffic at a busy road crossing in Delhi. The number of vehicles passing through the crossing each hour from six a.m. to twelve noon is shown in a horizontal bar graph. One unit of length stands for one hundred vehicles. The horizontal axis shows the number of vehicles from zero to twelve hundred. The vertical axis shows time intervals. The bar for eleven to twelve reaches six hundred. The bar for ten to eleven reaches seven hundred. The bar for nine to ten reaches eight hundred. The bar for eight to nine reaches one thousand. The bar for seven to eight reaches twelve hundred. The bar for six to seven reaches one hundred fifty. We can see that the maximum traffic at the crossing is shown by the longest bar, which is for the time interval seven to eight a.m. The bar graph shows that twelve hundred vehicles passed through the crossing at that time. The second longest bar is for eight to nine a.m., showing one thousand vehicles. The minimum traffic is shown by the smallest bar, for six to seven a.m., showing about one hundred fifty vehicles. The second smallest bar is for eleven a.m. to twelve noon, showing about six hundred vehicles. The total number of cars passing through the crossing during the two-hour interval eight to ten a.m. is about one thousand plus eight hundred, which equals eighteen hundred vehicles. [CHECKPOINT]
Let us answer the figure it out questions for the Delhi traffic graph. How many total cars passed through the crossing between six a.m. and noon? We add the values for each hour: one hundred fifty plus twelve hundred plus one thousand plus eight hundred plus seven hundred plus six hundred equals four thousand two hundred fifty vehicles. Why do you think so little traffic occurred during the hour of six to seven a.m., as compared to the other hours from seven a.m. to noon? Most people are still sleeping or just waking up, and schools and offices have not yet started, so fewer vehicles are on the road. Why do you think the traffic was the heaviest between seven and eight a.m.? This is peak morning commute time when students go to school and adults go to work. Why do you think the traffic was lesser and lesser each hour after eight a.m. all the way until noon? The morning rush has passed, most commuters have reached their destinations, and schools and offices are already in session. [CHECKPOINT]
Next is an example showing the population of India in crores. This bar graph shows the population of India in each decade over a period of fifty years. The numbers are expressed in crores. If you were to take one unit length to represent one person, drawing the bars will be difficult. Therefore, we choose the scale so that one unit represents ten crores. A bar of length five units represents fifty crores and of eight units represents eighty crores. Based on this bar graph, you may ask your friends questions like which decade had the highest population growth, or what was the population in a specific year. To find how much the population of India increased over fifty years, you subtract the initial population from the final population. To find the increase in each decade, you subtract the population of the previous decade from the current decade. Now we move to section four point four, Drawing a Bar Graph. In a previous example, Shri Nilesh prepared a frequency table representing the sweet preferences of the students in his class. Let us try to prepare a bar graph to present his data. First, we draw a horizontal line and a vertical line. On the horizontal line, we will write the name of each of the sweets, equally spaced, from which the bars will rise in accordance with their frequencies. On the vertical line we will write the frequencies representing the number of students. The sweets are jalebi with six students, gulab jamun with nine students, gujiya with thirteen students, barfi with three students, and rasgulla with seven students. Second, we must choose a scale. We decide how many students will be represented by a unit length of a bar so that it fits nicely on our paper. Here, we will take one unit length to represent one student. Third, for jalebi, we draw a bar having a height of six units. For gulab jamun, we draw a bar of nine units. For gujiya, we draw a bar of thirteen units. For barfi, we draw a bar of three units. For rasgulla, we draw a bar of seven units. Fourth, we get a bar graph with the vertical axis showing number of students from zero to fourteen, and the horizontal axis showing the sweets. The bars match the frequencies exactly. When the frequencies are larger and we cannot use the scale of one unit length equals one number, we need to choose a different scale like we did in the case of pictographs. [CHECKPOINT]
