Graphical Representations of data

Graphical Representation of Data 301 Graphical Representation of Data 28.1 INTRODUCTION Whenever verbal problems involving a certain situation is presented visually before the learners, it makes easier for the learner to understand the problem and attempt its solution. Similarly, when the data are presented pictorially (or graphically) before the learners, it makes the presentation eye-catching and more intelligible. The learners can easily see the salient features of the data and interpret them. There are many forms of representing data graphically. They are (i) Bar graphs (ii) Histograms (iii) Frequency polygons (iv) Ogive (v) Pictographs (vi) Pie charts In this lesson, we shall learn to read and draw Bar graphs, Histograms and Frequency polygons, and graphs related to day-to-day use, like temperature-time graph, velocity-time graph, pressure-volume graph, etc. Other form of graphs are beyond the scope of the present lesson. 28.2 OBJECTIVES After studying this lesson, the learner will be able to z draw bar charts for given data z draw a histogram and frequency polygon for given data z read and interpret given bar charts and histograms z read the relevant information from graphs relating to day-to-day activities, like (i) temperature-time graph302 Mathematics (ii) velocity-time graph (iii) pressure volume graph, etc. z draw graph relating to day-to-day activities, like the ones above. 28.3 EXPECTED BACKGROUND KNOWLEDGE z Knowledge of drawing and marking axes z Knowledge of drawing rectangles and plotting points z Practice of reading graphs. 28.4 BAR GRAPHS A bar graph is a graphical representation of frequency distributions of ungrouped data. It is a pictorial representation of the numerical data by a number of bars (rectangles) of uniform width erected vertically (or horizontally) with equal spacing between them. 28.4.1 Construction of Bar Graphs For the construction of bar graphs, we go through the following steps : Step 1 : We take a graph paper and draw two lines perpendicular to each other and call them horizontal and vertical axes. Step 2 : Along the horizontal axis, we take the values of the variables and along the vertical axis, we take the frequencies. Step 3 : Along the horizontal axis, we choose the uniform (equal) width of bars and the uniform gap between the bars, according to the space available. Step 4 : Choose a suitable scale to determine the heights of the bars. The scale is chosen according to the space available. Step 5 : Calculate the heights of the bars, according to the scale chosen and draw the bars. Step 6 : Mark the axes with proper labelling Let us take some examples to illustrate : Example 28.1 : The number of trees planted by an agency in different years is given below : Years 1997 1998 1999 2000 2001 2002 Total Number of 400 450 700 750 900 1500 4700 trees planted Solution : The bar graph is given below in Fig. 28.1 :Graphical Representation of Data 303 Fig. 28.1 Step 1 : We draw two perpendicular lines OX and OY. Step 2 : On OX, we represent years, from 1997–2002 and on OY we represent the number of trees planted. Step 3 : On OY, we start with 400 and marks points at equal intervals of 200. Step 4 : The height of the bars are calculated according to the number of trees. A kink (~) has been shown on the vertical axis showing that the marking on the vertical axis starts from zero but has been shown to start from 400 as the data needs. Examples 28.2 : The data below shows the number of students present in different classes on a particular day : Classes VI VII VIII IX X Number of 35 40 30 40 50 students present Represent the above data by a bar graph. Solution : The bar graph for the above data is shown in Fig. 28.2.304 Mathematics Fig. 28.2 Example 28.3 : The data regarding causes of accidents in factories are given below: Causes Percentage of Occurrence Faulty Machinery 30% Electrical Disturbance 20% Delay in repairs 35% Mechanical Failure 10% Others 5% Draw a bar graph to represent the above data. Solution : The bar graph representing the above data is shown in Fig. 28.3 below: Fig. 28.3Graphical Representation of Data 305 28.4.2 Interpretation of Bar graphs After drawing a bar graph, we can draw some conclusions, which is called interpreting bar graphs. Let us take some examples and do the same. Example 28.4 : Read the bar graphs given in Fig. 28.1, and answer the following questions : (i) In which year the maximum number of trees were planted ? (ii) What trend the number of trees planted show ? (iii) In which years the number of trees planted differ by 50 only ? Solution : After reading the bar graph, the answers to the above questions are as follows : (i) The maximum number (1500) of trees were planted in the year 2002, as in that year the height of the bar is maximum. (ii) The number of trees planted kept on increasing year after year (iii) (a) The years 1997 and 1998 (b) The years 1999 and 2000 Examples 28.5 : Read the bar graph given in Fig. 28.3 and answer the following questions : (i) Which cause is responsible for maximum accidents in factories ? Which is for minimum ? (ii) Can you think of one of the “other” causes ? (iii) How many percent of accidents could have been avoided by timely action? Solution : (i) Delay in repairs is responsible for maximum (35%) of accidents. “Other causes” are responsible for minimum number of accidents (ii) Carelessness of workers (iii) (35 + 20)% or 55% accidents could have been avoided by taking steps for timely repairs and provision of equipment which can control electrical disturbances. CHECK YOUR PROGRESS 28.1 1. Enlist the possible forms of representing a data graphically. 