if each value of a given group of observation is multiplied by 10, the standard detion of the resulting observation is –
First, standard deviation measures the spread of the data around the mean. If you multiply each observation by a constant, the spread should change proportionally. For example, if you have data points like 2, 4, 6 and multiply each by 10, they become 20, 40, 60. The spread between them increases, so the standard deviation should also increase by the same factor.
Wait, standard deviation is calculated using the square root of the variance. Variance is the average of squared differences from the mean. If each data point is multiplied by 10, the differences from the mean also get multiplied by 10. Squaring that would be 100 times, so variance becomes 100 times the original. Then taking the square root, the standard deviation would be 10 times the original. So multiplying by a constant k scales the standard deviation by |k|.
But wait, the options aren't given here. The user mentioned that the correct answer is the standard deviation multiplied by 10. So the answer should be that the standard deviation is multiplied by 10. Let me check the options again. The options A-D are missing, but the correct answer is probably option C or D. But the user didn't provide the options, so I need to focus on the explanation.
The core concept here is understanding how standard deviation is affected by linear transformations. Multiplying each data point by a constant scales the standard deviation by the absolute value of that constant. The variance would scale by the square of the constant, but since standard deviation is the square root of variance, it scales linearly with the constant.
The wrong options might be things like "remains the same" (which would be incorrect because scaling affects spread), "increases by 10 times the original mean" (which is not directly related to standard deviation), or "decreases by 10 times" (incorrect because multiplying by a positive constant doesn't decrease spread).
The clinical pearl here is that when data is scaled, standard deviation scales linearly, which is important in statistical analysis and understanding the effect of data transformations.
**Core Concept**
Standard deviation quantifies the dispersion of data around the mean. When all observations are multiplied by a constant (k), the standard deviation scales by the absolute value of that constant (|k|), while variance scales by k². This is because standard deviation is the square root of variance, which involves squared deviations from the mean.
**Why the Correct Answer is Right**
Multiplying each observation by 10 increases the spread of the dataset proportionally. The standard deviation, being a measure of spread, also increases by a factor of 10. For example, if original data points are *x₁, x₂, x₃*, multiplying by 10 yields *10x₁, 10x₂, 10x₃*. The mean becomes 10 times the original, and deviations from the mean (which determine standard deviation) also scale by 10. Thus, the new standard deviation is **10 ×