Measures of Central Tendency

Transcription

1 12 Measures of Central Tendency Quick Review 1. Mean, Mode and Median of Ungrouped Data (a) Mean: average of all data (b) Mode: the data with the highest frequency (c) Median: the middle data of an ordered set of data 2. Mean, Modal Class and Median of Grouped Data fx 1 1+ fx fx n n (a) Mean = f1 + f2 + + fn where f n is the frequency of the nth group and x n is the class mark of the nth group (b) Modal class: the class with the highest frequency (c) Median: read from cumulative frequency polygon / curve 3. Weighted Mean wx 1 1+ wx wx n n Weighted mean =, w1 + w2 + + wn where w n is the weight for the nth data 4. Comparison of Two Sets of Data E.g., The mean, median and mode of two sets of marks of students: Data Mean Median Mode Set Set Since the mean and median of set 2 are higher, we can conclude that the overall performance of students in set 2 is better than that of set 1. Concept Check Determine whether each of the following is true (1 4). 1. Mean is the best measure of central tendency when there is an extreme value in the data set. T / F 2. Consider the data set {0, 0, 1, 2, 2, 3, 3, 5}. When a datum {2} is added to the data set, the mean, mode and median are all equal to 2. T / F 3. Consider the following frequency distribution table of the studying time of a class of 30 students. Studying time (h) Frequency Then, the modal class is 10. T / F 4. If 4 is added to each of the datum in NF a data set, then the mean, mode and median are increased by 4. T / F 5. Find the mean, mode and median of the following data. (a) -2, 1, 1, 3, 5, 8, 12 (b) 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 5, 8, 8, 13 Ans: 1. F 2. T 3. F 4. T 5. Mean Mode Median (a) (b)

2 5. Relative Merits of Different Averages Mean is a good representation if there are no extreme values in a set of data, and it is used more often than the median and mode. Median is a good representation if there are extreme values in a set of data. Mode is a good representation for popular items. E.g., we would use the mode to represent popular size of shoes. Mistake Hunt 1. Find the median of the data set {13, 21, 5, 4, 18, 33, 17, 6, 4}. Median = the 5 th datum = 18 Before finding the median, we MUST first arrange the data in ascending order Level-up Practice Q. 6 of their values. The data set becomes {4, 4, 5, 6, 13, 17, 18, 21, 33}. Therefore, median = The following shows the frequency distribution table of the marks of a test in a class of 40 students. Marks Class mark Frequency Find the mean marks of the students in the test Mean = 5 = 65.5 Level-up Practice Q. 7 For grouped data, we cannot simply take the average of the class marks as the mean. We should use the formula in Quick Review 2. Hence the correct solution is as follows: Mean = 40 = Measures of Central Tendency 79

3 Study each of the following and determine whether the solution of each question is correct. If it is not, correct the solution. 1. Find the median of the data set {14, 21, 5, 19, 34, 18, 31, 21, 60, 24}. Median = the 5.5 th datum = = The following shows the frequency distribution table of the heights of 20 students in a class. Find the mean of the heights of 20 students in the class. Height (cm) Class mark (cm) Frequency Mean = What's Wrong? cm = cm Level-up Practice Multiple-choice Questions 1. The manager of a sports shoes company wants to find out the most popular shoes sizes. Which of the following statistics on the sales of shoes is suitable for the manager? A. Mean B. Mode C. Median D. Range 2. The mean of 10 data is calculated to be 48.7, with a datum 45 wrongly entered as 55. What is the correct mean of the 10 data? A B C D If the mean, mode and median of two sets of data are the same, which of the following must be true? I. All the data in both sets are the same. II. Two sets of data contain the same number of data. III. Both frequency curves of the two sets of data are symmetric. A. I only B. II and III only C. III only D. None of the above 4. A student scores 48, 27 and 63 in the Chinese, English and Mathematics exam whose weights are 3, 4 and 2 respectively. The weighted mean score of the student is A B. 42. C. 44. D The median of 10 numbers is x. If 3 is added to each datum, what is the new median? NF A. x - 3 B. x C. x + 1 D. x

4 Basic Questions 6. Find the mean, median and mode of the data set {23, 24, 26, 33, 34, 38, 38, 40}. 7. The following table shows the part-time working hours of 40 college students during last week. Working hour (h) Number of college students Find the mean working hours of the college students during last week. 8. The following table shows the weekly pocket money of 40 students in a class. Weekly pocket money ($) Frequency Find the modal class of the weekly pocket money of 40 students in the class. 9. Sam joined a dancing competition in the school. The following table shows the weight of each marking item and the marks that he got in these items. Marking item Skill Artistry Difficulty Weight Mark Find the weighted mean of Sam. Measures of Central Tendency 81

5 10. Calculate the mean, mode and median of the score in the table below. Score Number of students Enhanced Questions 11. 2, 3, 5, 5, a, b, 9, 9, 10, 12 The above data are arranged in ascending order. The median and mode are 8 and 9 respectively. (a) Find the values of a and b. (b) Find the mean of the data. 12. Consider the data 7, 5, 4, 9, 3, 4, 5, 7, 6, 4, 8, 5, 1, 2. (a) For the above set of data, mean = ; mode = ; median = (b) Multiply each datum by 2 and then add the result by 3. Find the new values of mean, mode and median. NF mean = ; mode = ; median = 13. The following stem-and-leaf diagram shows the studying time (in hours) of 30 students before the examination. Stem (10 hours) Leaf (1 hour)

6 (a) Find the mean, median and mode of the studying time. (b) If a student, whose studying time is 29 hours, is added to the above stem-and-leaf NF diagram, what will be the changes in the mean, mode and median of the studying time? Self-assessment 1. Susie buys 6 gifts for a party. The following shows the prices (in $) of the 6 gifts. 29.8, 18.4, 16, 29.8, 40, 28 Find the mean and median of the above data. 2. The marks of past tests of Cathy and the weights of the tests are shown below. Mark Weight Test Test Test Test Find the weighted mean mark of the tests for Cathy. 3. The mean, median and mode a set of data are 18, 19 and 20 respectively. If each datum is NF multiplied by 3, what is the mean, median and mode of the new data set? Measures of Central Tendency 83

7 Challenging Question The following frequency curves show the results of a test for F.3A and F.3B. (a) Fill in the blank with symbols > or <. Mean mark of 3A mean mark of 3B Median mark of 3A median mark of 3B Mode mark of 3A mode mark of 3B (b) By comparing the curves, which class performs better? Explain your answer. Internet Adventure Interesting Statistics Many statistical facts are interesting. In the following website, you can see a lot of interesting comics related to statistics. 84