Let us look at another example. The number of runs scored by Smriti in each of the eight matches are given in a table. Match one is eighty runs. Match two is fifty runs. Match three is ten runs. Match four is one hundred runs. Match five is ninety runs. Match six is zero runs. Match seven is ninety runs. Match eight is fifty runs. In this example, the minimum score is zero and the maximum score is one hundred. Using a scale of one unit length equals one run would mean that we have to go all the way from zero to one hundred runs in steps of one. This would be unnecessarily tedious. Instead, let us use a scale where one unit length equals ten runs. We mark this scale on the vertical line and draw the bars according to the scores in each match. We get a bar graph representing the data. The vertical axis shows runs from zero to one hundred in intervals of ten. The horizontal axis shows matches one through eight. Match one bar reaches eighty. Match two reaches fifty. Match three reaches ten. Match four reaches one hundred. Match five reaches ninety. Match six has no bar for zero. Match seven reaches ninety. Match eight reaches fifty. [CHECKPOINT]
Here is another example. The following table shows the monthly expenditure of Imran's family on various items. House rent is three thousand rupees. Food is three thousand four hundred rupees. Education is eight hundred rupees. Electricity is four hundred rupees. Transport is six hundred rupees. Miscellaneous is one thousand two hundred rupees. To represent this data in the form of a bar graph, we follow these steps. Draw two perpendicular lines, one horizontal and one vertical. Along the horizontal line, mark the items with equal spacing between them and mark the corresponding expenditures along the vertical line. Take bars of the same width, keeping a uniform gap between them. Choose a suitable scale along the vertical line. Let one unit length equal two hundred rupees, and then mark and write the corresponding values representing each unit length. Finally, calculate the heights of the bars for various items. House rent is three thousand divided by two hundred, which equals fifteen units. Food is three thousand four hundred divided by two hundred, which equals seventeen units. Education is eight hundred divided by two hundred, which equals four units. Electricity is four hundred divided by two hundred, which equals two units. Transport is six hundred divided by two hundred, which equals three units. Miscellaneous is one thousand two hundred divided by two hundred, which equals six units. Here is the bar graph we obtain. The vertical axis shows expenditure in rupees from two hundred to three thousand six hundred. The horizontal axis shows the items. The bars match the calculated unit heights. [CHECKPOINT]
Let us use this bar graph to answer questions. On which item does Imran's family spend the most and the second most? The tallest bar is for food at seventeen units, so food is the most. The second tallest bar is for house rent at fifteen units, so house rent is the second most. Is the cost of electricity about one-half the cost of education? Electricity is two units and education is four units. Two is exactly half of four, so yes. Is the cost of education less than one-fourth the cost of food? Education is four units. One-fourth of food is seventeen divided by four, which is four point two five units. Four is less than four point two five, so yes, education is less than one-fourth the cost of food. Now let us solve the figure it out questions for this section. First, Samantha visited a tea garden and collected data of insects and critters. Mites are six. Caterpillars are ten. Beetles are five. Butterflies are three. Grasshoppers are two. To prepare a bar graph, we draw a horizontal axis with the insect names and a vertical axis with numbers. We choose a scale of one unit equals one insect. The bar for mites reaches six. The bar for caterpillars reaches ten. The bar for beetles reaches five. The bar for butterflies reaches three. The bar for grasshoppers reaches two. Second, Pooja collected data on the number of tickets sold at the Bhopal railway station. Vidisha is twenty-four. Jabalpur is twenty. Seoni is sixteen. Indore is twenty-eight. Sagar is sixteen. She prepared a bar graph but a portion was erased. We need to reconstruct it. The number of tickets sold for Vidisha is twenty-four. The number of tickets sold for Jabalpur is twenty. The bar for Vidisha is six unit lengths and the bar for Jabalpur is five unit lengths. To find the scale, we divide the value by the unit length. Twenty-four divided by six equals four. Twenty divided by five equals four. So the scale is one unit length equals four tickets. To draw the correct bar for Sagar, which is sixteen tickets, we divide sixteen by four, which gives four unit lengths. We add the scale to the vertical axis by marking zero, four, eight, twelve, sixteen, twenty, twenty-four, twenty-eight. The bar for Seoni is sixteen tickets, which is four units. The bar for Indore is twenty-eight tickets, which is seven units. We check if the bars for Seoni and Indore are correct in the graph. If they do not match four and seven units respectively, we redraw them to the correct heights. [CHECKPOINT]
