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Worksheet
Averages
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Name
Question 1
Find the mean, median, and mode of the following data sets:
(a)\(3,\;7,\;5,\;2,\;8,\;5,\;4,\;9,\;5,\;2\)
(b)\(12,\;15,\;11,\;18,\;14,\;16,\;13\)
(a)
Sort: 2,2,3,4,5,5,5,7,8,9
Mean \(= frac{50}{10}=5\); Median \(= frac{5+5}{2}=5\); Mode \(=5\)
Answer
Mean \(5\), Median \(5\), Mode \(5\)
(b)
Sort: 11,12,13,14,15,16,18
Mean \(= frac{99}{7}= frac{99}{7}pprox14.1\); Median \(=14\); Mode: none
Answer
Mean \(pprox14.1\), Median \(14\), No mode
Question 2
The following frequency table shows scores in a test:
\[egin{array}{c|ccccc} ext{Score}&2&3&4&5&6\\hline ext{Frequency}&3&7&12&6&2\end{array}\]
(a)Find the mean score.
(b)Find the median score.
(c)Find the modal score.
(a)
Weighted mean
\(\dfrac{6+21+48+30+12}{30}=\dfrac{117}{30}=3.9\)
Answer
\(3.9\)
(b)
Cumulative freq: 3,10,22,28,30. Median = average of 15th and 16th values
Both in score 4 group
Answer
\(4\)
(c)
Answer
\(4\) (highest frequency)
Question 3
The mean of six numbers is 14. Five of the numbers are 10, 16, 12, 18, 15. Find the sixth number.
Sum of all six
\(6 imes14=84\); sum of five \(=71\)
Answer
\(84-71=13\)
Question 4
The grouped frequency table shows ages of visitors to a museum:
\[egin{array}{c|c} ext{Age (years)}& ext{Frequency}\\hline10\leq a<20&8\20\leq a<30&15\30\leq a<40&22\40\leq a<50&11\50\leq a<60&4\end{array}\]
(a)Estimate the mean age.
(b)State the modal class.
(a)
Midpoints: 15,25,35,45,55
\(ar{x}=\dfrac{8(15)+15(25)+22(35)+11(45)+4(55)}{60}=\dfrac{120+375+770+495+220}{60}=\dfrac{1980}{60}\)
Answer
\(33\) years
(b)
Answer
\(30\leq a<40\)
Question 5
When is the median a more appropriate average than the mean? Give an example.
When data is skewed or has outliers
The mean is pulled by extreme values; the median is resistant to outliers.
Example
House prices in a neighbourhood — a few very expensive properties inflate the mean, making the median a better measure of a "typical" price.
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