Chapter 4 Section 4.1The mean, mode, median and Range The idea of an average is extremely useful, because it enables you to compare one set of data with another set by comparing just two values their averages. There are several ways of expressing an average, but the most commonly used averages are the mean, mode, median and range. The mean The mean of a set of data is the sum of all the values in the set divided by the total number of values in the set. That is: This is what most people mean when they say average. Example 1 The ages of 11 players in a football team are: 21 23 20 27 25 24 25 30 21 22 28 What is the mean age of the team? Sum of all ages = 266 Total number in team = 11 Therefore, Mean age = 266 11 = 24.2 c.azzopardi.smc@gmail.com 1
Consolidation Find the mean for each set of data: a) 7 8 3 6 7 3 8 5 4 9 b) 47 3 23 19 30 22 c) 1.53 1.51 1.64 1.55 1.48 1.62 1.58 1.65 The Mode The mode is the value that occurs the most in a set of data. That is, it is the value with the highest frequency. The mode is a useful average because it is very easy to find and it can be applied to non-numerical data. For example, you can find the modal birthday month of the class. Example 1 What is the mode of the following? 1, 1, 3, 7, 10, 13 Which is the most popular number? c.azzopardi.smc@gmail.com 2
Consolidation 1) 3 4 7 3 2 4 5 3 4 6 8 4 2) 100 10 1000 10 100 1000 10 1000 100 1000 100 10 3) rain sun cloud sun rain fog snow rain fog sun snow sun 4) The frequency table shows the colours of eyes of the students in a class. Blue Brown Green Boys 4 8 1 Girls 8 5 2 a. How many students are in class? b. What is the modal eye colour for: i. boys ii. girls iii. the whole class c.azzopardi.smc@gmail.com 3
c. After two students join the class the modal eye colour for the whole class is blue. Which of the following statements are true? Both students had green eyes Both students had brown eyes Both students had blue eyes You cannot tell what their eye colours were The Median The median is the middle number of a set of data. A way to remember this is the following: MEDIAN MIDDLE Therefore the median is the middle number in a set of ordered numbers. We can also use the following formula: Median = ½ (n + 1) Example 1 Let try and find the median of the following numbers 1, 3, 7, 10, 13 Therefore the median is 7 c.azzopardi.smc@gmail.com 4
Special Case Let us find the median of the following set of numbers 18, 19, 21, 25, 27, 28 If we have two median numbers all we have to do is find the mean of the two numbers. Therefore; Sum := 21 + 25 = 46 Mean := 46/2 = 23 As a result, 23 is the median. To find the Median OR 1. Arrange the numbers in order from smallest to largest 2. Find the middle number 3. If you have two middle numbers find the mean of those two numbers You can use the formula to find the position of the MEDIAN number Consolidation 1) 14 8 6 16 4 12 10 4 18 16 6 2) 10 6 5 7 13 11 14 6 13 15 4 15 c.azzopardi.smc@gmail.com 5
3) The table shows the number of sandwiches sold in a shop over 25 days. Sandwiches sold 10 11 12 13 14 15 16 Frequency 2 3 6 4 3 4 3 a. What is the modal number of sandwiches sold? b. What is the median number of sandwiches sold? The Range The range for a set of data is the highest value of the set minus the lowest value. The range is not an average. It shows the spread of the data. It is, therefore, used when comparing two or more sets of similar data. You can also use it to comment on the consistency of two or more sets of data. Range = Highest value Lowest Value Consolidation 1) 3 8 7 4 5 9 10 6 7 4 c.azzopardi.smc@gmail.com 6
2) 1 0 4 5 3 2 5 4 2 1 0 1 3) In a golf tournament, the club chairperson had to choose either Maria or Fay to play in the first round. In the previous eight rounds, their scores were as follows. Maria s Scores : 75 92 80 73 72 88 86 90 Fay s Scores: 80 87 85 76 85 79 84 88 a) Calculate the mean score for each golfer b) Find the range of each golfer c) Which golfer would you choose to play in the tournament? Explain why. Support Exercise Pg 263 Exercsie 17A Nos 5, 7, 8, 9, 10 (Mean), 1, 2, 3, 4, 6 c.azzopardi.smc@gmail.com 7
