# Lesson outline 1: Mean, median, mode

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2 Pupils are asked to compute the mean, median and mode. (Give sufficient time to pupils to do the working.) Answers: mean 16.5 / mode 19 / median 17.5 Teacher calls on pupils to explain how they obtained their results. Expected to result in: Mean = (sum of all scores) (number of pupils). Illustrate the mean as the balancing point: Forces (deviations) at one side balance the forces (deviations) at the other side. Mode is the score with the highest frequency (the value that is in fashion, the most popular). Median is the value in the middle when the scores are placed in order (if odd number of observations) OR average of the middle two scores (if even number of observations). Half the number of observations are to the right of the median the other half to the left. Questions: Which of these three mean, median or mode do you feel can be used best to represent the set of scores? Justify your answer. DO NOT ANSWER the question at this stage; only make an inventory of the pupils opinions and their reasons, without further comment. Write the results on the chalkboard: Best measure to use mean median mode number of students in favour because Inform pupils that they are going to investigate how mean, mode and median behave, so as to make a decision on which measure might be best used in a certain context. Investigating (40 minutes) The following are covered in the pupil s worksheets (Worksheet for pupils is on a following page seven pages ahead) 122

3 a) Is mean, median, mode necessarily a value belonging to the set and/or a value that could be taken in reality? b) The effect on mean, median, mode of adding a zero value to the value set. c) The effect on mean / median/ mode of adding two values with equal but opposite deviations or unequal deviations from mean, mode, median. d) The effect on mean / median / mode of adding values equal to mean / median / mode. Pupils activity Teacher gives worksheets to pupils. In small groups pupils are to answer the questions individually, then next compare and discuss the following questions. A test was scored out of 20 (only whole marks were given) and 12 pupils scored: 19, 20, 17, 11, 19, 19, 15, 8, 15, 20, 17 and 18. Using the above or other pupils scores (using real scores obtained by the class for example) answer the following question: Q1) Must the mean / median / mode be a score attained by one of the pupils in the class? Justify your answer. Illustrate with examples and non examples. Note to the teacher: (i) mean The mean represents the scores but need not be one of scores itself, it might even be a score that is impossible ever to get. (ii) median The median will be a score of one of the pupils if number of scores is odd. If the number of scores is even the median will be a score nobody did get or even nobody ever can get. If the median is half way between 16 and 18, then 17 is a possible score although nobody did score 17; if the median is between 17 and 18 the median is 17.5, a score nobody can ever get as it is not a whole number. (iii) mode The mode is necessarily a score attained by several pupils. If all scores are different there is no mode. If certain scores have the same frequency a set of scores can have more than one mode (bimodal, trimodal, etc., distribution). Q2) Investigate how the mean / median / mode changes when a zero score is added to the following set of scores. a) 19, 20, 17, 11, 19, 19, 15, 8, 15, 20, 17 and 18. Mean is 16.5; median is 17.5 and mode is 19. b) 19, 20, 17, 11, 19, 19, 15, 8, 15, 20, 17, 18 and 11 Mean is 16.1; median is 17; mode is 19 c) 0, 19, 20, 17, 11, 0, 19, 19, 8, 15, 20, 17 and 18. Mean is 14.1; median 17; mode

8 4. A pupil did 4 small projects in mathematics on the topic of number patterns during the term scoring (out of 20) in order : 4, 16, 15 and 16. Which represents best the overall attainment level of the pupil on project work on number patterns mean, median or mode? Answer: median/mode Discuss: Is using the mean score to represent the work done in mathematics during a term a fair measure for the attainment of the pupil? 5. A house building company wanting to find out what type of houses they should build most often in a region carried out a survey in that region to find out the number of people in a family. Will they use mean, median or mode to decide what type of houses should be build most? Answer: mode 6. A car battery factory wants to give a guarantee to their customers as to the lifetime of their batteries, i.e., they want to tell the customer if you have a problem with the battery in the next??? months we will replace your battery with a new one. They checked the lifetime of 100 batteries. Will they use mean, median or mode to decide on the number of months to guarantee their batteries? Answer: mean 128

9 Worksheet for Pupils Investigation Question 1: A test was scored out of 20 (only whole marks were given) and 12 pupils scored: 19, 20, 17, 11, 19, 19, 15, 8, 15, 20, 17 and 18. Using the above or other pupils scores (using real scores obtained by the class for example) answer the following question: Has the mean, median or mode to be a score attained by one of the pupils in the class? Justify your answer. Illustrate with examples and non examples. Question set 2: Investigate how the mean, median and mode change when a zero score is added to the following set of scores. a) 19, 20, 17, 11, 19, 19, 15, 8, 15, 20, 17 and 18. b) 19, 20, 17, 11, 19, 19, 15, 8, 15, 20, 17, 18 and 11 c) 0, 19, 20, 17, 11, 0, 19, 19, 8, 15, 20, 17 and 18. Make a correct statement: 1. If to a set of scores a zero score is added the mean changes ALWAYS / SOMETIMES / NEVER 2. If to a set of scores a zero score is added the median changes ALWAYS / SOMETIMES / NEVER 3. If to a set of scores a zero score is added the mode changes ALWAYS / SOMETIMES / NEVER Question 3: A set of scores has a mean of 16. Without calculating the new mean state how the mean changes if two more scores are to be taken into account. a) the two scores are 14 and 18 b) the two scores are 15 and 17 c) the two scores are 14 and 17 d) the two scores are 12 and 20 e) the two scores are 12 and 19 Make a general statement about when the mean will change and when it will not change. 129

