Statistical Property: Standard Deviation Scaling
A fundamental property of standard deviation is how it changes when the data values are scaled (multiplied or divided by a constant).
- If each observation (\(x_i\)) in a dataset is multiplied by a constant (\(k\)), the new standard deviation (\(\sigma_{new}\)) becomes the absolute value of the constant multiplied by the original standard deviation (\(\sigma_{old}\)).
- Mathematically, if \(Y = kX\), then \(\sigma_Y = |k| \sigma_X\).
Calculating the New Standard Deviation
Given:
- Original standard deviation, \(\sigma_{old} = 10\).
- The constant multiplier, \(k = 5\).
Applying the property:
New standard deviation, \(\sigma_{new} = |k| \times \sigma_{old}\)
\(\sigma_{new} = 5 \times 10\)
\(\sigma_{new} = 50\)
Conclusion
When each of the 100 observations is multiplied by 5, the new standard deviation is 50.
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