| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Calculate range and interquartile range |
| Difficulty | Easy -1.2 This is a straightforward S1 question testing basic data handling skills: reading a stem-and-leaf diagram, finding quartiles using standard positions, applying a simple outlier formula, drawing a box plot, and calculating quartile skewness. All techniques are routine recall with no problem-solving or novel insight required. The multi-part structure is typical but each part is mechanical. |
| 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 |
| Stem | Leaf | |||||||||||
| 1 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 9 | 9 | 9 | (11) |
| 2 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | \(w\) | 9 | (10) | |
| 3 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 7 | 7 | 9 | (10) | |
| 4 | 1 | 1 | 2 | 3 | (4) | |||||||
| 5 | 1 | 9 | (2) | |||||||||
| 6 | 4 | (1) | ||||||||||
| \(Q _ { 1 }\) | \(Q _ { 2 }\) | \(Q _ { 3 }\) |
| \(x\) | 26 | \(y\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(w = 8\) | B1 | Cao. May be seen in table before part (a). \(w = 28\) is first B0 |
| \(x = 19\) | B1 | |
| \(y = 37\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{"37"} + 1 \times (\text{"37"} - \text{"19"}) [= 55]\) | M1 | Calculation for outliers using lower and upper quartile. Allow "their upper quartile" + "their IQR" i.e. \(37 + 18\) |
| 59 and 64 | A1ft | For identifying 59 and 64 as outliers from correct working. Ft identification of outlier(s) from "their 55". Answer only is M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Box with at least one whisker drawn | M1 | |
| 14 for lowest whisker, 26 for median, "19" and "37" for quartiles | A1ft | Ft their values for quartiles |
| Upper whisker at 51 plus outliers plotted (at least one outlier) | A1ft | Condone upper whisker at "their 55". NB award A0 if more than one whisker at either end |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\dfrac{(\text{"37"} - 26) - (26 - \text{"19"})}{(\text{"37"} - \text{"19"})}\) | M1 | For substituting their values into the formula |
| \(= 0.22\) (to 2 sf) | A1 | Allow \(\frac{2}{9}\) or \(0.\dot{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| E.g. "The mean uses all the data" | B1 | Correct reason supporting Landacre's use of mean or rejecting median. Allow comment on positive skew so mean > median. Comments on skewness/symmetry alone: B0. Mean includes outliers: B1. Condone "Median is not affected by outliers": B1. "Mean is more accurate": B0 |
# Question 1:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $w = 8$ | B1 | Cao. May be seen in table before part (a). $w = 28$ is first B0 |
| $x = 19$ | B1 | |
| $y = 37$ | B1 | |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{"37"} + 1 \times (\text{"37"} - \text{"19"}) [= 55]$ | M1 | Calculation for outliers using lower and upper quartile. Allow "their upper quartile" + "their IQR" i.e. $37 + 18$ |
| 59 and 64 | A1ft | For identifying 59 and 64 as outliers from correct working. Ft identification of outlier(s) from "their 55". Answer only is M0A0 |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Box with at least one whisker drawn | M1 | |
| 14 for lowest whisker, 26 for median, "19" and "37" for quartiles | A1ft | Ft their values for quartiles |
| Upper whisker at 51 plus outliers plotted (at least one outlier) | A1ft | Condone upper whisker at "their 55". NB award A0 if more than one whisker at either end |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\dfrac{(\text{"37"} - 26) - (26 - \text{"19"})}{(\text{"37"} - \text{"19"})}$ | M1 | For substituting their values into the formula |
| $= 0.22$ (to 2 sf) | A1 | Allow $\frac{2}{9}$ or $0.\dot{2}$ |
## Part (e)
| Answer | Mark | Guidance |
|--------|------|----------|
| E.g. "The mean uses all the data" | B1 | Correct reason supporting Landacre's use of mean or rejecting median. Allow comment on positive skew so mean > median. Comments on skewness/symmetry alone: B0. Mean includes outliers: B1. Condone "Median is not affected by outliers": B1. "Mean is more accurate": B0 |
---\begin{enumerate}
\item The company Seafield requires contractors to record the number of hours they work each week. A random sample of 38 weeks is taken and the number of hours worked per week by contractor Kiana is summarised in the stem and leaf diagram below.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Stem & \multicolumn{12}{|l|}{Leaf} \\
\hline
1 & 4 & 4 & 4 & 5 & 5 & 5 & 6 & 6 & 9 & 9 & 9 & (11) \\
\hline
2 & 1 & 2 & 2 & 3 & 3 & 4 & 4 & 4 & $w$ & 9 & & (10) \\
\hline
3 & 2 & 3 & 4 & 4 & 5 & 6 & 7 & 7 & 7 & 9 & & (10) \\
\hline
4 & 1 & 1 & 2 & 3 & & & & & & & & (4) \\
\hline
5 & 1 & 9 & & & & & & & & & & (2) \\
\hline
6 & 4 & & & & & & & & & & & (1) \\
\hline
\end{tabular}
\end{center}
Key : 3|2 means 32
The quartiles for this distribution are summarised in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
$Q _ { 1 }$ & $Q _ { 2 }$ & $Q _ { 3 }$ \\
\hline
$x$ & 26 & $y$ \\
\hline
\end{tabular}
\end{center}
(a) Find the values of $w , x$ and $y$
Kiana is looking for outliers in the data. She decides to classify as outliers any observations greater than
$$Q _ { 3 } + 1.0 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
(b) Showing your working clearly, identify any outliers that Kiana finds.\\
(c) Draw a box plot for these data in the space provided on the grid opposite.\\
(d) Use the formula
$$\text { skewness } = \frac { \left( Q _ { 3 } - Q _ { 2 } \right) - \left( Q _ { 2 } - Q _ { 1 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }$$
to find the skewness of these data. Give your answer to 2 significant figures.
Kiana's new employer, Landacre, wishes to know the average number of hours per week she worked during her employment at Seafield to help calculate the cost of employing her.\\
(e) Explain why Landacre might prefer to know Kiana's mean, rather than median, number of hours worked per week.
Turn over for a spare grid if you need to redraw your box plot.
\hfill \mbox{\textit{Edexcel S1 2022 Q1 [11]}}