Third, Chinu listed the various means of transport that passed across the road in front of his house from nine a.m. to ten a.m. The list contains bikes, cars, buses, auto rickshaws, bicycles, bullock carts, and scooters. We prepare a frequency distribution table by counting each. Bikes appear fifteen times. Cars appear seven times. Buses appear five times. Auto rickshaws appear nine times. Bicycles appear eight times. Bullock carts appear two times. Scooters appear eight times. The means of transport used the most is the bike with fifteen occurrences. If you were there to collect this data, you would stand at a safe spot, use a tally sheet with categories for each transport type, and make a tally mark each time a vehicle passes. After the hour, you would count the tallies to get the frequencies. Fourth, roll a die thirty times and record the number you obtain each time. Prepare a frequency distribution table using tally marks. Since this is a theoretical exercise, we will show how to do it. You would list numbers one through six. You would roll the die thirty times and mark tallies. Then you count. The number that appeared the minimum number of times is the one with the fewest tallies. The number that appeared the maximum number of times is the one with the most tallies. The numbers that appeared an equal number of times are those with matching tally counts. [CHECKPOINT]
Fifth, Faiz prepared a frequency distribution table of data on the number of wickets taken by Jaspreet Bumrah in his last thirty matches. Zero wickets in two matches. One wicket in four matches. Two wickets in six matches. Three wickets in eight matches. Four wickets in three matches. Five wickets in five matches. Six wickets in one match. Seven wickets in one match. What information is this table giving? It gives the frequency of matches in which Bumrah took a specific number of wickets. What may be the title of this table? Frequency Distribution of Wickets Taken by Bumrah in Last Thirty Matches. What caught your attention in this table? The highest frequency is for three wickets, meaning he most often took three wickets per match. In how many matches has Bumrah taken four wickets? Three matches. Mayank says that to know the total number of wickets, we have to add the numbers zero, one, two, three, up to seven. Can Mayank get the total number of wickets taken in this way? No, because adding the wicket counts zero through seven ignores how many times each wicket count occurred. We must multiply each wicket count by its frequency. How would you correctly figure out the total number of wickets taken by Bumrah in his last thirty matches? We calculate zero times two plus one times four plus two times six plus three times eight plus four times three plus five times five plus six times one plus seven times one. This equals zero plus four plus twelve plus twenty-four plus twelve plus twenty-five plus six plus seven, which equals ninety wickets. [CHECKPOINT]
Sixth, a pictograph shows the number of tractors in five different villages. The scale is one tractor symbol equals one tractor. Village A has six tractors. Village B has five tractors. Village C has eight tractors. Village D has three tractors. Village E has six tractors. Which village has the smallest number of tractors? Village D with three. Which village has the most tractors? Village C with eight. How many more tractors does Village C have than Village B? Eight minus five equals three more tractors. Komal says Village D has half the number of tractors as Village E. Village D has three. Village E has six. Three is exactly half of six. So yes, she is right. Seventh, the number of girl students in each class of a school is depicted by a pictograph. The scale is one symbol equals four girls. Class one has three symbols, so twelve girls. Class two has four symbols, so sixteen girls. Class three has five symbols, so twenty girls. Class four has three and a half symbols, so fourteen girls. Class five has two and a half symbols, so ten girls. Class six has four symbols, so sixteen girls. Class seven has three symbols, so twelve girls. Class eight has one and a half symbols, so six girls. Which class has the least number of girl students? Class eight with six girls. What is the difference between the number of girls in class five and six? Class five has ten, class six has sixteen. The difference is sixteen minus ten equals six girls. If two more girls were admitted in class two, the new total would be eighteen girls. Since each symbol is four girls, we would add half a symbol to the existing four symbols, making four and a half symbols. How many girls are there in class seven? Three symbols times four equals twelve girls. [CHECKPOINT]
Eighth, Mudhol Hounds are largely found in North Karnataka's Bagalkote and Vijaypura districts. The government took an initiative to protect this breed. The number of Mudhol dogs in six villages are as follows. Village A has eighteen. Village B has thirty-six. Village C has twelve. Village D has forty-eight. Village E has eighteen. Village F has twenty-four. We prepare a pictograph. A useful scale or key would be one symbol equals six dogs, because all numbers are multiples of six. For Village B with thirty-six dogs, we divide thirty-six by six, which gives six symbols. Kamini said that the number of these dogs in Village B and Village D together will be more than the number of these dogs in the other four villages. Village B has thirty-six and Village D has forty-eight. Their sum is eighty-four. The other four villages are A, C, E, F. Their counts are eighteen, twelve, eighteen, twenty-four. Their sum is seventy-two. Eighty-four is greater than seventy-two. So yes, Kamini is right. Ninth, a survey of one hundred twenty school students was conducted to find out which activity they preferred to do in their free time. Playing is forty-five. Reading story books is thirty. Watching TV is twenty. Listening to music is ten. Painting is fifteen. We draw a bar graph with a scale of one unit length equals five students. The vertical axis goes from zero to fifty. The horizontal axis lists the activities. The bar for playing reaches nine units. The bar for reading reaches six units. The bar for watching TV reaches four units. The bar for listening to music reaches two units. The bar for painting reaches three units. The activity preferred by most students other than playing is reading story books with thirty students. [CHECKPOINT]
