| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Standard combined mean and SD |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas for mean and variance from summary statistics, followed by combining two samples using weighted averages. Part (a) requires direct substitution into memorized formulas (mean = Σt/n, variance = Σt²/n - mean²), and part (b) uses the standard combined mean formula and variance combination formula. No problem-solving insight is needed—just careful arithmetic and formula recall, making it easier than average. |
| Spec | 2.02g Calculate mean and standard deviation5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks |
|---|---|
| (a) mean = \(\frac{427}{20} = 21.35\) minutes; variance = \(\frac{11077}{20} - 21.35^2 = 98.0\) minutes\(^2\) (3sf) | M1 A1 M2 A1 |
| (b) for 2\(^{nd}\) sample: \(\frac{\sum f}{30} = 18.5 \therefore \sum f = 30 \times 18.5 = 555\); \(\frac{\sum f^2}{30} - 18.5^2 = 8.2^2 \therefore \sum f^2 = 30(8.2^2 + 18.5^2) = 12284.7\); for combined sample: mean = \(\frac{427+555}{50} = 19.6\) minutes (3sf); variance = \(\frac{11077+12284.7}{50} - 19.6^2 = 81.5\) minutes\(^2\) (3sf) | M1 M2 A1 M1 A1 M1 A1 |
| (13) |
**(a)** mean = $\frac{427}{20} = 21.35$ minutes; variance = $\frac{11077}{20} - 21.35^2 = 98.0$ minutes$^2$ (3sf) | M1 A1 M2 A1 |
**(b)** for 2$^{nd}$ sample: $\frac{\sum f}{30} = 18.5 \therefore \sum f = 30 \times 18.5 = 555$; $\frac{\sum f^2}{30} - 18.5^2 = 8.2^2 \therefore \sum f^2 = 30(8.2^2 + 18.5^2) = 12284.7$; for combined sample: mean = $\frac{427+555}{50} = 19.6$ minutes (3sf); variance = $\frac{11077+12284.7}{50} - 19.6^2 = 81.5$ minutes$^2$ (3sf) | M1 M2 A1 M1 A1 M1 A1 |
| | | (13) |
---4. A company offering a bicycle courier service within London collected data on the delivery times for a sample of jobs completed by staff at each of its two offices.
The times, $t$ minutes, for 20 deliveries handled by the company's Hammersmith office were summarised by
$$\Sigma t = 427 , \text { and } \Sigma t ^ { 2 } = 11077$$
\begin{enumerate}[label=(\alph*)]
\item Find the mean and variance of the delivery times in this sample.
The company's Holborn office handles more business, so the delivery times for a sample of 30 jobs handled by this office was taken. The mean and standard deviation of this sample were 18.5 minutes and 8.2 minutes respectively.
\item Find the mean and variance of the delivery times of the combined sample of 50 deliveries.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [13]}}