The standard deviation of the set of numbers 1, 4, 5, 7, 8, 10, 12, 13, 15, 17…

Standard deviation (SD) measures how spread out a set of values is around its mean. Two transformation rules govern how SD changes: adding or subtracting the same constant to every value shifts the mean by that constant but leaves each value's distance from the mean exactly the same, so SD stays unchanged (shift-invariance); multiplying every value by a constant k, by contrast, scales the SD by |k|.

1. The original set {1, 4, 5, 7, 8, 10, 12, 13, 15, 17} has mean x̄ and standard deviation SD(xi) = 4.85, as given.

2. Adding 10 to every value creates a new set with terms yi = xi + 10, so the new mean becomes ȳ = x̄ + 10.

3. Each value's deviation from its own mean is unaffected by the shift: yi − ȳ = (xi + 10) − (x̄ + 10) = xi − x̄.

4. Standard deviation is built from these squared deviations, and since every deviation in the new set exactly equals the corresponding deviation in the original set, the new standard deviation equals the original one.

Cross-check with a small example: take {1, 3}. Mean = 2, deviations = −1 and +1, giving SD = 1. Add 10 to each value to get {11, 13}: mean = 12, deviations = −1 and +1 again, so SD = 1 — unchanged. The same shift-invariance applies to the given 10-number set.

So the standard deviation of the new set remains 4.85.