Adding the same number to every value moves the mean by that amount and leaves the standard deviation unchanged. Multiplying every value by a factor multiplies the standard deviation by the size of that factor, regardless of its sign.
The 2027 SEC G3 Mathematics K310 syllabus includes calculating standard deviation and using it to compare datasets. These examples use short lists to show what happens to the distances from the mean.
- Write down how each value changes.
- Track the mean separately from the spread.
- A constant addition leaves the spread unchanged.
- A multiplication scales the spread by the absolute factor.
CHAPTER 1 OF 4 · Back to contents
Measure spread from the centre
For the scores 4, 6 and 8, the mean is 6. Their deviations from the mean are −2, 0 and 2. Standard deviation uses the squares of these deviations, so the signs do not cancel out the spread.
Add 5 to every score. The mean rises by 5, and each score still has the same distance from the mean. The standard deviation stays the same.
Double every score instead. The mean doubles and each distance from the mean doubles. The squared deviations become four times as large. Taking the square root makes the standard deviation twice as large.
CHAPTER 2 OF 4 · Back to contents
Worked example | Shift the scores, then compare a scale change
Compare the original scores with scores obtained by adding 5 and scores obtained by doubling first and then adding 5. Use the standard deviation of the listed data, with the sum of squared deviations divided by the number of values before taking the square root.
| Rule | Scores | Mean | Standard deviation |
|---|---|---|---|
| Original | 4, 6, 8 | 6 | √(8/3) |
| Add 5 | 9, 11, 13 | 11 | √(8/3) |
| Double, then add 5 | 13, 17, 21 | 17 | 2√(8/3) |
In the third row, the deviations from 17 are −4, 0 and 4. Their squared sum is 32. Dividing by 3 and taking the square root gives √(32/3), which equals 2√(8/3).
You can check the order of the operations with one value. For an original score of 4, doubling first and adding 5 gives 13. Adding 5 first and doubling gives 18.
Contents · Previous chapter · Next chapter
CHAPTER 3 OF 4 · Back to contents
Practise with a negative multiplier
Multiply each original score by −3 and add 20. The new scores are 8, 2 and −4. The mean is −3 × 6 + 20 = 2.
Their deviations are 6, 0 and −6. The order has reversed and the distances from the mean are three times as large. The standard deviation is 3√(8/3).
In general, if every value x becomes ax + b, the mean follows the same rule as a value. The standard deviation is multiplied by |a|.
Contents · Previous chapter · Next chapter
CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
What if I change only one score?
Recalculate the spread. The shortcuts apply when the same rule changes every score.
How does the variance change when scores double?
Variance is the square of standard deviation, so it becomes four times as large.
What if the multiplier is zero?
Every transformed value equals the same constant. The standard deviation is zero.
Contents · Previous chapter · Continue with the related guide