| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Compare distributions using stem-and-leaf |
| Difficulty | Easy -1.2 This is a routine S1 statistics question testing standard skills: reading stem-and-leaf diagrams, finding quartiles using position formulas, applying the outlier definition, and drawing/comparing box plots. All parts follow textbook procedures with no problem-solving or novel insight required. The median condition in part (a) is straightforward arithmetic. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| Totals | Birch | Maple | Totals | |
| (2) | 98 | 2 | 57789 | (5) |
| (8) | 99965311 | 3 | 0266899 | (7) |
| (9) | 988763111 | 4 | \(111 \boldsymbol { k } 78\) | (6) |
| (9) | 777543210 | 5 | 0123444 | (7) |
| (3) | 765 | 6 | 346 | (3) |
| (3) | 654 | 7 | 07 | (2) |
| (1) | 5 | 8 | 00 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(k = 3\) | B1 | Cao |
| Answer | Marks | Guidance |
|---|---|---|
| \(Q_1 = 39\), \(Q_3 = 57\) | B1 B1 | B1 for \(Q_1\) correct; B1 for \(Q_3\) correct |
| Answer | Marks | Guidance |
|---|---|---|
| \("57" + 1.5 \times ("57" - "39")\) or \("39" - 1.5 \times ("57" - "39")\) | M1 | For either method correct or a correct value (ft their \(Q_1\) and \(Q_3\)) |
| \(84\) and \(12\), therefore only 1 outlier \([85]\) | A1 | Both limits correct and statement about the outlier or outlier given |
| Answer | Marks | Guidance |
|---|---|---|
| Box drawn with 2 whiskers | M1 | For box drawn with only 2 whiskers, only one at each end (condone median line missing) |
| Upper whisker ending at 76 (or 84 ft upper outlier limit) and lower whisker ending at 28 | M1 | |
| \(Q_1\), \(Q_2\) and \(Q_3\) plotted, with \(Q_2 = 48\) and ft their \(Q_1\) and \(Q_3\) | M1 | |
| Fully correct box plot with outlier correctly shown – must be only 1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| A correct difference of the medians with supporting figures e.g. On average Birch trees grow slightly taller as the median is larger \(48 > 45\) oe or a correct difference of the spread with supporting figures e.g. Maple has a greater spread/variation of heights as the range is larger \(55 > 48\) (excluding outlier) oe, or Birch has greater range \(57 > 55\) (with outlier) oe | B1ft | For a correct comment referring to heights, with reference to a correctly named statistic. Must include the figures compared. Allow 'grow more/bigger' to imply taller. Ignore any reference to skew. SC If Q1/Q3 are incorrect then allow a ft comment about spread referring to the difference in IQR if compared to 18 |
| Answer | Marks | Guidance |
|---|---|---|
| \(36\), \(a < x\) where \(43\), \(x\), \(45\) or \(54\), \(2a\), \(80\) | M1 | For either range correct. Allow \(72 \leqslant 2a \leqslant 80\) or \(27 \leqslant a \leqslant 40\) for \(54\), \(2a\), \(80\). Condone \(<\) rather than \(\leqslant\). May be seen as separate inequalities e.g. \(2a \leqslant 80\), \(2a \geqslant 54\). A final answer of \(36\), \(a\), \(40\) or \(36 < a < 40\) implies M1 |
| \(36\), \(a\), "43" and \(54\), \(2a\), \(80\) | A1ft | For both ranges correct ft their \(k\). Allow \(72 \leqslant 2a \leqslant 80\) or \(27 \leqslant a \leqslant 40\). Condone \(<\) rather than \(\leqslant\). May be seen as separate inequalities |
| \(36\), \(a\), \(40\) | A1 | Allow 36 to 40 or 36, 37, 38, 39, 40. NB It is possible to get M1A0A1 |
# Question 1:
## Part (a)
| $k = 3$ | B1 | Cao |
|---|---|---|
**Total: (1)**
## Part (b)
| $Q_1 = 39$, $Q_3 = 57$ | B1 B1 | B1 for $Q_1$ correct; B1 for $Q_3$ correct |
|---|---|---|
**Total: (2)**
## Part (c)
| $"57" + 1.5 \times ("57" - "39")$ or $"39" - 1.5 \times ("57" - "39")$ | M1 | For either method correct or a correct value (ft their $Q_1$ and $Q_3$) |
|---|---|---|
| $84$ **and** $12$, therefore only 1 outlier $[85]$ | A1 | Both limits correct and statement about the outlier or outlier given |
**Total: (2)**
## Part (d)
| Box drawn with 2 whiskers | M1 | For box drawn with only 2 whiskers, only one at each end (condone median line missing) |
|---|---|---|
| Upper whisker ending at 76 (or 84 ft upper outlier limit) and lower whisker ending at 28 | M1 | |
| $Q_1$, $Q_2$ and $Q_3$ plotted, with $Q_2 = 48$ and ft their $Q_1$ and $Q_3$ | M1 | |