2. What are the steps needed to represent a data by a Bar Graph ?306 Mathematics 3. For the data on expenditure of a company over different heads, draw a bar graph : Head Percentage of Expenditure Salary of Employees 45% Travelling Allowance 15% Rent of Premises 20% Machinery and materials 10% Other expenditure 10% 4. Given below are data on causes of strikes in mills : Causes Percentage (i) Non fulfillment of economic demands 45 (ii) Overwork 20 (iii) Rivalry in unions 20 (iv) Non-congenial working conditions 10 (v) Others 5 Draw a bar graph depicting the above data. 5. From the bar graph given below, answer the following questions : Fig. 28.4 (i) The names of two steel plants which produced maximum steel in the country during the time period. (ii) What percentage of steel was produced in “other” plants ?Graphical Representation of Data 307 (iii) The steel plant at Durgapur produced how much less steel than at Bokaro? (iv) What percentage of total steel under discussion was produced at Bhalai, Durgapur and Bokaro steel plants ? 28.5 HISTOGRAMS AND FREQUENCY POLYGONS A histogram is a graphical representation of a continuous frequency distribution i.e. grouped frequency distributions. It is a graph, including vertical rectangles, with no space between the rectangles. The class-intervals are taken along the horizontal axis and the respective class frequencies on the vertical axis using suitable scales on each axis. For each class, a rectangle is drawn with base as width of the class and height as the class frequency. The area of the rectangles must be proportional to the frequencies of the respective classes. A frequency polygon is the join of the mid-points of the tops of the adjoining rectangles. The mid-points of the first and the last classes are joined to the mid-points of the classes preceding and succeding respectively at zero frequency to complete the polygon. Let us illustrate these with the help of examples. Examples 28.6 : The following is the frequency distribution of weights of 30 students of class IX of a school. Draw a histogram to represent the data. Classes : 45–50 50–55 55–60 60–65 65–70 Total Frequency : 3 7 12 5 3 30 Solution : For drawing a histogram we go through the steps similar to those of a bar graph. They are given below : Step 1 : On a paper, we draw two perpendicular lines and call them horizontal and vertical axes. Step 2 : Along the horizontal axis, we take classes of equal width : 45–50, 50–55, ...... As the axis starts from 45–50, we take one interval 40–45 before it and put a kink on axis before that Step 3 : Choose a suitable scale on the vertical axis to represent the frequency. It can start from 0 to 12, with a step of 2, i.e., 0, 2, 4, 6, ...., 12, 14 Step 4 : Draw the rectangles as shown in Fig. 28.5.308 Mathematics Fig. 28.5 shows the histogram required. Note : A frequency polygon has been shown in dotted lines, as explained in the steps shown above. Example 28.7 : The daily earnings of 100 shopkeepers are given below : Daily earnings 200-300 300-400 400-500 500-600 600-700 700-800 800-900 (in Rs) No. of shops 3 12 15 30 25 12 3 Draw a histogram and a frequency polygon to represent the above data. Solution : Following the steps suggested in Example 28.6, the histogram and frequency polygon representing the above data are given below in Fig. 28.6 Fig. 28.6Graphical Representation of Data 309 Example 28.8 : Draw a frequency polygon for the following data : Pocket 0–50 50–100 100–150 150–200 200–250 250–300 allowance (in rupees) Number of 16 25 13 26 15 5 students Solution : To draw a frequency polygon without-drawing a histogram we go through the following steps : Step 1 : Draw two lines perpendicular to each other. Step 2 : Find the class-marks of different classes. They are 25, 75, 125, 175, 225, 275 Step 3 : Plot the ordered pairs A (25, 16), B (75, 25), C(125, 13), D (175, 26), E(225, 15) and F(275, 5) Step 4 : Join the points A, B, C, D, E and F and complete the polygon as explained before The frequency polygon is given below : Fig. 28.7 28.5.1 Reading a Histogram Let us explain it with the help of an example Example 28.9. The following histogram shows the monthly wages (in rupees) of workers in a factory310 Mathematics Fig. 28.8 (i) Find the maximum number of workers getting a wage. (ii) Find the least wage and highest wage with no. of workers earning them (iii) How many workers get a monthly wage of Rs. 8000 or less ? Solution : (i) The maximum number of workers is 25 getting wages between Rs (7000 – 8000). (ii) The least wage is between Rs (4000 – 5000) and 4 workers are getting that. The corresponding figures for highest wage are Rs (9000 – 10000) and four workers get that (iii) 50 workers get a wage of Rs 8000 or less as Rs (4000 – 5000) – 4 workers Rs (5000 – 6000) – 10 workers Rs (6000 – 7000) – 12 workers Rs (7000 – 8000) – 24 workers Total – 50 CHECK YOUR PROGRESS 28.2 1. What is the difference between a bar graph and a histogram ? 