Standard Deviation After a Unit Conversion: Why Shifts Do Not Change Spread

When every observation changes by y = ax + b, the standard deviation becomes |a| times the old standard deviation. Adding a constant changes the location, not the spread.

Think in distances from the mean

The new mean is aμ + b. Subtract it from a transformed observation: (ax + b) − (aμ + b) = a(x − μ). The shift cancels. Squared deviations are multiplied by a²; taking the square root gives |a|. This derivation works for either population or sample standard deviation when the same convention is retained.

The absolute-value sign matters. Standard deviation cannot be negative. Reflecting every value through zero reverses order but preserves distances apart. A transformation with a = 0 puts every observation at one point, so its standard deviation becomes zero.

Worked example: seconds to minutes

Times of 60, 120 and 180 seconds have mean 120 seconds. Their squared deviations are 3,600, 0 and 3,600. Treating these three entries as the whole population gives variance 7,200/3 = 2,400 seconds² and standard deviation √2,400 = 20√6 seconds.

Divide every observation by 60 to express the same times as 1, 2 and 3 minutes. The standard deviation is also divided by 60: √6/3 minutes, approximately 0.8165 minutes. The variance is divided by 60², giving 2/3 minutes². Keeping the numerical standard deviation unchanged while changing its unit would describe a different amount of variation.

A shift example you can picture

Suppose every pupil receives exactly 5 additional marks. If there is no cap or other adjustment, every difference between pupils stays the same. The mean rises by 5, but standard deviation stays unchanged. If marks are capped at a maximum, however, the transformation is no longer the same linear rule for all pupils. Recalculate from the adjusted values.

• Multiply all values by 4: standard deviation × 4, variance × 16.
• Subtract 9 from all values: standard deviation unchanged.
• Multiply all values by −3: standard deviation × 3, variance × 9.

Common mistake: mixing mean and spread rules

The mean follows aμ + b, while standard deviation follows |a|σ. Do not add b to the standard deviation. Write units alongside the answer, and distinguish the standard deviation from its square. Rounding a converted result too early can also create a mismatch when you convert it back to the original unit.

Shift se jagah badalti hai; scale se spread badalta hai.

Try three questions, then check the working

1. A set has standard deviation 12 cm. Express it in metres.

   Every value is divided by 100. Standard deviation = 12/100 = 0.12 m. Variance would be divided by 10,000.

2. Standard deviation is 4. Every x becomes 10 − 2x. Find the new value.

   The multiplier is −2, so the new standard deviation is |−2| × 4 = 8. The shift of 10 contributes nothing to spread.

3. Variance is 9. Every value is tripled. Find new variance and standard deviation.

   Old standard deviation is √9 = 3. New variance is 3² × 9 = 81, and new standard deviation is 3 × 3 = 9.

Build the next step

• Mean from a Frequency Table
• Statistics Class 10 Notes: Mean, Median, Mode Formulas and Solved Examples

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Reference and authorship

The examples and practice questions here are original, with calculations checked separately. Definitions and related background: OpenStax: Measures of the Spread of the Data. Prepared with AI assistance and editorial arithmetic review; no textbook affiliation is claimed.