| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Construct back-to-back stem-and-leaf from raw data |
| Difficulty | Easy -1.2 This is a straightforward multi-part question testing basic statistical skills: constructing a stem-and-leaf diagram from given data (routine), finding IQR by identifying quartiles (standard procedure), and calculating variance using the computational formula with coded data. All parts involve direct application of well-practiced techniques with no problem-solving or conceptual challenges beyond standard S1 level. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Thaters School | 38 | 43 | 48 | 52 | 54 | 56 | 57 | 58 | 58 | 61 | 62 | 66 | 75 |
| Whitefay Park School | 45 | 47 | 53 | 56 | 56 | 61 | 64 | 66 | 69 | 73 | 75 | 78 | 83 |
| Answer | Marks | Guidance |
|---|---|---|
| \[\begin{array}{r | c | l}\text{Thaters School} & & \text{Whitefay Park School}\\ 8 & 3 & \\ 8\ 3 & 4 & 5\ 7\\ 8\ 8\ 7\ 6\ 4\ 2 & 5 & 3\ 6\ 6\\ 6\ 2\ 1 & 6 & 1\ 4\ 6\ 9\\ 5 & 7 & 3\ 5\ 8\\ & 8 & 3\end{array}\] |
| Key: \(8\ | \ 4\ | \ 5\) represents 48 minutes for Thaters School and 45 minutes for Whitefay Park School |
| Correct stem | B1 | Can be upside down, ignore extra values |
| Correct Thaters School labelled on left, leaves in order right to left, lined up vertically, no commas | B1 | |
| Correct Whitefay Park School labelled on right, in order left to right, lined up vertically, no commas | B1 | |
| Correct key for *their* diagram, both teams identified, 'minutes' stated at least once | B1 | SC if 2 separate diagrams drawn, SCB1 if both keys meet criteria |
| Answer | Marks | Guidance |
|---|---|---|
| \(LQ = 50\), \(UQ = 61.5\) | B1 | Both quartiles correct |
| \(IQ\ \text{range} = 61.5 - 50 = 11.5\) | B1 | FT \(61 \leqslant UQ \leqslant 62 - 48 \leqslant LQ \leqslant 52\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Sigma(x-60)^2 = (-15)^2+(-13)^2+(-7)^2+(-4)^2+(-4)^2+1^2+4^2+6^2+9^2+13^2+23^2+15^2+18^2\) | M1 | Summing squares with at least 5 correct unsimplified terms |
| \(= 1856\) | A1 | Exact value |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Var} = \frac{\Sigma(x-60)^2}{13} - \left(\frac{\Sigma(x-60)}{13}\right)^2\) | M1 | Using two coded values in correct formula (variance or sd) |
| \(\text{Var} = \frac{\textit{their}\ 1856}{13} - \left(\frac{46}{13}\right)^2 = 130\) | A1 | Correct answer; SC if correct variance obtained by another method give SCB1 |
## Question 7:
### Part 7(i):
$$\begin{array}{r|c|l}\text{Thaters School} & & \text{Whitefay Park School}\\ 8 & 3 & \\ 8\ 3 & 4 & 5\ 7\\ 8\ 8\ 7\ 6\ 4\ 2 & 5 & 3\ 6\ 6\\ 6\ 2\ 1 & 6 & 1\ 4\ 6\ 9\\ 5 & 7 & 3\ 5\ 8\\ & 8 & 3\end{array}$$
Key: $8\ |\ 4\ |\ 5$ represents 48 minutes for Thaters School and 45 minutes for Whitefay Park School
Correct stem | B1 | Can be upside down, ignore extra values
Correct Thaters School labelled on left, leaves in order right to left, lined up vertically, no commas | B1 |
Correct Whitefay Park School labelled on right, in order left to right, lined up vertically, no commas | B1 |
Correct key for *their* diagram, both teams identified, 'minutes' stated at least once | B1 | SC if 2 separate diagrams drawn, SCB1 if both keys meet criteria
**Total: 4 marks**
### Part 7(ii):
$LQ = 50$, $UQ = 61.5$ | B1 | Both quartiles correct
$IQ\ \text{range} = 61.5 - 50 = 11.5$ | B1 | FT $61 \leqslant UQ \leqslant 62 - 48 \leqslant LQ \leqslant 52$
**Total: 2 marks**
### Part 7(iii):
$\Sigma(x-60)^2 = (-15)^2+(-13)^2+(-7)^2+(-4)^2+(-4)^2+1^2+4^2+6^2+9^2+13^2+23^2+15^2+18^2$ | M1 | Summing squares with at least 5 correct unsimplified terms
$= 1856$ | A1 | Exact value
**Total: 2 marks**
### Part 7(iv):
$\text{Var} = \frac{\Sigma(x-60)^2}{13} - \left(\frac{\Sigma(x-60)}{13}\right)^2$ | M1 | Using two coded values in correct formula (variance or sd)
$\text{Var} = \frac{\textit{their}\ 1856}{13} - \left(\frac{46}{13}\right)^2 = 130$ | A1 | Correct answer; SC if correct variance obtained by another method give SCB1
**Total: 2 marks**7 The times in minutes taken by 13 pupils at each of two schools in a cross-country race are recorded in the table below.
\begin{center}
\begin{tabular}{ | l | l | l | l | l | l | l | l | l | l | l | l | l | l | }
\hline
Thaters School & 38 & 43 & 48 & 52 & 54 & 56 & 57 & 58 & 58 & 61 & 62 & 66 & 75 \\
\hline
Whitefay Park School & 45 & 47 & 53 & 56 & 56 & 61 & 64 & 66 & 69 & 73 & 75 & 78 & 83 \\
\hline
\end{tabular}
\end{center}
(i) Draw a back-to-back stem-and-leaf diagram to illustrate these times with Thaters School on the left.\\
(ii) Find the interquartile range of the times for pupils at Thaters School.\\
The times taken by pupils at Whitefay Park School are denoted by $x$ minutes.\\
(iii) Find the value of $\Sigma ( x - 60 ) ^ { 2 }$.\\
(iv) It is given that $\Sigma ( x - 60 ) = 46$. Use this result, together with your answer to part (iii), to find the variance of $x$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE S1 2019 Q7 [10]}}