Question

The mean and variance of 10 observation were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is ________.

Answer 1

Correct Answer: 2

Given that the incorrect mean (M) and variance (V) of the observations are 15 each:

1. First, calculate the incorrect sum of the observations using the formula: ΣX = M × n = 15 × 10 = 150

2. Correct the sum by replacing the incorrect observation (25) with the correct observation (15). Thus, the corrected sum is: ΣXcorrect = 150 - 25 + 15 = 140 

3. Compute the correct mean: Mcorrect = ΣXcorrect / n = 140 / 10 = 14

4. Use the variance formula: V = (ΣX²/n) - (M²) to find the correct ΣX²:

Given V = 15, replace with the incorrect data:

15 = (ΣX² / 10) - (15²)

Solving for ΣX² gives: ΣX² = 15 × 10 + 225 = 375

5. Correct the ΣX² by replacing 25² with 15²:

ΣX²correct = 375 - 625 + 225 = 225

6. Calculate the correct variance:

Vcorrect = (ΣX²correct / n) - (Mcorrect)²

= (225 / 10) - 14² = 22.5 - 196 = 1.5

7. Finally, find the correct standard deviation, which is the square root of the correct variance:

SDcorrect = √1.5 ≈ 1.22

The correct standard deviation is approximately 1.22, which confirms it fits within the expected range: 2,2.

Answer 2

The correct answer is 2
Given,
\(\frac{\sum_{i=1}^{10}x_i}{10}= 15\ \ .....(1)\)
\(⇒\) \(\sum_{i=1}^{10} x_i = 150\)
and \(\frac{\sum_{i=1}^{10} x_{i}^{2}}{10} - 15^2 = 15\)
\(⇒\) \(\sum_{i=1}^{10} x_{2i} = 2400\)
Replacing 25 by 15 we get
\(⇒\) \(\sum_{i=1}^{9} (x_i + 25) = 150\)
\(⇒\)\(\sum_{i=1}^{9} x_i = 125\)
∴ Correct mean
= \(\frac{\sum_{i=1}^{9}{x_i + 15}}{10} = \frac{125 + 15}{10}\)
= 14
Similarly,
\(\sum_{i=1}^{2} x_{i}^{2} = 2400 - 25^2\)
= 1775
∴ Correct variance = \(\frac{\sum_{i=1}^{9} x_{i}^{2} + 15^2}{10} - 14^2\)
\(= \frac{1775+225}{10}-14^2\)
= 4
∴ Correct S.D. \(= \sqrt4\)
= 2