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Work in pairs on the following questions and be prepared to take part in a discussion with another pair. (1) What does the word “average” mean to you? Why might it be useful to know the average value? When is it not helpful to know the average value? I’m average. I’m the mean. Only about I’m average. 30% of us are above We are average. I’m the middle average. We are the mode. one. I’m the Everyone above us median. is above average. 105cm 105cm 112cm 129cm 137cm 178cm 193cm (2) The mean, median and mode are all averages (measures of central tendency). They all have different values for the people shown above. Discuss how you think these averages are defined (how were they calculated or found?) (3) Which, if any, is the best type of average to use to describe the height of this group of people? Explain your choice. (4) Think of a situation in which the mode is the most useful type of average to use. Think of another situation in which it is the least useful type of average to use. (5) Think of one major disadvantage of the mean as an average. Think of a situation where the values of the mean and the median are very likely to be close. (6) Why do you think the mean is used as an average more often than the median or mode? (7) Which do you think will be larger and why, the mean or the median? See Ms. Makunja when you have shared as a pair. From a list of raw data: Follow these steps to find the mean height and the median height (in cm) of the seven people on page 39: 105, 105, 112, 129, 137, 178, 193 STAT EDIT 1:Edit ENTER Highlight heading L1 CLEAR ENTER then carefully type the seven values into list L1 (note: they don’t have to be arranged in any particular order) STAT CALC 1:1-Var Stats ENTER 2nd 1 ENTER To see x 137 (the mean is 137 cm) and n=7 (you entered 7 values) and then scroll down to see Med = 129 (the median is 129 cm) From a frequency table: Here are the scores achieved by 20 students in a test: 3, 8, 9, 1, 4, 2, 7, 6, 5, 9, 10, 3, 4, 6, 2, 8, 7, 6, 3, 7 Here is the data organised into a frequency table: Score out of ten 1 2 3 4 5 6 7 8 9 10 (x) Frequency (f) 1 2 3 2 1 3 3 2 2 1 Follow these steps to find the mean the median scores of the group of students: STAT EDIT 1:Edit ENTER Highlight heading L1 CLEAR ENTER then type the integers from 1 to 10 into list L1 Highlight heading L2 CLEAR ENTER Then type the frequencies shown in the table (1, 2, 3, 2, and so on) into list L2 STAT CALC 1:1-Var Stats ENTER 2nd 1 , (comma key is above the 7) 2nd 2 ENTER To see x 5.5 (the mean is 5.5 cm) and n=20 (you entered 20 values) and then scroll down to see Med = 6 (the median is 6 cm) Question Which of the measures of central tendency is more affected by the presence of an outlier? Give an example of data sets where outliers make a big difference. Group Activity Work in pairs, discussing the following 4 questions. You have 20 minutes to do this and then you should be ready to participate in a whole class discussion. (1) Here again are the scores achieved by 20 students in a test: Score out of ten (x) 1 2 3 4 5 6 7 8 9 10 Frequency (f) 1 2 3 2 1 3 3 2 2 1 We’ve already learnt how to find the mean score using the TI83 but how could the mean be obtained without the calculator? Explain your method and why it works. For help with this you may refer to your text, page 432 and read example 6. (2) Look again at the test scores of the 20 students above. You should have found that the mean score is 5.5 so we would expect the median to be similar. To find the median we usually list the values in ascending order and then locate the one in the middle position. We could do that with this data. We know that the first value is 1, the next two are 2 and so on: 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10 Since there are 20 values (an even number) there will be a middle pair (the 10 th and 11th values). Counting along the list we find that the 10th value is 6 and the 11th value is also 6. So the median is in the middle of 6 and 6, so the median is 6. We didn’t have to write out the whole list to locate the 10th and 11th values. We could have obtained this information from the frequency table. How? For help with this section you may refer to you text, page 432, and read examples 8 and 9. (3) Finding the mean of grouped data The following data summarises the distances travelled by a fleet of 190 buses before experiencing a major breakdown. Distance d (in 40<d 60 60<d80 80<d100 100<d120 120<d140 140<d220 thousands of miles) Midpoint (x) 50 70 90 110 130 180 Frequency (f) 32 25 34 46 33 20 Discuss in your group: Is distance a discrete or a continuous variable? Look at the second class, 60 < d 80. What exactly does this mean? What are the upper and lower class boundaries? Show how the midpoints have been calculated It is not possible to say exactly how far each of the 190 buses travelled before experiencing a major breakdown using this data. Why not? Why do you think we are calculating midpoints of classes? Can you see how we are going to find the mean distance travelled before a breakdown? Why is this an estimate of the mean distance and not the actual mean distance? For help with this section you may refer to you text, page 440, and read example 10. (4) Use your knowledge gained from above to find estimation for: a. the modal class b. the median c. the mean age of the bus drivers. How could you use your TI83 to obtain this estimate for the mean from the grouped frequency table shown? Ages of bus drivers data, to the nearest year: Age (years) 21-25 26-30 31-35 36-40 41-45 46-50 51-55 Frequency (f) 11 14 32 27 29 17 7 Age (years) Frequency (f) Midpoint (x) fx Cumulative (f) 21-25 11 23 (f) 26-30 14 28 31-35 32 33 36-40 27 38 41-45 29 43 46-50 17 48 51-55 7 53 Using correct IB notation write a formula for calculating the mean of a set of data from a frequency table. See Ms. Makunja when you have generalized your formula. Practice: Exercise 18B.1 Page 429. Q. 4, 5, 7, 10, 13 Exercise 18B.2 Q. 14, 20, 22 Exercise 18B.3 Q 2, 7 (by hand and check with technology)

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posted: | 11/12/2011 |

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