Question

The standard deviation of 100 observations is 10. If 20 is added to each observation, then what will be the new standard deviation?

• A. 10
• B. 15
• C. 20
• D. 25

Hint:

Adding or subtracting a constant shifts the entire distribution but does not change its spread, so variance and standard deviation remain the same. However, multiplying or dividing by a constant $k$ will multiply the standard deviation by $|k|$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the standard deviation of a dataset and asked to find the new standard deviation after a constant value (20) is added to every single observation.
Step 2: Key Formula or Approach:
Standard deviation is a measure of dispersion (spread) around the mean. The property of standard deviation states that if a constant $c$ is added to or subtracted from each observation in a dataset, the standard deviation remains completely unchanged.
Mathematically, if $Y_i = X_i + c$, then $\sigma_Y = \sigma_X$.
Step 3: Detailed Explanation:
Let the original observations be $x_1, x_2, \dots, x_{100}$.
The original standard deviation is $\sigma_x = 10$.
When 20 is added to each observation, the new observations become $y_i = x_i + 20$.
The mean of the new observations will increase by 20 ($\bar{y} = \bar{x} + 20$).
The deviation of each new observation from the new mean is: \[ y_i - \bar{y} = (x_i + 20) - (\bar{x} + 20) = x_i - \bar{x} \] Since the deviations from the mean are identical to the original deviations, the variance and the standard deviation remain unaffected.
Therefore, the new standard deviation is still 10.
Step 4: Final Answer:
The new standard deviation will be 10.