Section 4.2 The Use of Frequency Tables When a lot of information has been gathered, it is often convenient to put it together in a frequency table. From this table you can then find the values of the three averages and the range. Example 1 A survey was done on the number of people in each car leaving a shopping centre. The results are summarized in the table below. Number of people in each car 1 2 3 4 5 6 Frequency 45 198 121 76 52 13 For the number of people in a car, calculate: a) the mode b) the median c) the mean a) The modal number of people in a car is easy to spot. It is the number with the largest frequency, which is 198. Hence, the modal number of people in a car is 2. b) The median number of people in a car is found by working out where the middle of the set of numbers is located. First, add up frequencies to get the total number of cars surveyed, which comes to 505. Next calculate the middle position: (505 + 1) 2 = 253 Now add the frequency across the table to find which group contains the 253 rd item. The 243 rd item is the end of the group with 2 in a car. Therefore, the 253 rd item must be in the group with 3 in a car. Hence, the median number of people in a car is 3. c.azzopardi.smc@gmail.com 8
c) To calculate the mean number of people in a car, multiply the number of people in the car by the frequency. This is best done in an extra column. Add these to find the total number of people and divide by the total frequency (the number of cars surveyed). Number in Car Frequency Number in these cars 1 45 1 45 = 45 2 198 2 198 = 396 3 121 3 121 = 363 4 76 4 76 = 304 5 52 5 52 = 260 6 13 6 13 = 78 TOTAL 505 1446 Hence, the mean number of people in a car is 1446 505 = 2.9 (to 1 decimal place) Consolidation Find the i. mode, ii. median and iii. mean from each frequency table below. 1. A survey of the shoe size of all boys in one year of a school gave these results. Shoe Size 4 5 6 7 8 9 10 Number Students of 12 30 34 35 23 8 3 Mode: c.azzopardi.smc@gmail.com 9
Median: Mean: Shoe Size Frequency Total Shoe Size 4 5 6 7 8 9 10 TOTAL 2. A school did a survey on how many times in a week students arrived late at school. These are the findings. Number of times late 0 1 2 3 4 5 Frequency 481 34 23 15 3 4 Mode: c.azzopardi.smc@gmail.com 10
Median: Mean: Number of times late Frequency Total times late 0 1 2 3 4 5 TOTAL Support Exercise Pg 267 Exercise 17B Nos 1 6 Section 4.3 Frequency Tables with Grouped Data Sometimes the information you are given is grouped in some way. Normally, grouped data are continuous data, which is data that can have any value within a range of values (e.g. height, mass, time). In these situations, the mean can only be estimated as you do not have all the information. c.azzopardi.smc@gmail.com 11
Discrete data is data that consists of separate numbers, for example, goals scored, marks in a test, number of children and shoe size. In both cases, when using a grouped table to estimate the mean, first find the midpoint of the interval by adding the two end-values and then dividing by two. Example 1 Pocket money, p ($) 0 < p 1 1 < p 2 2 < p 3 3 < p 4 4 < p 5 Number of students 2 5 5 9 15 a) Write down the modal class. b) Calculate an estimate of the mean weekly pocket money. a) The modal class is easy to pick out, since it is simply the one with the largest frequency. Here the modal class is $4 to $5. b) To estimate the mean, assume that each person in each class has the midpoint amount, then build up the following table. To find the midpoint value, the two end-values are added together and then divided by two. Pocket money, p ($) Frequency (f) Midpoint (m) f m 0 < p 1 2 0.5 2 0.5 = 1 1 < p 2 5 1.5 5 1.5 = 7.5 2 < p 3 5 2.5 5 2.5 = 12.5 3 < p 4 9 3.5 9 3.5 = 31.5 4 < p 5 15 4.5 15 4.5 = 67.5 Totals Σf = 36 Σ (m f) = 120 c.azzopardi.smc@gmail.com 12
The estimated mean will be $120 36 = $3.33 (rounded to the nearest cent) Note: You cannot find the median or range from a grouped table since you do not know the actual values. Example 2 For the table of values given below, find: i) the modal group ii) an estimate for the mean. x 0 < x 10 10 < x 20 20 < x 30 30 < x 40 40 < x 50 Frequency 4 6 11 17 9 i) The modal group: ii) Mean: x Frequency (f) Midpoint (m) f m Total Σf = Σ f m = Mean : Consolidation c.azzopardi.smc@gmail.com 13
