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The number of cars parked in a carpark is recorded each day for thirteen days. The dataset is listed in ascending order:
\(12,15,15,17,19,20,22,24,25,27,28,30\ \text{and}\ m.\)
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a. \(11.5\)
b. \(44\)
a. \(\text{Dataset has 13 numbers.}\)
\(\text{Median = 8th value = 22}\)
\(Q_1=\dfrac{15+17}{2}=16,\quad Q_3=\dfrac{27+28}{2}=27.5\)
\(IQR=27.5-16 = 11.5\)
b. \(\text{Find the upper fence:}\)
\(\text{Upper fence} = Q_3+1.5\times IQR=27.5+1.5(11.5)=44.75\)
\(\text{Since $m$ is a whole number and must be below 44.75.}\)
\(m_\text{max}=44\)
A dataset has an interquartile range \( (IQR)\) of 18.
The upper quartile \((Q_3)\) is 45.
What is the maximum value that would NOT be classified as an outlier?
\(D\)
\(1.5 \times \text{IQR} = 1.5 \times 18 = 27\)
\(\text{Upper boundary} = Q_3 + 27 = 45 + 27 = 72\)
\(\text{Values above 72 are outliers.}\)
\(\therefore\ \text{Maximum value that is not an outlier} = 72\)
\(\Rightarrow D\)
The following data shows the ages of people attending a fitness class:
16, 24, 26, 28, 30, 32, 34, 36, 52, 62
How many outliers are in this dataset?
\(C\)
\(\text{Arrange in order (already ordered)}\ n = 10 \)
\(\text{Median }(Q_2) = \dfrac{30 + 32}{2} = 31 \)
\(Q_1 = 26\)
\(Q_3 =36\)
\(\text{IQR} = 36-26 = 10 \)
\(1.5 \times \text{IQR} = 1.5 \times 10 = 15\)
\(\text{Lower boundary} = 26-15 = 11 \)
\(\text{Upper boundary} = 36+15 = 51\)
\(\text{Values 52 and 62 are both above 51} \therefore\ 2 \text{ outliers}\)
\(\Rightarrow C\)
A dataset has a lower quartile (Q1) of 25 and an upper quartile (Q3) of 35.
Which of the following values would be classified as an outlier?
\(A\)
\(\text{IQR}=Q_3-Q_1\)
\(=35-25=10\)
\(\text{Lower boundary}=25-1.5\times 10=10\)
\(\text{Upper boundary}=35+1.5\times 10=50\)
\(\text{Check each value:} \)
\(2 < 10 \text{ (outlier)} \)
\(10 = 10 \text{ (on boundary, not an outlier)}\)
\(48 < 50 \text{ (not an outlier)} \)
\(50 = 50 \text{ (on boundary, not an outlier)}\)
\(\therefore\ 2\ \text{is the only outlier}\)
\(\Rightarrow A\)
A dataset has the following values arranged in order:
12, 15, 18, 21, 24, 27, 30, 33, 36
What is the interquartile range (IQR) of this dataset?
\(B\)
\(\text{Number of data values: }n=9\)
\(\text{Median }(Q_2): =24\)
\(\text{Lower half of data: }12, 15, 18, 21\)
\(Q_1=\dfrac{15+18}{2}\ =16.5\)
\(\text{Upper half of data: } 27, 30, 33, 36\)
\(Q_3=\dfrac{30+33}{2}\ =31.5\)
\(\text{IQR}=Q_3-Q_1\)
\(=31.5-16.5=15\)
\(\Rightarrow C\)
The five-number summary of a dataset is given.
Lowest score = 1
Lowest quartile (`Q_1`) = 4
Median (`Q_2`) = 7
Upper quartile (`Q_3`) = 10
Highest score = 20
Is 20 an outlier? Justify your answer with calculations. (2 marks)
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`text(20 is an outlier – See Worked Solution)`
`IQR = Q_3-Q_1 = 10-4=6`
| `text(Upper Fence)` | `= Q_3 + 1.5 xx IQR` |
| `=10 + 1.5 xx 6=19` |
`text{S}text{ince 20 > 19, 20 is an outlier.}`
The heights, in centimetres, of 10 players on a basketball team are shown.
