### Stem-and-leaf diagram, mean, median

Question Sample Titled 'Stem-and-leaf diagram, mean, median'

Noah conducts a survey on the numbers of micro films watched in a month by students. Questionnaires are sent out and twenty three of them are returned. The stem-and-leaf diagram below shows the numbers of micro films recorded in the twenty three questionnaires:

Stem (tens)Leaf (units)
 ${0}$ ${1}$ ${2}$
 ${0}$ ${0}$ ${0}$ ${2}$ ${3}$ ${4}$ ${4}$ ${4}$ ${4}$ ${5}$ ${5}$ ${6}$ ${6}$ ${7}$ ${8}$ ${8}$ ${8}$ ${0}$ ${1}$ ${1}$ ${3}$ ${8}$ ${4}$

 (a) Find the mean and the median of the numbers of micro films recorded in the twenty three questionnaires. (2 marks) (b) Noah receives six more questionnaires. He finds that the mean of the numbers of micro films recorded in these six questionnaires is ${7}$. It is found that the numbers of micro films recorded in three of these six questionnaires are ${8}$ , ${8}$ and ${9}$ .  (4 marks) (i) Write down the mean of the numbers of micro films recorded in the twenty nine questionnaires. (ii) Is it possible that the median of the numbers of micro films recorded in the twenty nine questionnaires is the same as the median found in (a)? Explain your answer.

 (a) The mean$={7}$ 1A The median$={6}$ 1A (b) (i) The mean$={7}$ 1A (ii) Let ${a}$ , ${b}$ and ${c}$ be the numbers of micro films recorded in the three other questionnaires. Note that $\dfrac{{{a}+{b}+{c}+{8}+{8}+{9}}}{{6}}={7}$ 1M Therefore, we have ${a}+{b}+{c}={17}$ If the two medians are the same, then we have ${a}\le{6}$ , ${b}\le{6}$ and ${c}\le{6}$ . 1M i.e. ${a}+{b}+{c}\le{18}$ . It is possible since ${a}+{b}+{c}={17}$ . Thus, it is possible that the two medians are the same.  1A f.t.

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