Question 2 - A-Level Maths

CAIE S1 2018 June — Question 2 7 marks

Exam Board	CAIE
Module	S1 (Statistics 1)
Year	2018
Session	June
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Measures of Location and Spread
Type	Construct stem-and-leaf then find median and quartiles
Difficulty	Easy -1.2 This is a straightforward data handling question requiring basic statistical skills: reading ordered data to find median and quartiles (n=55 makes positions easy: 14th, 28th, 42nd), then applying a standard outlier formula. The data is already sorted and the stem-and-leaf construction is routine. No problem-solving or conceptual insight required—pure procedural recall.
Spec	2.02f Measures of average and spread

2 In a survey 55 students were asked to record, to the nearest kilometre, the total number of kilometres they travelled to school in a particular week. The results are shown below.

5	5	9	10	13	13	13	15	15	15	15
16	18	18	18	19	19	20	20	20	20	21
21	21	21	23	25	25	27	27	29	30	33
35	38	39	40	42	45	48	50	50	51	51
52	55	57	57	60	61	64	65	66	69	70

On the grid, draw a box-and-whisker plot to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{246c92f4-7603-43ff-8533-042a4be99a69-04_512_1596_900_262} An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
Show that there are no outliers.

Show mark scheme Show mark scheme source

Question 2(i):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(LQ = 18\), Median \(= 25\), \(UQ = 50\)	B1	Median correct
B1	LQ and UQ correct
[Box plot drawn]	B1	Quartiles and median plotted as box graph with linear scale, min 3 values
[Whiskers drawn]	B1ft	Whiskers drawn to correct endpoints with linear scale, not through box, not joining at top or bottom of box. Ft their UQ and LQ. Whiskers must be with ruler. If scale non-linear or non-existent, SCB1 if all 5 data values (quartiles and end points) have values shown and all are correct numerically and fulfil the 'box' and 'whiskers ruled line' requirements
[Axis label]	B1	Label to include 'distance or travelled' and 'km,' allow 'total km', linear scale, numbered at least \(5-70\)
Total: 5

Question 2(ii):

Method 1:

Answer	Marks	Guidance
\(1.5 \times IQR = 48\)	M1	Attempt to find \(1.5 \times\) their IQR and add to UQ or subtract from LQ
\(LQ - 48 = -ve\) (i.e. \(< 0\)), \(UQ + 48 = 98\) (i.e. \(> 70\))
Hence no outliers	A1	Correct conclusion from correct working, need both ends. No need to state comparisons

Method 2:

Answer	Marks	Guidance
\(LQ - 5 = 13 (< 48)\), \(70 - UQ = 20 (< 48)\)	M1	Compare their \(1.5 \times IQR (= 48) >\) gap (20) between UQ and max 70 or LQ and min 5
Hence no outliers	A1	Correct conclusion from correct working, need both ends. No need to state comparisons

## Question 2(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $LQ = 18$, Median $= 25$, $UQ = 50$ | **B1** | Median correct |
| | **B1** | LQ and UQ correct |
| [Box plot drawn] | **B1** | Quartiles and median plotted as box graph with linear scale, min 3 values |
| [Whiskers drawn] | **B1ft** | Whiskers drawn to correct endpoints with linear scale, not through box, not joining at top or bottom of box. Ft their UQ and LQ. Whiskers must be with ruler. If scale non-linear or non-existent, SCB1 if all 5 data values (quartiles and end points) have values shown and all are correct numerically and fulfil the 'box' and 'whiskers ruled line' requirements |
| [Axis label] | **B1** | Label to include 'distance or travelled' and 'km,' allow 'total km', linear scale, numbered at least $5-70$ |
| | **Total: 5** | |

## Question 2(ii):

**Method 1:**

$1.5 \times IQR = 48$ | M1 | Attempt to find $1.5 \times$ their IQR and add to UQ **or** subtract from LQ |

$LQ - 48 = -ve$ (i.e. $< 0$), $UQ + 48 = 98$ (i.e. $> 70$) | | |

Hence no outliers | A1 | Correct conclusion from correct working, need both ends. No need to state comparisons |

**Method 2:**

$LQ - 5 = 13 (< 48)$, $70 - UQ = 20 (< 48)$ | M1 | Compare their $1.5 \times IQR (= 48) >$ gap (20) between UQ and max 70 **or** LQ and min 5 |

Hence no outliers | A1 | Correct conclusion from correct working, need both ends. No need to state comparisons |

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Show LaTeX source

2 In a survey 55 students were asked to record, to the nearest kilometre, the total number of kilometres they travelled to school in a particular week. The results are shown below.

\begin{center}
\begin{tabular}{ r r r r r r r r r r r }
5 & 5 & 9 & 10 & 13 & 13 & 13 & 15 & 15 & 15 & 15 \\
16 & 18 & 18 & 18 & 19 & 19 & 20 & 20 & 20 & 20 & 21 \\
21 & 21 & 21 & 23 & 25 & 25 & 27 & 27 & 29 & 30 & 33 \\
35 & 38 & 39 & 40 & 42 & 45 & 48 & 50 & 50 & 51 & 51 \\
52 & 55 & 57 & 57 & 60 & 61 & 64 & 65 & 66 & 69 & 70 \\
\end{tabular}
\end{center}

(i) On the grid, draw a box-and-whisker plot to illustrate the data.\\
\includegraphics[max width=\textwidth, alt={}, center]{246c92f4-7603-43ff-8533-042a4be99a69-04_512_1596_900_262}

An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.\\
(ii) Show that there are no outliers.\\

\hfill \mbox{\textit{CAIE S1 2018 Q2 [7]}}

This paper (7 questions)

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Q1 3 Q2 7 Q3 6 Q4 8 Q5 8 Q6 8 Q7 10

5	5	9	10	13	13	13	15	15	15	15
16	18	18	18	19	19	20	20	20	20	21
21	21	21	23	25	25	27	27	29	30	33
35	38	39	40	42	45	48	50	50	51	51
52	55	57	57	60	61	64	65	66	69	70

5	5	9	10	13	13	13	15	15	15	15
16	18	18	18	19	19	20	20	20	20	21
21	21	21	23	25	25	27	27	29	30	33
35	38	39	40	42	45	48	50	50	51	51
52	55	57	57	60	61	64	65	66	69	70

5	5	9	10	13	13	13	15	15	15	15
16	18	18	18	19	19	20	20	20	20	21
21	21	21	23	25	25	27	27	29	30	33
35	38	39	40	42	45	48	50	50	51	51
52	55	57	57	60	61	64	65	66	69	70