2. Write various steps in the construction of a Histogram.Graphical Representation of Data 311 3. Draw a histogram for the following frequency distribution : Height of students 135–140 140–145 145–150 150–155 155–160 160–165 (in cm) No. of students 3 5 12 7 5 3 Also draw a frequency polygon for the above data on the same sheet 4. Draw a frequency polygon for the data in Question 3 on a separate paper. 5. Draw a histogram and a frequency polygon for the following grouped data: Annual income 4-6 6-8 8-10 10-12 12-14 14-16 16-18 18-20 (in ten thousand rupees) No. of families 25 20 15 15 13 7 3 2 in a locality 6. Interpret the data represented by the following histogram by answering the following questions : Fig. 28.9 Shirt sale in a week in a shop. (i) The least number of shirts were sold in which class ? (ii) The maximum number of shirts were sold in which class ? (iii) How many shirts were sold upto the 42 shirt size ? (iv) How many shirts of size 44–66 were sold ?312 Mathematics 28.6 GRAPHS RELATED TO DAY-TO-DAY ACTIVITIES In addition to histograms and frequency polygons, we are sometimes faced with graphs of other types. When a patient is admitted in a hospital with fever the doctor/nurses prepare a temperature-time graph, which can be referred to any time for reference. Similarly, the velocity time graph and pressure-volume graph are of day-to-day use. We shall learn to draw these graphs and interpret them in the sections below : 28.6.1 Temperature-Time Graph-Reading and Construction Example 28.10 : The body temperature of a patient admitted in a hospital with typhoid fever at different times of a day are given below : Time of the day 7 9 11 13 15 17 19 21 23 hrs hrs hrs hrs hrs hrs hrs hrs hrs Temperature 102 103 104 103 101 100 99 100 99 (in °F) Draw a graph to represent the above data. Solution : The graph of the above data is given in Fig. 28.10. The graph has been obtained by joining the points corresponding to pairs, like (7, 102), (9, 103), ........., (23, 99) in the rectangular system of coordinates, by line-segments. Fig. 28.10 Note : While drawing the graph it has been assumed that during the time interval in between times, the same trend was present.Graphical Representation of Data 313 Example 28.11 : If the medicine was given to the patient at 9 hours, whose temperature-time graph is shown in Fig. 28.10, answer the following questions : (i) At what time of the day was the temperature highest ? At what time lowest? (ii) After how much time, the action of medicine had started ? (iii) What trend do you observe from the above graph ? Solution : (i) The temperature of the patient was highest at 11 hours and lowest at 19 hrs and 23 hrs. (ii) The action of the medicine started 2 hours after the medicine was given as the temperature started falling after that. (iii) The administered medicine suited the patient as the temperature constantly fell after that, with the exception of period between 19 hrs and 21 hrs when it became slightly higher at 100°F but again fell after that 28.6.2 Velocity Time Graph During a journey from one place to other, the speeds of vehicles keep on changing according to traffic congestions. This can be very well shown by a velocity-time graph. Let us illustrate it with the help of example : Example 28.12. During a journey from city A to city B by car the following data regarding the time and velocity of the car was recorded : Time of the day 6 7 8 9 10 11 12 13 14 15 16 17 (in hours) Velocity 60 60 45 50 60 50 45 60 50 65 40 50 (in km/hour) Represent the above data by a velocity time graph. Solution : As before the graph can be obtained by plotting the ordered pairs (6, 60), (7, 60), ... (15, 65), ..., (17, 50) in the rectangular system of coordinates and then by joining them by line-segments.314 Mathematics Fig. 28.11 Example 28.13 : Read the velocity-time graph given in Fig. 28.11 and reply the following questions : 1. At what time duration of the day, the velocity of the car (i) was lowest ? was highest ? (ii) constant (iii) went on increasing (iv) went on decreasing 2. What was the average speed of the car in the journey ? Solution : 1 (i) At 16 hours; At 15 hours (ii) The velocity was constant at 60 km/hour between 6 hours and 7 hours (iii) Between 8 hours to 10 hours (iv) Between 10 hours to 12 hours 2. The average speed of the car was 60 60 45 50 60 50 45 60 50 65 40 50 12 F + + + + + + + + + + + H G I K J km/hour = 635 12 or 