170, 180, 185, 188, 192, 193, 193, 194, 196, 202
Is the height of the shortest player on the team considered an outlier? Justify your answer with calculations. (3 marks)
`text(See Worked Solutions)`
`Q_1 = 185, \ Q_3 = 194`Mean mark 51%.
COMMENT: The last statement must be made to achieve full marks here!
`IQR = 194 – 185 = 9`
`text(Shortest player = 170)`
`text(Outlier height:)`
| `Q_1 – 1.5 xx IQR ` | `= 185 – 1.5 xx 9` |
| `= 171.5` |
`:.\ text(S)text(ince 170 < 171.5, 170 is an outlier.)`
A cumulative frequency table for a data set is shown.
\begin{array} {|c|c|}
\hline
\ \ \ \ \ \ \ \textit{Score}\ \ \ \ \ \ \ & \ \ \ \ \ \textit{Cumulative}\ \ \ \ \ \\ & \textit{frequency} \\
\hline
\rule{0pt}{2.5ex} \text{1} \rule[-1ex]{0pt}{0pt} & 5 \\
\hline
\rule{0pt}{2.5ex} \text{2} \rule[-1ex]{0pt}{0pt} & 9 \\
\hline
\rule{0pt}{2.5ex} \text{3} \rule[-1ex]{0pt}{0pt} & 16 \\
\hline
\rule{0pt}{2.5ex} \text{4} \rule[-1ex]{0pt}{0pt} & 20 \\
\hline
\rule{0pt}{2.5ex} \text{5} \rule[-1ex]{0pt}{0pt} & 34 \\
\hline
\rule{0pt}{2.5ex} \text{6} \rule[-1ex]{0pt}{0pt} & 42 \\
\hline
\end{array}
What is the interquartile range of this data set? (2 marks)
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`2`
`text(42 data points ⇒ median) = text(21st + 22nd)/2`♦♦ Mean mark 27%.
`text(Q)_1= 11text(th data point) = 3`
`text(Q)_3= 32text(nd data point) = 5`
`:.\ text(IQR)= 5-3=2`
A set of data has a lower quartile (`Q_L`) of 10 and an upper quartile (`Q_U`) of 16.
What is the maximum possible range for this set of data if there are no outliers? (2 marks)
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`24`
`IQR = 16-10 = 6`
`text(If no outliers,)`
| `text(Upper limit)` | `= Q_U + 1.5 xx IQR` |
| `= 16 + 1.5 xx 6= 25` |
| `text(Lower limit)` | `= Q_L-1.5 xx IQR` |
| `= 10-1.5 xx 6= 1` |
`:.\ text(Maximum range)= 25-1=24`
In a small business, the seven employees earn the following wages per week:
\(\$300, \ \$490, \ \$520, \ \$590, \ \$660, \ \$680, \ \$970\)
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What effect will this have on the standard deviation? (1 mark)
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a. \(\text{See Worked Solutions.} \)
b. \(\text{The standard deviation will remain the same.}\)
a. \(300, 490, 520, 590, 660, 680, 970\)
\(\text{Median}= 590\)
\(Q_1= 490,\ Q_3= 680\)
\(IQR= 680-490 = 190\)
\(\text{Outlier if \$970 is greater than:} \)
\(Q_3 + 1.5 \times IQR = 680 + 1.5 \times 190 = \$965 \)
\(\therefore\ \text{The wage \$970 per week is an outlier.}\)
b. \(\text{All values increase by \$20, but so too does the mean.} \)
\(\text{Therefore the spread about the new mean will not change} \)
\(\text{and therefore the standard deviation will remain the same.} \)
This set of data is arranged in order from smallest to largest.
`5, \ 6, \ 11, \ x, \ 13, \ 18, \ 25`
The range is six less than twice the value of `x`.
Which one of the following is true?
`D`
`5, 6, 11, x, 13, 18, 25`
| `text(Range)` | `= 2x-6` |
| `25-5` | `= 2x-6` |
| `2x` | `= 26` |
| `x` | `= 13` |
| `:.\ text(Median)` | `= 13` |
`Q_1 = 6\ \ \ \ \ Q_3 = 18`
`:.\ text(IQR) = 12`
`=> D`