| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | October |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Find median and quartiles from stem-and-leaf diagram |
| Difficulty | Easy -1.2 This is a straightforward S1 question testing standard procedures: reading a stem-and-leaf diagram, finding median/quartiles using position formulas for n=19, calculating IQR, applying the 1.5×IQR outlier rule, drawing a box plot, and computing mean/standard deviation from given summations. All steps are routine recall and mechanical calculation with no problem-solving or conceptual insight required. Easier than average A-level maths. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| 10 | 1 | 2 | |||||||
| 11 | 2 | 7 | 7 | 8 | 8 | ||||
| 12 | 0 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 7 |
| 13 | 1 | 2 | 9 |
| Answer | Marks | Guidance |
|---|---|---|
| 6(a) | \(Q_1 = 117\), \(Q_2 = 122\), \(Q_3 = 125\) | B1, B1, B1 |
| 6(b) | \(\text{IQR} = 125 - 117 = 8\) | B1ft |
| 6(c) | Upper limit: \(125 + 1.5 \times 8 = 125 + 12 = 137\); Lower limit: \(117 - 12 = 105\); Outliers are: 101, 102 and 139 | M1A1, A1ft |
| 6(d) | Box and two whiskers (Must be on grid, otherwise 0/4); \(Q_1\), \(Q_2\) and \(Q_3\) all plotted correctly, ft their values from (a); Lower whisker plotted at 105 or 112 and outliers at 101 and 102; Upper whisker plotted at 137 or 132 and outlier at 139. (Whiskers must be compatible e.g. if lower whisker at 105 upper must be at 137) | M1, A1ft, B1, B1 |
| 6(e) | \([\overline{x} =] 121\) | B1 |
| \([\sigma_x] = \sqrt{\frac{279709}{19} - \overline{x}^2} = \sqrt{14721.526\ldots - 14641} = \sqrt{80.526\ldots} =\) awrt 8.97 (Allow \(s = 9.2195\ldots\) or awrt 9.22) | M1, A1 | M1 for correct expression including square root, ft their mean; A1 for awrt 8.97 (Allow \(s = 9.2195\ldots\) or awrt 9.22) |
| 6(f) | \(\overline{x} + 2.7 \times \sigma_x =\) awrt 145; \(\overline{x} - 2.7 \times \sigma_x =\) awrt 96.8 (allow 97) | M1A1 |
| 6(g) | Probably not suitable...since: data is skewed or (f) says no outliers, (c) says 3 or (a) says median = 122, (c) says mean = 121 | B1 |
**6(a)** | $Q_1 = 117$, $Q_2 = 122$, $Q_3 = 125$ | B1, B1, B1 |
**6(b)** | $\text{IQR} = 125 - 117 = 8$ | B1ft |
**6(c)** | Upper limit: $125 + 1.5 \times 8 = 125 + 12 = 137$; Lower limit: $117 - 12 = 105$; Outliers are: 101, 102 and 139 | M1A1, A1ft | M1 for correct expression for either limit, ft their values; 1st A1 for both 137 and 105 identified; 2nd A1ft for correctly identifying all outliers using their limits. Must have scored M1. This 2nd A1ft can be awarded if all outliers seen on box plot in (d)
**6(d)** | Box and two whiskers (Must be on grid, otherwise 0/4); $Q_1$, $Q_2$ and $Q_3$ all plotted correctly, ft their values from (a); Lower whisker plotted at 105 or 112 and outliers at 101 and 102; Upper whisker plotted at 137 or 132 and outlier at 139. (Whiskers must be compatible e.g. if lower whisker at 105 upper must be at 137) | M1, A1ft, B1, B1 | M1 for box and two whiskers (Must be on grid, otherwise 0/4); A1ft for $Q_1$, $Q_2$ and $Q_3$ all plotted correctly, ft their values from (a); 1st B1 for lower whisker at 105 or 112 and outliers at 101 and 102; 2nd B1 for upper whisker at 137 or 132 and outlier at 139. (Whiskers must be compatible)
**6(e)** | $[\overline{x} =] 121$ | B1 |
| $[\sigma_x] = \sqrt{\frac{279709}{19} - \overline{x}^2} = \sqrt{14721.526\ldots - 14641} = \sqrt{80.526\ldots} =$ awrt 8.97 (Allow $s = 9.2195\ldots$ or awrt 9.22) | M1, A1 | M1 for correct expression including square root, ft their mean; A1 for awrt 8.97 (Allow $s = 9.2195\ldots$ or awrt 9.22)
**6(f)** | $\overline{x} + 2.7 \times \sigma_x =$ awrt 145; $\overline{x} - 2.7 \times \sigma_x =$ awrt 96.8 (allow 97) | M1A1 | M1 for attempt to calculate one limit, ft their mean and s.d.; A1 for both limits correct allow awrt 145 and awrt 96.8 or 97
**6(g)** | Probably not suitable...since: data is skewed or (f) says no outliers, (c) says 3 or (a) says median = 122, (c) says mean = 121 | B1 | For stating, or implying, normal not suitable and giving at least one supporting reason. A calculation or description of skewness is not required but if present must be correct. For their values or their box plot. "Not normal since data is not continuous" is B0\begin{enumerate}
\item The stem and leaf diagram gives the blood pressure, $x \mathrm { mmHg }$, for a random sample of 19 female patients.
\end{enumerate}
\begin{center}
\begin{tabular}{ l | c c c c c c c c c }
10 & 1 & 2 & & & & & & & \\
11 & 2 & 7 & 7 & 8 & 8 & & & & \\
12 & 0 & 2 & 2 & 3 & 4 & 4 & 5 & 5 & 7 \\
13 & 1 & 2 & 9 & & & & & & \\
\end{tabular}
\end{center}
Key: 10 | 1 means blood pressure of 101 mmHg\\
(a) Find the median and the quartiles for these data.\\
(b) Find the interquartile range ( $Q _ { 3 } - Q _ { 1 }$ )
An outlier is a value that is greater than $Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$ or less than $Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$\\
(c) Showing your working clearly, identify any outliers for these data.\\
(d) On the grid on page 21 draw a box and whisker plot to represent these data. Show any outliers clearly.
The above data can be summarised by
$$\sum x = 2299 \text { and } \sum x ^ { 2 } = 279709$$
(e) Calculate the mean and the standard deviation for these data.
For a random sample taken from a normal distribution, a rule for determining outliers is: an outlier is more than $2.7 \times$ standard deviation above or below the mean.\\
(f) Find the limits to determine outliers using this rule.\\
(g) State, giving a reason based on some of the above calculations, whether or not a normal distribution is a suitable model for these data.
\includegraphics[max width=\textwidth, alt={}, center]{8ff7539e-fa44-4388-af8c-80656f081528-21_2281_73_308_15}\\
Turn over for a spare diagram if you need to redraw your plot.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{8ff7539e-fa44-4388-af8c-80656f081528-24_2639_1830_121_121}
\end{center}
\hfill \mbox{\textit{Edexcel S1 2016 Q6 [17]}}