Shifting every value moves the centre but not the spread, and scaling every value scales both. You can predict the new mean and standard deviation without recalculating.
This lesson closes the main sequence in dispersion of ungrouped data. It builds on comparing data sets using centre and spread.
What are the rules?
Use the data 4, 6, 8, 10, 12, with mean 8, variance 8 and standard deviation 2.83.
| Change to every value | New mean | New standard deviation | New variance |
|---|---|---|---|
| Add 5 | 8 + 5 = 13 | 2.83 (unchanged) | 8 (unchanged) |
| Multiply by 3 | 8 × 3 = 24 | 2.83 × 3 = 8.49 | 8 × 9 = 72 |
| Subtract 2 | 6 | 2.83 (unchanged) | 8 (unchanged) |
Adding moves the whole set along the number line, so the gaps between values stay the same. Multiplying stretches the gaps, so the spread grows by the same factor.
Worked example: confirming the rule
Multiply each value by 3 to get 12, 18, 24, 30, 36.
- Mean = 120 ÷ 5 = 24.
- Deviations: −12, −6, 0, 6, 12, squared: 144, 36, 0, 36, 144, total 360.
- Variance = 360 ÷ 5 = 72, and σ = √72 = 8.49.
This matches the prediction. The variance is 9 times the original, because 3² = 9.
Adding or removing a value
Add a value equal to the mean. Add 8 to the list to get 4, 6, 8, 8, 10, 12.
The mean is still 48 ÷ 6 = 8. The sum of squared deviations is still 40, but it is divided by 6, so the variance is 40 ÷ 6 = 6.67 and the spread decreased.
Remove an extreme value. Take 12 out to leave 4, 6, 8, 10.
The mean becomes 28 ÷ 4 = 7. The deviations are −3, −1, 1, 3, squared 9, 1, 1, 9, total 20, so the variance is 5. The spread decreased, because the largest deviation left.
The slips to avoid
A common slip is to add 5 to the standard deviation when 5 is added to each value. Adding a constant does not change the gaps, so the spread stays put.
Another slip is to multiply the variance by k instead of k². When each value is multiplied by 3, the standard deviation is multiplied by 3, but the variance is multiplied by 9. You can test a prediction on invented data using the descriptive statistics explorer.
Check yourself
The data 2, 3, 5, 6, 9 has mean 5 and variance 6. Predict the mean and variance after (a) subtracting 2 from each value, and (b) multiplying each value by 2.
Answer
(a) The mean is 5 − 2 = 3, and the variance stays 6, because shifting does not change the gaps.
(b) The mean is 5 × 2 = 10, and the variance is 6 × 2² = 24.
Check (b): the values are 4, 6, 10, 12, 18 with mean 10. Deviations: −6, −4, 0, 2, 8 squared 36, 16, 0, 4, 64, total 120, and 120 ÷ 5 = 24.
What to study next
Test the whole chapter with the dispersion of ungrouped data practice set. Then look at choosing an appropriate summary of a dataset to decide which measure suits which question.
For a teacher to go through these rules with you, see online one-to-one Mathematics tuition.