1) Jason brought 100 pebbles back from the beach and found their masses, recording each mass to the nearest gram. His results are summarized in the table below: Mass (m) 40 < m 60 60 < m 80 80 < m 100 100 < m 120 120 < m 140 140 < m 160 Frequency 5 9 22 27 26 11 i) Mode: ii) Mean: Mass Frequency (f) Midpoint (m) f m Total 2) A gardener measured the heights of all his roses to the nearest centimeter and summarized his results as follows: Height (cm) 10-14 15-18 19-22 23-26 27-40 Frequency 21 57 65 52 12 a) How many roses did the gardener have? c.azzopardi.smc@gmail.com 14
b) What is the modal class of the roses? c) What is the estimated mean height of the roses? Height (cm) Frequency (f) Midpoint (m) f m 10-14 15-18 19-22 23-26 27-40 Total Support Exercise Pg 275 Exercise 17D Nos 1 5 Section 4.4 Drawing and Interpreting Bar Charts A bar chart consists of a series of bars or blocks of the same width, drawn either vertically or horizontally from an axis. The heights and lengths of the bars always represent frequencies. Example 1 c.azzopardi.smc@gmail.com 15
Frequency Form 4 The grouped frequency table below shows the marks of 24 students in a test. Draw a bar chart for the data. Marks 1-10 11-20 21-30 31-40 41-50 Frequency 2 3 5 8 6 8 7 6 5 4 3 2 1 0 1-10 11-20 21-30 31-40 41-50 Mark Note: Both axes are labeled The class intervals are written under the middle of each bar The bars are separated by equal spaces By using a dual bar chart, it is easy to compare two sets of related data, as Example 2 shows. c.azzopardi.smc@gmail.com 16
Temperature ( F) Form 4 Example 2 This dual bar chart shows the average daily maximum temperature for England and Turkey over a fivemonth period. 100 90 80 70 60 50 40 30 Key 20 England 10 Turkey 0 April May June July August Month In which month was the difference between temperatures in England and Turkey the greatest? Note: You must always include a key to identify the two different sets of data. c.azzopardi.smc@gmail.com 17
Frequency Form 4 Consolidation 1) For her survey on fitness, Samina asked a sample of people, as they left a sports centre, which activity they had taken part in. She then drew a bar chart to show her data. 20 18 16 14 12 10 8 6 4 2 0 Squash Weight Training Badminton Aerobics Basketball Swimming Activity a. Which was the most popular activity? b. How many tool part in Samina s survey? c.azzopardi.smc@gmail.com 18
2) The frequency table below shows the levels achieved by 100 students in their A levels. Grade F E D C B A Frequency 12 22 24 25 15 2 a. Draw a suitable bar chart to illustrate the data. b. What fraction of the students achieve a Grade C or Grade B? c.azzopardi.smc@gmail.com 19
3) This table shows the number of point Mark and Joseph were each awarded in eight rounds of a general knowledge quiz. Round 1 2 3 4 5 6 7 8 Mark 7 8 7 6 8 6 9 4 Joseph 6 7 6 9 6 8 5 6 a. Draw a dual bar chart to illustrate the data. b. Comment on how well each of them did in the quiz. Support Exercise Handout c.azzopardi.smc@gmail.com 20
Section 4.5 Interpreting Pie Charts Pictograms, bar charts and line graphs are easy to draw but they can be difficult to interpret when there is a big difference between the frequencies or there are only a few categories. In these cases, it is often more convenient to illustrate the data on a pie chart. In a pie chart the whole of a data is represented by a circle (the pie ) and each category of it is represented by a sector of the circle. The angle of each sector is proportional to the frequency of the category it represents. So a pie chart cannot show individual frequencies, like a bar chart can, it can only show proportions. Calculating the frequency that each sector represents 1) Measure angles from the pie chart 2) Find the fraction of the whole circle 3) Multiply this fraction with the total number of items to calculate the frequency. Example 1 A chocolate firm asked 1440 students which type of chocolate they preferred. The pie chart showed the following results: Milk Chocolate - 150 White Chocolate - 120 Fruit and nut - 90 This information can be recorded into a table and the frequencies for each type are calculated. c.azzopardi.smc@gmail.com 21
Chocolate Angle Working Frequency Milk 150 600 students White 120 480 students Fruit and Nut 90 360 students TOTAL 360 1440 students Consolidation 1) 300 passengers have boarded a train at Waterloo Station in London. The following angles where given on a pie chart: Southampton - 120 Bournemouth - 90 Parkstone - 36 Branksome - 54 Poole - 60 Town Angle Working Frequency TOTAL c.azzopardi.smc@gmail.com 22
