| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2003 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Outliers from raw data |
| Difficulty | Moderate -0.8 This is a straightforward S1 question testing standard descriptive statistics procedures: calculating mean, drawing stem-and-leaf, finding quartiles, applying the 1.5×IQR outlier rule, drawing a box plot, and commenting on skewness. All steps are routine recall and application of formulas with no problem-solving or insight required. Easier than average A-level due to being purely procedural. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x} = \frac{20+15+\ldots+17}{14} = \frac{312}{14} = 22.2857\ldots\) | M1 A1 | (awrt 22.3) (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Bags of crisps | 1 | 0 means 10 |
| 0 | 5 | |
| 1 | 0 | 1 3 5 7 |
| 2 | 0 | 0 5 |
| 3 | 0 | 1 3 |
| 4 | 0 | 2 |
| B1 B1 B1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(Q_2 = 20; \quad Q_1 = 13; \quad Q_3 = 31\) | B1; B1; B1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(1.5 \times \text{IQR} = 1.5 \times (31-13) = 27\) | B1 | |
| \(31 + 27 = 58; \quad 13-27 = -14\) | M1 | both |
| No outliers | A1 B1 | (3 marks) |
| Answer | Marks |
|---|---|
| scale and label | B1 |
| \(Q_1 = 13, \quad Q_2 = 20, \quad Q_3 = 31\) | B1 ft |
| Whiskers 5, 42; | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(Q_2 - Q_1 = 7; \quad Q_3 - Q_2 = 11; \quad Q_3 - Q_2 > Q_2 - Q_1\) | M1 | |
| Positive skew | A1 | (2 marks) |
| (13 marks) |
**(a)**
| $\bar{x} = \frac{20+15+\ldots+17}{14} = \frac{312}{14} = 22.2857\ldots$ | M1 A1 | (awrt 22.3) (2 marks) |
**(b)**
| Bags of crisps | 1 | 0 means 10 | Totals | | |
|---|---|---|---|---|---|
| 0 | 5 | | | (1) | Label and key |
| 1 | 0 | 1 3 5 7 | | (5) | 2 correct rows |
| 2 | 0 | 0 5 | | (3) | All correct |
| 3 | 0 | 1 3 | | (3) | |
| 4 | 0 | 2 | | (2) | |
| | | | | B1 B1 B1 | (3 marks) |
**(c)**
| $Q_2 = 20; \quad Q_1 = 13; \quad Q_3 = 31$ | B1; B1; B1 | (3 marks) |
**(d)**
| $1.5 \times \text{IQR} = 1.5 \times (31-13) = 27$ | B1 | |
| $31 + 27 = 58; \quad 13-27 = -14$ | M1 | both |
| No outliers | A1 B1 | (3 marks) |
**(e)**
| scale and label | B1 | |
| $Q_1 = 13, \quad Q_2 = 20, \quad Q_3 = 31$ | B1 ft | |
| Whiskers 5, 42; | B1 | |
**(f)**
| $Q_2 - Q_1 = 7; \quad Q_3 - Q_2 = 11; \quad Q_3 - Q_2 > Q_2 - Q_1$ | M1 | |
| Positive skew | A1 | (2 marks) |
| | | (13 marks) |
---6. The number of bags of potato crisps sold per day in a bar was recorded over a two-week period. The results are shown below.
$$20,15,10,30,33,40,5,11,13,20,25,42,31,17$$
\begin{enumerate}[label=(\alph*)]
\item Calculate the mean of these data.
\item Draw a stem and leaf diagram to represent these data.
\item Find the median and the quartiles of these data.
An outlier is an observation that falls either $1.5 \times$ (interquartile range) above the upper quartile or $1.5 \times$ (interquartile range) below the lower quartile.
\item Determine whether or not any items of data are outliers.
\item On graph paper draw a box plot to represent these data. Show your scale clearly.
\item Comment on the skewness of the distribution of bags of crisps sold per day. Justify your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2003 Q6 [16]}}