| Fully correct box plot with outlier correctly shown – must be only 1 | A1 | |
**Total: (4)**
## Part (e)
| A correct difference of the **medians** with supporting figures e.g. On average Birch trees grow slightly **taller** as the **median** is larger $48 > 45$ oe **or** a correct difference of the **spread** with supporting figures e.g. Maple has a greater spread/variation of **heights** as the **range** is larger $55 > 48$ (excluding outlier) oe, or Birch has greater range $57 > 55$ (with outlier) oe | B1ft | For a correct comment **referring to heights**, with reference to a **correctly named statistic**. Must **include the figures** compared. Allow 'grow more/bigger' to imply taller. Ignore any reference to skew. **SC If Q1/Q3 are incorrect then allow a ft comment about spread referring to the difference in IQR if compared to 18** |
|---|---|---|
**Total: (1)**
## Part (f)
| $36$, $a < x$ where $43$, $x$, $45$ **or** $54$, $2a$, $80$ | M1 | For either range correct. Allow $72 \leqslant 2a \leqslant 80$ or $27 \leqslant a \leqslant 40$ for $54$, $2a$, $80$. Condone $<$ rather than $\leqslant$. May be seen as separate inequalities e.g. $2a \leqslant 80$, $2a \geqslant 54$. A final answer of $36$, $a$, $40$ or $36 < a < 40$ implies M1 |
|---|---|---|
| $36$, $a$, "43" and $54$, $2a$, $80$ | A1ft | For both ranges correct ft their $k$. Allow $72 \leqslant 2a \leqslant 80$ or $27 \leqslant a \leqslant 40$. Condone $<$ rather than $\leqslant$. May be seen as separate inequalities |
| $36$, $a$, $40$ | A1 | Allow 36 to 40 or 36, 37, 38, 39, 40. **NB** It is possible to get M1A0A1 |
**Total: (3)**
### Question 1 Total: **13 marks**\begin{enumerate}
\item A researcher is investigating the growth of two types of tree, Birch and Maple. The height, to the nearest cm, a seedling grows in one year is recorded for 35 Birch trees and 32 Maple trees. The results are summarised in the back-to-back stem and leaf diagram below.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Totals & \multicolumn{2}{|c|}{Birch} & Maple & Totals \\
\hline
(2) & 98 & 2 & 57789 & (5) \\
\hline
(8) & 99965311 & 3 & 0266899 & (7) \\
\hline
(9) & 988763111 & 4 & $111 \boldsymbol { k } 78$ & (6) \\
\hline
(9) & 777543210 & 5 & 0123444 & (7) \\
\hline
(3) & 765 & 6 & 346 & (3) \\
\hline
(3) & 654 & 7 & 07 & (2) \\
\hline
(1) & 5 & 8 & 00 & (2) \\
\hline
\end{tabular}
\end{center}
Key: 5 | 6 | 3 means 65 cm for a Birch tree and 63 cm for a Maple tree\\
The median height that these Maple trees grow in one year is 45 cm .\\
(a) Find the value of $\boldsymbol { k }$, used in the stem and leaf diagram.\\
(b) Find the lower quartile and the upper quartile of the height grown in one year for these Birch trees.
The researcher defines an outlier as an observation that is
$$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { or less than } Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
(c) Show that there is only one outlier amongst the Birch trees.
The grid on page 3 shows a box plot for the heights that the Maple trees grow in one year.\\
(d) On the same grid draw a box plot for the heights that the Birch trees grow in one year.\\
(e) Comment on any difference in the distributions of the growth of these Birch trees and the growth of these Maple trees.\\
State the values of any statistics you have used to support your comment.
The researcher realises he has missed out 4 pieces of data for the Maple trees. The heights each seedling grows in one year, to the nearest cm, in ascending order, for these 4 Maple trees are $27 \mathrm {~cm} , a \mathrm {~cm} , 48 \mathrm {~cm} , 2 a \mathrm {~cm}$.
Given that there is no change to the box plot for the Maple trees given on page 3\\
(f) find the range of possible values for $a$
Show your working clearly.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-03_1243_1659_1464_210}
\end{center}
Only use this grid if you need to redraw your answer for part (d)\\
\includegraphics[max width=\textwidth, alt={}, center]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-05_1154_1643_1503_217}\\
(Total for Question 1 is 13 marks)\\
\hfill \mbox{\textit{Edexcel S1 2024 Q1 [13]}}