| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2003 |
| Session | November |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Identify outliers using IQR rule |
| Difficulty | Moderate -0.8 This is a straightforward statistics question requiring standard calculations (mean, SD, median, IQR) from given data, applying a provided outlier formula, and drawing box plots. All techniques are routine S1 procedures with no problem-solving insight needed—easier than average A-level maths. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers2.02i Select/critique data presentation |
| 299 | 1050 | 2315 | 999 | 485 |
| 350 | 169 | 1015 | 650 | 830 |
| 99 | 2100 | 689 | 550 | 475 |
| Answer | Marks | Guidance |
|---|---|---|
| Part (a) | ||
| \(\sum x = 12075; \sum x^2 = 15499685\) | ||
| \(\therefore \bar{x} = \frac{12075}{15} = 805\) | B1 | |
| \(sd = \sqrt{\frac{15499685}{15} - 805^2} = 620.71491\) | M1 | |
| \(621\) | A1 | (3 marks) |
| (NB Using \(n-1\) gives \(642.50125\ldots\)) | ||
| Part (b) | ||
| \(99, 169, 299, 350, 475, 485, 550, 650, 689, 830, 999, 1015, 1050, 2100, 2315\) | M1 | Attempt to order |
| \(\therefore Q_2 = 650\) | A1 | |
| \(\therefore IQR = Q_3 - Q_1 = 1015 - 350 = 665\) | M1 | Attempt at \(Q_3 - Q_1\) |
| \(665\) | A1 | (4 marks) |
| Part (c) | ||
| \(Q_3 + 1.5(Q_3 - Q_1) = 1015 + 1.5 \times 665 = 2012.5\) | M1 | Use of given outlier formula |
| \(\therefore 2100\) and \(2315\) are outliers | A1 | |
| \(Q_1 - 1.5(Q_3 - Q_1) = 350 - 1.5 \times 665 < 0\) | A1 | |
| \(\therefore\) No outliers | A1 | (3 marks) |
| Part (d) | ||
| Boxplot | M1 | |
| Scales & Labels | A1 | |
| Website | A1 | |
| Shop | A1 | (4 marks) |
| Part (e) | ||
| Median website > median shop | ||
| Website negative skew; shop approx symmetrical (Ignoring outliers) | ||
| Ranges approximately equal | ||
| Shop \(Q_3 <\) Website \(Q_3 \Rightarrow\) shop sales low value | ||
| Website sales more variable in value | Any two sensible comments | B1, B1 |
**Part (a)** | |
$\sum x = 12075; \sum x^2 = 15499685$ | |
$\therefore \bar{x} = \frac{12075}{15} = 805$ | B1 |
$sd = \sqrt{\frac{15499685}{15} - 805^2} = 620.71491$ | M1 |
$621$ | A1 | (3 marks)
(NB Using $n-1$ gives $642.50125\ldots$) | |
**Part (b)** | |
$99, 169, 299, 350, 475, 485, 550, 650, 689, 830, 999, 1015, 1050, 2100, 2315$ | M1 | Attempt to order |
$\therefore Q_2 = 650$ | A1 |
$\therefore IQR = Q_3 - Q_1 = 1015 - 350 = 665$ | M1 | Attempt at $Q_3 - Q_1$ |
$665$ | A1 | (4 marks)
**Part (c)** | |
$Q_3 + 1.5(Q_3 - Q_1) = 1015 + 1.5 \times 665 = 2012.5$ | M1 | Use of given outlier formula |
$\therefore 2100$ and $2315$ are outliers | A1 |
$Q_1 - 1.5(Q_3 - Q_1) = 350 - 1.5 \times 665 < 0$ | A1 |
$\therefore$ No outliers | A1 | (3 marks)
**Part (d)** | |
Boxplot | M1 |
Scales & Labels | A1 |
Website | A1 |
Shop | A1 | (4 marks)
**Part (e)** | |
Median website > median shop | |
Website negative skew; shop approx symmetrical (Ignoring outliers) | |
Ranges approximately equal | |
Shop $Q_3 <$ Website $Q_3 \Rightarrow$ shop sales low value | |
Website sales more variable in value | Any two sensible comments | B1, B1 | (2 marks)
---6. A travel agent sells holidays from his shop. The price, in $\pounds$, of 15 holidays sold on a particular day are shown below.
\begin{center}
\begin{tabular}{ r r r r r }
299 & 1050 & 2315 & 999 & 485 \\
350 & 169 & 1015 & 650 & 830 \\
99 & 2100 & 689 & 550 & 475 \\
\end{tabular}
\end{center}
For these data, find
\begin{enumerate}[label=(\alph*)]
\item the mean and the standard deviation,
\item the median and the inter-quartile range.
An outlier is an observation that falls either more than $1.5 \times$ (inter-quartile range) above the upper quartile or more than $1.5 \times$ (inter-quartile range) below the lower quartile.
\item Determine if any of the prices are outliers.
The travel agent also sells holidays from a website on the Internet. On the same day, he recorded the price, $\pounds x$, of each of 20 holidays sold on the website. The cheapest holiday sold was $\pounds 98$, the most expensive was $\pounds 2400$ and the quartiles of these data were $\pounds 305 , \pounds 1379$ and $\pounds 1805$. There were no outliers.
\item On graph paper, and using the same scale, draw box plots for the holidays sold in the shop and the holidays sold on the website.
\item Compare and contrast sales from the shop and sales from the website.
\section*{END}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2003 Q6 [16]}}