Distinguish scaling from combining independent errors
Multiplying every observation by a constant scales its own spread. Adding two different varying measurements together raises a separate covariance question: the variance of a sum depends on both component variances and their relationship. It is generally incorrect to add their standard deviations as though that were another unit conversion. If the components are independent, variances add; if they are related, a covariance term is needed. The shared affine transformation rule does not supply that missing relationship.
A percentage transformation also needs a fixed denominator to be a common scaling. Dividing each observation by a different baseline changes individual factors, so one shared multiplier cannot predict the resulting standard deviation. State whether a conversion uses the same reference for every observation. This is particularly useful when comparing normalized scores or relative changes, where visually similar percent labels can conceal different transformations of the underlying measurements.
Start with the affine transformation rule
If every observation changes by y=a+bx, the transformed standard deviation is |b| times the original standard deviation. Its variance is b² times the original variance. Adding the same constant a changes the mean but not spread. These statements hold for both sample and population formulas when the observation set, weights and denominator choice stay consistent. They describe transforming each existing observation, not appending additional observations to the list.
Verify a shift with a complete example
Take 2, 4 and 6. Their mean is 4 and their sample standard deviation is 2, because squared deviations total 8 and the sample divisor is 2. Add 10 to every value to obtain 12, 14 and 16. The mean becomes 14, while deviations remain −2, 0 and 2. Sample standard deviation stays 2. The raw magnitudes increased, but their distances from the correspondingly shifted mean did not.
Scale every observation by the same factor
Multiply the original list by three to obtain 6, 12 and 18. Its sample standard deviation is 6, three times the original 2. Sample variance changes from 4 to 36, a factor of nine. Each deviation was multiplied by three before being squared. Confusing the standard-deviation factor with the variance factor creates substantial errors in unit conversions and in comparisons of rescaled measurements.
Handle negative factors and zero
Multiplying 2,4,6 by −2 gives −4,−8,−12. The mean becomes −8, and the sample standard deviation is positive 4. A reflection reverses ordering but cannot create a negative distance-based spread. If b=0, all transformed values equal a and spread becomes zero, provided the chosen calculation has enough observations. The absolute value in |b| is essential; multiplying a standard deviation directly by a negative factor gives an invalid result.
Apply the rule to real unit conversions
A temperature conversion F=32+(9/5)C combines a shift and a scale. A Celsius standard deviation of 2 corresponds to 3.6 Fahrenheit degrees; the 32 offset contributes nothing to spread. Variance is multiplied by (9/5)²=3.24. Likewise, converting minutes to seconds multiplies standard deviation by 60 and variance by 3,600. Preserve the distinction between a temperature reading and a temperature difference when labeling the units of spread.
| Values | Standard deviation | Variance |
|---|---|---|
| 2,4,6 | 2 | 4 |
| 12,14,16 | 2 | 4 |
| 6,12,18 | 6 | 36 |
| −4,−8,−12 | 4 | 16 |
All rows use sample calculations. Each transformed list contains the same number of observations.
Recognize operations outside the rule
Adding a new observation is different from adding a constant to every observation. Appending 10 to 2,4,6 changes the list's size and mean, so sample spread must be recomputed. Squaring, logging or rounding each value is also not generally an affine transformation with one shared constant factor. Rounding can erase small differences; a nonlinear conversion can change relative distances. Do not apply the simple |b| identity to those operations without checking their mathematics.
Use paired calculator runs as a check
Run the Standard Deviation Calculator in sample mode for 2,4,6 and again for 6,12,18. Confirm standard deviations 2 and 6 and variances 4 and 36. Then test −4,−8,−12 for standard deviation 4. Keep the model fixed across runs: comparing sample SD before a conversion with population SD afterward adds a denominator change and prevents the pure scaling identity from matching.
Report transformations and avoid false precision
Save the equation, original units, transformed units and denominator choice with the comparison. Calculate from unrounded original data where available; transforming rounded summary statistics can introduce small discrepancies against a recomputation. A useful check is reversible: dividing the transformed standard deviation by |b| recovers the original when b is nonzero. The transformation rule makes unit changes predictable, but it does not establish measurement quality, remove outliers or justify comparing incompatible populations.