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If all the observations are multiplied by 2, then
- a)New SD would be also multiplied by 2
- b)New SD would be half of the previous SD
- c)New SD would be increased by 2
- d)New SD would be decreased by 2
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If all the observations are multiplied by 2, thena)New SD would be als...
Explanation:
Standard deviation is a measure of the variability or dispersion of a set of observations. It is calculated as the square root of the variance. When all the observations are multiplied by a constant, the mean changes by the same constant but the variance changes by the square of the constant.
Let's assume that the original set of observations is {X1, X2, X3, ..., Xn}, and the constant by which all the observations are multiplied is k. The new set of observations would be {kX1, kX2, kX3, ..., kXn}.
To calculate the new standard deviation, we need to calculate the new variance first.
New Variance = [(kX1 - kμ)² + (kX2 - kμ)² + ... + (kXn - kμ)²]/n
= k²[(X1 - μ)² + (X2 - μ)² + ... + (Xn - μ)²]/n
= k²Var(X)
where μ is the mean of the original set of observations and Var(X) is the variance of the original set of observations.
Therefore, the new standard deviation would be:
New SD = √[k²Var(X)]
= k√Var(X)
= kSD
Therefore, the new standard deviation would be multiplied by the same constant by which all the observations are multiplied. Hence, the correct answer is option 'A'.
Standard deviation is a measure of the variability or dispersion of a set of observations. It is calculated as the square root of the variance. When all the observations are multiplied by a constant, the mean changes by the same constant but the variance changes by the square of the constant.
Let's assume that the original set of observations is {X1, X2, X3, ..., Xn}, and the constant by which all the observations are multiplied is k. The new set of observations would be {kX1, kX2, kX3, ..., kXn}.
To calculate the new standard deviation, we need to calculate the new variance first.
New Variance = [(kX1 - kμ)² + (kX2 - kμ)² + ... + (kXn - kμ)²]/n
= k²[(X1 - μ)² + (X2 - μ)² + ... + (Xn - μ)²]/n
= k²Var(X)
where μ is the mean of the original set of observations and Var(X) is the variance of the original set of observations.
Therefore, the new standard deviation would be:
New SD = √[k²Var(X)]
= k√Var(X)
= kSD
Therefore, the new standard deviation would be multiplied by the same constant by which all the observations are multiplied. Hence, the correct answer is option 'A'.
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If all the observations are multiplied by 2, thena)New SD would be also multiplied by 2b)New SD would be half of the previous SDc)New SD would be increased by 2d)New SD would be decreased by 2Correct answer is option 'A'. Can you explain this answer?
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