Tenth, students and teachers of a primary school decided to plant tree saplings. The bar graph shows the number of saplings planted each day. Monday is sixty. Tuesday is forty-five. Wednesday is thirty-three. Thursday is forty-five. Friday is fifty-seven. Saturday is sixty-nine. Sunday is forty-five. The total number of saplings planted on Wednesday and Thursday is thirty-three plus forty-five, which equals seventy-eight. The total number of saplings planted during the whole week is sixty plus forty-five plus thirty-three plus forty-five plus fifty-seven plus sixty-nine plus forty-five, which equals three hundred fifty-four. The greatest number of saplings were planted on Saturday with sixty-nine. The least number of saplings were planted on Wednesday with thirty-three. Why do you think that is the case? Saturday might have been a weekend with more volunteers available, while Wednesday might have had school exams or bad weather. More saplings might be planted on days with better weather or more free time. You could figure this out by checking weather records or school schedules for that week. Eleventh, the number of tigers in India went down drastically between nineteen hundred and nineteen seventy. Project Tiger was launched in nineteen seventy-three. Starting in two thousand six, the exact number was tracked. The frequency table shows two thousand six as fourteen hundred. Two thousand ten as seventeen hundred. Two thousand fourteen as twenty-two hundred. Two thousand eighteen as three thousand. Two thousand twenty-two as three thousand seven hundred. The bar graph has mistakes. The bars must match the exact values. The horizontal axis should show the number of tigers from zero to four thousand. The vertical axis should show the years. The bar for two thousand six should reach fourteen hundred. The bar for two thousand ten should reach seventeen hundred. The bar for two thousand fourteen should reach twenty-two hundred. The bar for two thousand eighteen should reach three thousand. The bar for two thousand twenty-two should reach three thousand seven hundred. We check the graph and adjust any bars that do not match these exact heights. [CHECKPOINT]
Let us review the summary points. Facts, numbers, measures, observations and other descriptions of things that convey information about those things is called data. Data can be organised in a tabular form using tally marks for easy analysis and interpretation. Frequencies are the counts of the occurrences of values, measures or observations. Pictographs represent data in the form of pictures, or objects or parts of objects. Each picture represents a frequency which can be one or more than one. This is called the scale and it must be specified. Bar graphs have bars of uniform width. The length or height indicates the total frequency of occurrence. The scale that is used to convert length or height to frequency must be specified. Choosing the appropriate scale for a pictograph or bar graph is important to accurately and effectively convey the desired information or data and to also make it visually appealing. Other aspects of a graph also contribute to its effectiveness and visual appeal such as how colours are used, what accompanying pictures are drawn, and whether the bars are horizontal or vertical. These aspects correspond to the artistic and aesthetic side of data handling and presentation. However, making visual representations of data fancy can also sometimes be misleading. By reading pictographs and bar graphs accurately, we can quickly understand and make inferences about the data presented. Now we move to section four point five, Artistic and Aesthetic Considerations. In addition to the steps described in previous sections, there are also some other more artistic and aesthetic aspects one can consider when preparing visual presentations of data to make them more interesting and effective. First, when making a visual presentation of data such as a pictograph or bar graph, it is important to make it fit in the intended space. This can be controlled, for example, by choosing the scale appropriately, as we have seen earlier. It is also desirable to make the data presentation visually appealing and easy-to-understand, so that the intended audience appreciates the information being conveyed. Let us consider an example. Here is a table naming the tallest mountain on each continent, along with the height of each mountain in meters. Asia has Everest at eight