Stat 218 - Day 3 Measures of Center Measures of Center: Mean, Median Mean: arithmetic average Median: middle value Position (n+1)/2 if n is odd, average of position n/2 and n/2+1 if n is even Example: Rower weights (rowers04.mtw) (a) Before doing any calculations, how do you expect the mean and median weights of the rowers to compare? Explain. (b) Calculate the mean and median of the rower weights ((MTB> mean c2, MTB> median c2). How do they compare? Why does this make sense? (c) Eliminate the coxswain. How does this affect the mean and median? (d) Eliminate the lightweights. How does this affect the mean and median? (e) Make the heaviest rower much heavier, first by 20 pounds and then by 1000 pounds (as if the number entered were a typographical error). How does this affect the mean and median? A numerical summary is resistant if it is not very affected by outliers. (f) Which is a resistant measure of center- mean or median? (g) If a distribution is skewed to the right, how do you expect the mean to compare to the median? Explain. (h) Repeat (f) for a distribution that is skewed to the left.(i) Repeat (f) for a distribution that is symmetric. Example: Cancer pamphlets Researchers in Philadelphia investigated whether pamphlets containing information for cancer patients are written at a level that the cancer patients can comprehend. They applied tests to measure the reading levels of 63 cancer patients and also the readability levels of 30 cancer pamphlets (based on such factors as the lengths of sentences and number of polysyllabic words). These numbers correspond to grade levels, but patient reading levels of under grade 3 and above grade 12 are not determined exactly. The tallies in the following table indicate the number of patients at each reading level and the number of pamphlets at each readability level. patients’ Pamphlets’ reading level tally Readability tally under 3 6 6 3 3 4 7 3 4 4 8 8 5 3 9 4 6 3 10 1 7 2 11 1 8 6 12 4 9 5 13 2 10 4 14 1 11 7 15 2 12 2 16 1 above 12 17 TOTAL 30 TOTAL 63 (a) Explain why the form of the data do not allow one to calculate the mean reading skill level of a patient. (b) Determine the median reading level of a patient and the median readability level of a pamphlet. (c) How do these medians compare? Are they fairly close? (d) Does the closeness of these medians indicate that the pamphlets are well matched to the patients’ reading levels?(e) What proportion of the patients do not have the reading skill level necessary to read even the simplest pamphlet in the study? Example: Planetary measurements The following table lists the average distance from the sun (in millions of miles) and diameter (in miles) for the nine planets of our solar system. planet distance diameter Mercury 36 3,030 Venus 67 7,520 Earth 93 7,926 Mars 142 4,217 Jupiter 484 88,838 Saturn 887 74,896 Uranus 1,765 31,762 Neptune 2,791 30,774 Pluto 3,654 1,428 (a) Calculate the median distance from the sun. (Be careful to report the appropriate units with your answer.) (b) Without doing any calculations, would you expect the mean distance to be close to, less than, or greater than the median distance? Explain. (c) If a classmate calculates the median diameter to be 88,838 miles, what do you think would be the most likely cause of his/her mistake? Example: (a) A real estate agent notes that the mean housing price for an area is $425,780 and concludes that half of the houses in the area cost more than that. Is this conclusion valid? Explain.(b) A businesswoman calculates that the median cost of the five business trips that she took in a month is $600 and concludes that the total cost must have been $3000. Is this conclusion valid? Explain. If it is not, demonstrate this with a hypothetical example. (c) A company executive concludes that an accountant must have made a mistake because she reported that 90% of the company’s employees earn less than the mean salary. Is this conclusion valid? Explain. If it is not, demonstrate this with a hypothetical example. (d) I claimed that when I moved from Pennsylvania to California in 2001, the mean IQ dropped in both states. Is this possible? If so, how could it happen? (e) Does is make sense to calculate the mean day of the week on which students were born?