52.92 km/hour.Graphical Representation of Data 315 28.6.3 Pressure-Volume Graph For a fixed quantity of a gas at a constant temperature, is there any relation between pressure and volume of the gas ? Let us see that from the following example : Example 28.14 : The following data pertains to pressure and volume of a fixed quantity of gas : Pressure (p) 60 90 45 30 75 (in Newton) Volume (v) 90 60 120 180 72 (in cm3 ) Draw a graph to represent the above data. Solution : Fig. 28.12 The graph is obtained by joining the plot of the ordered pairs (60, 90), (90, 60), ..... (75, 72) by free hand curve. Example 28.15 : Read the above graph, given in Fig. 28.12, and answer the following questions : 1. Full in the blank : (i) As volume increases, the corresponding pressure ... (ii) As pressure decreases, the volume ... (iii) Pressure × Volume = ...316 Mathematics 2. What will be the pressure when volume is 100 cm3 ? 3. What will be the volume, when the pressure is 100 Newton ? Solution : 1 (i) Decreases (ii) Increases (iii) Constant = 5400 2. We know that pv = 5400 ∴ When volume = 100, p = 54 Newtons as can be seen from the graph at point A 3. When p = 100 Newtons, v = 54 cm3 as can be seen from the graph at the point B. CHECK YOUR PROGRESS 28.3 Represent the data given in each of the questions below graphically : 1. For a town, the maximum temperature for the following months are given below : Months March April May June July August September October Maximum temperature 35 38 38 42 45 40 38 35 (in °C) 2. The body temperatures of a patient admitted in a hospital are given below: Time of the day 8 9 10 11 12 13 14 15 16 17 (in hours) Temperature (in °F) 103 104 105 102 102 100 99 99 100 98 3. The speeds of a car going from station A to station B at different times of the day are given below : Time of the day (in hours) 7 8 9 10 11 12 13 14 15 Speed (in km/hour) 45 45 50 60 60 75 60 60 50 4. The data on pressure and volume of a gas are given below : Pressure (in Newtons) 60 80 50 30 40 20 Volume (in cm3 ) 40 30 48 80 60 120Graphical Representation of Data 317 5. For question No. 1, read the graph and reply the following questions : (i) Find the range of temperature for given months (ii) Which month had the least temperature ? (iii) Which month had the highest temperature ? (iv) In which month was the temperature less than 40°F ? (v) Can you predict the temperature for the next two months ? 6. Read the graph of Question No. 2 and answer the following question : (i) At what time of the day was the temperature of the patient maximum? (ii) If the medicine takes at least two hours to show the effect, at what time of the day was the medicine given ? LET US SUM UP z Bar graphs are the graphical representation of ungrouped frequency data. z Histograms and frequency polygons are the graphical representation of continuous grouped frequency data. z The graphical representation of data from day-to-day life is the join of points corresponding to ordered pairs represented by the data. z The graphical representations show the trends readily and at a glance only. TERMINAL EXERCISE Draw the bar graph for the following data in each case : 1. Height of samplings (in m) 0.5 0.75 1.0 1.25 1.50 1.75 2.00 No. of samplings 15 18 25 40 12 8 7 2. Weight (in kg) 7 8 9 10 11 12 15 No. of baskets of apples 4 5 7 8 5 4 3 3. Number of parcels 120 150 80 60 40 50 received in a post office Weight of parcels (in kg) 1 2 3 4 5 6 4. Interpret the data given in Question 1 and 2.318 Mathematics Draw a histogram and frequency polygon for the data in each case below: 5. Weight (in kg) 40–45 45–50 50–55 55–60 60–65 65–70 No. of students 4 7 8 9 6 3 in the class 6. Daily earning 100-120 120-140 140-160 160-180 180-200 (in rupees) No. of workers 5 7 8 3 2 7. Read the graphs for Question Nos. 5 and 6 and interpret them. 8. The minimum temperatures of a town for a year are given below : Month Jan Feb March April May June July Aug Sep Oct Nov Dec Min. temp. 12 14 16 20 20 24 25 24 22 18 16 12 (in °C) Draw a graph to represent the above data and interpret it. 9. A man left New Delhi for Lucknow by car at 7 AM. The speed of the car at different times of the day is given below : Time of the day 7 9 11 13 15 17 19 21 (in hour) Velocity (in km/hour) 45 50 60 65 70 60 55 40 Represent the above data by a velocity time graph and answer the following questions At what time was the velocity of the car (i) Maximum (ii) Minimum (iii) Between (50–60) km/hour (iv) Can you give hypothesis regarding the places where speed is extreme ? 10. The following data pertains to a gas in a container : Pressure (in Newton) 100 80 50 40 125 200 Volume (in cm3 ) 40 50 80 100 32 20 Represent the above data by a pressure volume graph. What relation do you find between pressure and volume from the data ?Graphical Representation of Data 319 ANSWERS Check Your Progress 28.1 5. (i) Bhilai and Rourkela (ii) 8.33% (iii) 50 (in ten thousand tonnes) (iv) 62.5% Check Your Progress 28.2 6. (i) (44–46) size (ii) (40–42) size (iii) 700 (iv) 50