2) 180 people were asked about their favourite type of music. Use the angles taken from a pie chart, calculate the total number of people who chose each of the five types of music. Rock - 144 R n B - 60 Pop - 80 Dance - 40 Classical - 36 Music Angel Working Frequency TOTAL We are not always given the angle of the sector; we could be given the frequency and asked to find the angle of the sector. Calculating the angle that each frequency represents 1) Calculate the frequency of a particular sector from the pie chart 2) Find the fraction of the whole frequency 3) Multiply this fraction with the total degrees. c.azzopardi.smc@gmail.com 23
Example 2 The following table shows the eye colours of a group of 36 people. Find by how many degrees each sector is going to be represented with. Colour of Eyes Number of people Working Angle Brown 12 120 Blue 15 150 Green 6 60 Other 3 30 Total 36 360 Consolidation 1) The following table illustrates the choice of ice-cream flavor that customers had from an icecream shop. Find the angles which will be used on the pie chart. Flavour Frequency Working Angle Vanilla 35 Strawberry 20 Chocolate 22 Raspberry Ripple 13 TOTAL c.azzopardi.smc@gmail.com 24
2) The chart shows the sales of ice creams at a school party. The total number of ice-creams sold was 600. a) What fraction of the ice-creams sold where chocolate? b) How many ice-creams sold where chocolate? c) What fraction of the ice-creams is vanilla? d) How many vanilla ice-creams were sold? c.azzopardi.smc@gmail.com 25
e) How many strawberry ice-creams were sold? f) How many mint ice-creams were sold? 3) A store records how many CDs of different types of music they sold in one week. They show the results in a pie chart. There were 25 classical CDs sold and these had an angle of 45 on the pie chart. a) What fraction, in its simplest form, of CDs sold were classical? b) How many CDs were sold altogether that week? c) There were 75 pop CDs sold that week. What angle would be used on the pie chart to show sales of pop CDs? Support Exercise Handout c.azzopardi.smc@gmail.com 26
Section 4.6 Drawing Pie Charts Example 1 20 people were surveyed about their preferred drink. The replies are shown in the table below: Drink Tea Coffee Milk Cola Frequency 6 7 4 3 Show the results on a pie chart. First we must know what the total number of people observed were if you are not told in the question. 6 + 7 + 4 + 3 = 20 people Second, we must work out the size of the angle which will represent the drink choice. Tea : Coffee: Milk: Cola: Third, draw the pie chart sector by sector. c.azzopardi.smc@gmail.com 27
Milk, 72 Cola, 54 Tea, 108 Tea Coffee Milk Cola Coffee, 126 Note: You should always label the sectors of the pie chart You should always write the angle on each sector of the pie chart Consolidation 1. Eight students are asked about how they travel to school. Use the data in the following table to make up a pie chart. c.azzopardi.smc@gmail.com 28
Method of travel to school Number of students Walk 3 Car 2 Bike 2 Bus 1 Working: Pie Chart: a) Which transport method is the most common? b) From your pie chart, what proportion of students use green transport? c.azzopardi.smc@gmail.com 29
2. Joseph asked 180 boys what was their favourite sport. Here are the results. Sport Football Rugby Cricket Basketball Other Frequency 74 25 18 37 26 a) Draw a pie chart for these results with radius 6cm. Sport Football Rugby Cricket Basketball Other Frequency 74 25 18 37 26 Angle Joseph also asked 90 girls for their favourite sport. In a pie chart showing the results, the angle for Tennis was 84. b) How many of the girls said that Tennis was their favourite sport? c.azzopardi.smc@gmail.com 30
3. 16 people were asked how many portions of fruit and vegetables they ate per day. The results are shown below. Draw a pie chart for it: Number of portions Frequency None 3 Two 5 Five 6 Seven 4 a) You should eat at least 5 portions. What fraction of people is not eating healthily? b) Can you write this fraction any other way? Support Exercise Handout c.azzopardi.smc@gmail.com 31