| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Adding data values |
| Difficulty | Moderate -0.3 This is a straightforward application of coding/transformation formulas for mean and standard deviation. Part (i) requires standard algebraic manipulation of Σ(x-45) to find mean and variance, while part (ii) extends this by incorporating one additional value. The techniques are routine for S1 students who have practiced these formulas, making it slightly easier than average but not trivial due to the two-part structure and need for careful arithmetic. |
| Spec | 2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Mean \(= 45 - \frac{148}{36} = 40.9\) or \(\frac{1472}{36}\) EITHER Var \(= \frac{3089}{36} - \left(\frac{-148}{36}\right)^2 = 68.9\) sd \(= 8.30\) | B1, M1, A1 [3] | Correct answer; Expanding \(\Sigma(x-45)^2\) with at least 2 terms correct and solving, then substituting their \(\Sigma x^2\) in correct variance formula with their mean² subst numerically; Correct answer |
| OR \(\Sigma x^2 = 3089 - 36 \times 45^2 + 90 \times 1472 = 62669\) Var \(= \frac{62669}{36} - \left(\frac{1472}{36}\right)^2\) sd \(= 8.30\) | M1, A1 | |
| (ii) New \(\Sigma(x-45) = -148 - 16 = -164\) New \(\Sigma(x-45)^2 = 3089 + 16^2 = 3345\) | M1, M1 | Adding their coded new value to \(-148\); Adding their (coded value)² to \(3089\) |
| New sd \(= \sqrt{\frac{3345}{37 - \left(\frac{-164}{37}\right)^2}} = 8.41\) | M1, A1 [4] | Subst in coded var formula, can have one of \(29\) and one of \(-16\) here; Correct answer |
| OR \(\Sigma x = 36 \times 45 - 148 = 1472\) New \(\Sigma x = 1472 + 29 = 1501\) \(\Sigma x^2 = 3089 - 36 \times 45^2 + 90 \times 1472 = 62669\) New \(\Sigma x^2 = 62669 + 29^2 = 63510\) | M1, M1 | Finding \(\Sigma x\) and adding \(29\); Finding \(\Sigma x^2\) and adding \(29^2\), at least 2 terms of \(3089, 36 \times 45^2, 90 \times 1472\) |
| New sd \(= \sqrt{\frac{63510}{37}-(1501/37)^2} = 8.41\) | M1, A1 | Subst their values in correct var formula; Correct answer |
**(i)** Mean $= 45 - \frac{148}{36} = 40.9$ or $\frac{1472}{36}$ EITHER Var $= \frac{3089}{36} - \left(\frac{-148}{36}\right)^2 = 68.9$ sd $= 8.30$ | B1, M1, A1 [3] | Correct answer; Expanding $\Sigma(x-45)^2$ with at least 2 terms correct and solving, then substituting their $\Sigma x^2$ in correct variance formula with their mean² subst numerically; Correct answer
OR $\Sigma x^2 = 3089 - 36 \times 45^2 + 90 \times 1472 = 62669$ Var $= \frac{62669}{36} - \left(\frac{1472}{36}\right)^2$ sd $= 8.30$ | M1, A1 |
**(ii)** New $\Sigma(x-45) = -148 - 16 = -164$ New $\Sigma(x-45)^2 = 3089 + 16^2 = 3345$ | M1, M1 | Adding their coded new value to $-148$; Adding their (coded value)² to $3089$
New sd $= \sqrt{\frac{3345}{37 - \left(\frac{-164}{37}\right)^2}} = 8.41$ | M1, A1 [4] | Subst in coded var formula, can have one of $29$ and one of $-16$ here; Correct answer
OR $\Sigma x = 36 \times 45 - 148 = 1472$ New $\Sigma x = 1472 + 29 = 1501$ $\Sigma x^2 = 3089 - 36 \times 45^2 + 90 \times 1472 = 62669$ New $\Sigma x^2 = 62669 + 29^2 = 63510$ | M1, M1 | Finding $\Sigma x$ and adding $29$; Finding $\Sigma x^2$ and adding $29^2$, at least 2 terms of $3089, 36 \times 45^2, 90 \times 1472$
New sd $= \sqrt{\frac{63510}{37}-(1501/37)^2} = 8.41$ | M1, A1 | Subst their values in correct var formula; Correct answer3 A sample of 36 data values, $x$, gave $\Sigma ( x - 45 ) = - 148$ and $\Sigma ( x - 45 ) ^ { 2 } = 3089$.\\
(i) Find the mean and standard deviation of the 36 values.\\
(ii) One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.
\hfill \mbox{\textit{CAIE S1 2011 Q3 [7]}}