Adding a constant to every data point increases the mean by , but the standard deviation remains unchanged. Subtracting a constant has the corresponding effect: the mean decreases, while the standard deviation still does not change.
Suppose the original data values are , with mean . After adding to every value, the new mean is
Therefore, the entire distribution shifts by .
The standard deviation measures how far data values are spread from their mean. For each transformed value, its deviation from the new mean is
Every deviation is therefore unchanged. Consequently, the variance, which is based on squared deviations, and the standard deviation are also unchanged.
Consider the data set . Its mean is , and its population standard deviation is
Adding gives . The new mean is , while the standard deviation remains .
Notice that the transformed data set has moved five units to the right, but the gaps between corresponding values remain unchanged. On a box plot or histogram, the shape and horizontal spread would be identical; only the labels showing location would shift by five.
The same result applies to sample standard deviation. Although sample variance divides by , its numerator still uses squared deviations from the sample mean. Since each deviation is unchanged after the shift, both sample variance and sample standard deviation remain unchanged.
| Statistic | Effect of adding |
|---|---|
| Mean | Becomes |
| Variance | Unchanged |
| Standard deviation | Unchanged |
A common misconception is that standard deviation increases because every value becomes larger. This is incorrect: standard deviation measures spread, not the overall size or location of the data.
This is assessed in IB Mathematics: Analysis and Approaches, SL 4.3. State both effects explicitly and justify the unchanged standard deviation by explaining that all deviations from the mean remain the same.