10 Question 4: A set has a median score of 16. Without calculating the new median state how the median changes if two more scores are added to the set as follows: a) the two scores are 14 and 18 b) the two scores are 15 and 17 c) the two scores are 14 and 15 d) the two scores are 8 and 20 e) the two scores are 12 and 19 f) the two scores are 18 and 19 Make a general statement about when will the median change, and when will it not change. Question 5: A set has a mode score of 16. Without calculating the new mode state how the mode changes if two more scores are added to the set as follows: a) the two scores are 14 and 18 b) the two scores are 14 and 15 c) the two scores are 18 and 19 Make a general statement about when will the mode change, and when will it not change. Question 6: A set of scores has a mean of 16. Without calculating the new mean, state how the mean changes if two more scores equal to the mean are added. Question 7: A set of scores has a median of 16. Without calculating the new median, state how the median changes if two more scores equal to the median are added. Question 8: A set of scores has a mode of 16. Without calculating the new mode, state how the mode changes if two more scores equal to the mode are added. Question 9: Answer question 6, 7 and 8 if only ONE value equal to mean, median and mode respectively were added. 130

11 Question 10: Write down a data set of the ages of 12 people travelling in a bus with a) mean 24 b) median 24 c) mode 24 Compare the data sets each member in your group has written down. Are all the same? Why are there differences? How can different data sets have the same mean (median / mode)? Which set is the best? Why? A baby is born in the bus, making now 21 passengers the last one with age 0. Compute the change in mean / median / mode of your data set. The grand-grand parents (age 90 and 94) of the newborn enter the bus making up a total of 23 passengers. Compute the change in mean / median / mode of your data set. The 0 and the 90 / 94 are called outliers they are far from the mean / mode/ median. Comparing your results how do outliers affect the mean / median / mode? Make a correct statement: 1. If to a set of ages outliers are added the mean changes ALWAYS / SOMETIMES / NEVER 2. If to a set of ages outliers are added the median changes ALWAYS / SOMETIMES / NEVER 3. If to a set of ages outliers are added the mode changes ALWAYS / SOMETIMES / NEVER Which of the three measures is most affected? 131

13 Lesson outline 2: Mean, median, mode A look at the average wage The scenario for this lesson idea is a manufacturing and marketing company, in which the notion of average wage is considered from different points of view. The purpose is to show how a selection of the mean, the median, or the mode gives different answers to the same question. Materials The question A look at average wage below can either be photocopied or copied onto the blackboard. Students are allowed to use a calculator. Classroom organisation Students can work individually, in pairs or in small groups. The following information is to be given to pupils: Question: A look at average wage The Head of the Union Mr. Motswiri in the Matongo Manufacturing and Marketing Company was negotiating with Ms. Kelebogile Matongo, the president of the company. He said, The cost of living is going up. Our workers need more money. No one in our union earns more than P a year. Ms. Matongo replied, It s true that costs are going up. It s the same for us we have to pay higher prices for materials, so we get lower profits. Besides, the average salary in our company is over P I don t see how we can afford a wage increase at this time. That night the union official conducted the monthly union meeting. A sales clerk spoke up. We sales clerks make only P a year. Most workers in the union make P a year. We want our pay increased at least to that level. The union official decided to take a careful look at the salary information. He went to the salary administration. They told him that they had all the salary information on a spreadsheet in the computer, and printed off this table: Type of job Number employed Salary Union member President 1 P No Vice president 2 P No Plant Manager 3 P No Foreman 12 P9 000 Yes Workman 30 P7 500 Yes Payroll clerk 3 P6 750 Yes Secretary 6 P6 000 Yes Sales Clerk 10 P5 000 Yes Security officer 5 P4 000 Yes TOTAL 72 P The union official calculated the mean: 133

14 P MEAN = P11 065,97 72 Hmmmm, Mr. Motswiri thought, Miss Matongo is right, but the mean salary is pulled up by those high executive salaries. It doesn t give a really good picture of the typical worker s salary. Then he thought, The salary clerk is sort of right. Each of the thirty workmen makes P That is the most common salary the mode. However, there are thirty-six union members who don t make P and of those, twenty-four make less. Finally, the union head said to himself, I wonder what the middle salary is? He thought of the employees as being lined up in order of salary, low to high. The middle salary (it s called the median) is midway between employee 36 and employee 37. He said, employee 36 and employee 37 each make P7500.-, so the middle salary is also P Questions: 1. If the twenty-four lowest salaried workers were all moved up to P7500.-, what would be a) the new median? b) the new mean? c) the new mode? 2. What salary position do you support, and why? Activity 1: Presentation of the scenario Review with students the problem setting and the salary information. Ask students to identify how the mean, median, and the mode are used in the problem description. Use question 1 to review one way in which pay raises may be distributed. Answers: a) New median P7 500 Pose these questions: b) New mean P c) New mode: P7 500 Which measures of central tendency stayed the same? Which measures of central tendency changed? Why? If you changed only one or two salaries, which measure of central tendency will be sure to change? [The mean, since its calculation includes all values.] If you changed only one or two salaries, which measure of central tendency will be most likely to stay the same? [The mode is most likely to stay the same, because it is the most frequently occurring salary, and only one or two salaries are being changed.] If you change only one or two salaries, how likely is the median to change? [It depends. If the median is embedded in the middle of several 134

