| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate mean from coded sums |
| Difficulty | Moderate -0.8 This is a straightforward statistics question requiring basic ordering of data, finding median/IQR from a list, and calculating mean/standard deviation using a calculator. Part (a) only requires ordering known values with inequalities, part (b) tests conceptual understanding at a basic level, and part (c) is direct calculation with all values given. No complex problem-solving or novel insight required. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| Special Case: Identification that LQ = 18 and UQ = 34 | M1, A1, A2, (A1) | May be implied by correct median or correct IQR. Ignore any reference to \(a\) and \(b\). CAO × 3 (all correct). Both CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Two values (25 and 37) of mode. No unique value. Sparse data. Many different values | B1 | Or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| \(a\) and \(b\) (two values) unknown. Impossible to calculate. Cannot be calculated | B1 | Or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Special Case: Evidence of \(\frac{\sum x}{15}\) | B1, B1, (M1) | CAO. AWFW (9.423 & 9.754). Treat rounding of a correct stated answer to an integer as ISW |
## Part (a)
Ordering values gives: (a) 14 15 18 20 25 25 26 27 29 32 34 37 37 (b)
Median = 26
IQR = 34 − 18 = 16
Special Case: Identification that LQ = 18 and UQ = 34 | M1, A1, A2, (A1) | May be implied by correct median or correct IQR. Ignore any reference to $a$ and $b$. CAO × 3 (all correct). Both CAO | 4
## Part (b)(i)
Two values (25 and 37) of mode. No unique value. Sparse data. Many different values | B1 | Or equivalent | —
## Part (b)(ii)
$a$ and $b$ (two values) unknown. Impossible to calculate. Cannot be calculated | B1 | Or equivalent | 2
## Part (c)
Mean = $\frac{\sum x}{n} = \frac{390}{15} = 26$
If not identified, assume order is $\bar{x}$ then $s$
SD ($\sum x^2 = 11472$) = 9.4 to 9.8
Special Case: Evidence of $\frac{\sum x}{15}$ | B1, B1, (M1) | CAO. AWFW (9.423 & 9.754). Treat rounding of a correct stated answer to an integer as ISW | 2
---2 Lizzie, the receptionist at a dental practice, was asked to keep a weekly record of the number of patients who failed to turn up for an appointment. Her records for the first 15 weeks were as follows.
$$\begin{array} { l l l l l l l l l l l l l l l }
20 & 26 & 32 & a & 37 & 14 & 27 & 34 & 15 & 18 & b & 25 & 37 & 29 & 25
\end{array}$$
Unfortunately, Lizzie forgot to record the actual values for two of the 15 weeks, so she recorded them as $a$ and $b$. However, she did remember that $a < 10$ and that $b > 40$.
\begin{enumerate}[label=(\alph*)]
\item Calculate the median and the interquartile range of these 15 values.
\item Give a reason why, for these data:
\begin{enumerate}[label=(\roman*)]
\item the mode is not an appropriate measure of average;
\item the standard deviation cannot be used as a measure of spread.
\end{enumerate}\item Subsequent investigations revealed that the missing values were 8 and 43 .
Calculate the mean and the standard deviation of the 15 values.
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2010 Q2 [8]}}