thousand eight hundred forty-eight meters. South America has Aconcagua at six thousand nine hundred sixty-two meters. North America has Denali at six thousand one hundred ninety-four meters. Africa has Kilimanjaro at five thousand eight hundred ninety-five meters. Europe has Elbrus at five thousand six hundred forty-two meters. Antarctica has Vinson Massif at four thousand eight hundred ninety-two meters. Australia has Koscuiszko at two thousand two hundred twenty-eight meters. How much taller is Mount Everest than Mount Koscuiszko? We subtract: eight thousand eight hundred forty-eight minus two thousand two hundred twenty-eight equals six thousand six hundred twenty meters. Are Mount Denali and Mount Kilimanjaro very different in height? Denali is six thousand one hundred ninety-four and Kilimanjaro is five thousand eight hundred ninety-five. The difference is only two hundred ninety-nine meters, so they are not very different. This is not so easy to quickly discern from a large table of numbers. We can convert the table of numbers into a bar graph, where each value is drawn as a horizontal box. These are longer or shorter depending on the number they represent. This makes it easier to compare the heights of all these mountains at a glance. [CHECKPOINT]
However, since the boxes represent heights, it is better and more visually appealing to rotate the picture, so that the boxes grow upward, vertically from the ground like mountains. A bar graph with vertical bars is also called a column graph. Columns are the pillars you find in a building that hold up the roof. Below is a column graph for our table of tallest mountains. From this column graph, it becomes easier to compare and visualise the heights of the mountains. In general, it is more intuitive, suggestive and visually appealing to represent heights, that are measured upwards from the ground, using bar graphs that have vertical bars or columns. Similarly, lengths that are parallel to the ground, for example distances between location on Earth, are usually best represented using bar graphs with horizontal arcs. Let us solve the figure it out questions. If you wanted to visually represent the data of the heights of the tallest persons in each class in your school, would you use a graph with vertical bars or horizontal bars? You would use vertical bars because height is measured upwards from the ground, making vertical bars more intuitive and visually suggestive. If you were making a table of the longest rivers on each continent and their lengths, would you prefer to use a bar graph with vertical bars or with horizontal bars? You would prefer horizontal bars because river lengths are measured horizontally across the land, so horizontal bars better represent the concept of length. [CHECKPOINT]
When data visualisations such as bar graphs are further beautified with more extensive artistic and visual imagery, they are called information graphics or infographics for short. The aim of infographics is to make use of attention-attracting and engaging visuals to communicate information even more clearly and quickly, in a visually pleasing way. As an example, let us go back to the table listing the tallest mountain on each continent. We drew a bar graph with vertical bars rather than horizontal bars, to be more indicative of mountains. But instead of rectangles, we could use triangles, which look a bit more like mountains. And we can add a splash of colour as well. While this infographic might look more appealing and suggestive at first glance, it does have some issues. The goal of our bar graph earlier was to represent the heights of various mountains using bars of the appropriate heights but the same widths. The purpose of using the same widths was to make it clear that we are only comparing heights. However, in this infographic, the taller triangles are also wider. Are taller mountains always wider? The infographic is implying additional information that may be misleading and may or may not be correct. Sometimes going for more appealing pictures can also accidentally mislead. Taking this idea further, and to make the picture even more visually stimulating and suggestive, we can further change the shapes of the mountains to make them look even more like mountains, and add other details, while attempting to preserve the heights. For example, we can create an imaginary mountain range that contains all these mountains. Is the infographic below better than the column graph with rectangular columns of equal width? The mountains look more realistic, but is the picture accurate? For example, Everest appears to be twice as tall as Elbrus. What is five thousand six hundred forty-two times two? It is eleven thousand two hundred eighty-four. But Everest is only eight thousand eight hundred forty-eight. So the visual scaling is inaccurate. While preparing visually-appealing presentations of data, we also need to be careful that the pictures we draw do not mislead us about the facts. In general, it is important to be careful when making or reading infographics, so that we do not mislead our intended audiences and we, ourselves, are not misled. [CHECKPOINT]
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]