16 Section F: Mean, median and mode for grouped discrete data A 60 item multiple choice test was tried in a class with 43 pupils. The results are represented in the following frequency distribution. No. of correct answers Frequency A mode or median cannot be obtained from this frequency table. You can only read off the class interval that contains the mode and the class interval that contains the median. The class interval with the highest frequency is 21-30: the modal class. The median is the 22nd observation, i.e., the score of the 22nd student; that falls in the class interval If the number of data is large but discrete (for example the scores of 2000 pupils in an examination marked out of 60) or continuous (the time taken to run 100 m) data is best placed in groups or intervals. The scores could be grouped in five intervals: 1-10, 11-20, 21-30, 31-40, 41-50, and as in the distribution table above for 43 students only. By grouping data some information is lost. For example in the class 1-10 there are 4 students, and we no longer can see what their actual scores were (did all score 10?). For calculation purposes the mid-interval value (average of lower bound and upper bound value of the interval) is used. From a grouped frequency table you can find: the modal class (the class with the highest frequency) the interval in which the median is found an estimate of the mean A calculation of the mean using mid-interval values. [ ] sum of mid interval value frequency Estimate of the mean = sum of the frequencies Example The table gives the end of year examination mark of 200 students (maximum mark was 50) and the calculation to obtain an estimate of the mean. Mark Frequency Mid-value Mid-value Frequency TOTAL

17 An estimate for the mean is 6000 = 30. The estimated mean is The modal class is Most students scored in the range The median is the 1 2 ( ) = 1001 th term, i.e., the average of the 100th 2 and 101st term. As the actual value of these terms is unknown you can only give the interval in which these values are: the interval Calculator Your calculator can help you with these and longer calculations in statistics. Actual procedures vary with the brand of calculator, but if yours has statistics capability, the general way to use it is as follows. For more details, consult the instructions that came with the calculator. 1. Place it in Statistics Mode (if it has such a mode). 2. Clear out any previously stored statistical data from the memory registers. 3. Now enter individual data values: a) for single values key in each value, followed by a press of the DATA or ENTER or + key. b) for grouped values some calculators allow the entry of grouped data by having separate entry keys for the group frequency (or count) and for that group s average value. Consult your documentation. 4. Once all data points have been entered, a press of the X key (or equivalent) will display the mean of all the entered data. In most calculators there are other keys for the sample and population standard deviations. 137

18 Self mark exercise 3 1. The histogram illustrates the mass of fish caught in the Sashe dam one day. a) Make the grouped frequency table. b) How many fish were caught in all? c) What is the modal class? d) In what class interval is the median mass? e) Calculate an estimate for the mean mass of fish caught (use your frequency table). 2. The diameters of a batch of oranges were measured and the results displayed in a histogram. Continued on next page 138

19 a) How many orange had a diameter d cm in the range 8 d < 9? b) Make the grouped frequency table corresponding to the histogram. c) How many oranges in total were measured? d) What is the modal class interval? e) Calculate an estimate of the mean length of the diameter of the oranges (use your frequency table). 3. The height of fruit trees was measured one year after planting. The result is displayed in the histogram below. As the data, length of the trees, is continuous and taken to the nearest cm the first class has as class boundaries 99.5 and The class boundaries are taken half-way between the class intervals. a) How many trees had a height h cm in the range cm? b) Make the grouped frequency table corresponding to the histogram.the first class interval is , the second , etc. c) How many fruit trees in total were measured? d) What is the modal class interval? e) Calculate an estimate of the mean height of the fruit trees from the grouped frequency table. 4. The ages of pupils in Sefhare CJSS were represented in a histogram. Although age can be considered to be continuous, you are 14 from the day you turn 14 until the day you turn 15. The class interval has as lower bound 14 and as upper bound 15. Continued on next page 139

20 a) What are the modal classes? (a bi-modal distribution) b) How many pupils are in their 16th year? c) Make a grouped frequency table, first class 11 age < 12. d) Calculate an estimate of the mean age of the pupils in the school (use the frequency table). 5. The number of water pipes of different lengths made in a factory during a month are shown in the histogram below. a) How many pipes with length L m, 14 L < 18 were made? b) What is the modal class? c) In which class is the median length of pipe? d) How many pipes were produced during the month? e) Make an estimate for the total length of pipe produced. f) Make an estimate for the mean length of pipe produced. Continued on next page 140

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