A set of n values X₁,X₂,...,Xₙ has a standard deviation of 5. The standard deviation of n values X₁−7,X₂−7,...,Xₙ−7 will be:

Correct — A. Subtracting 7 from every value shifts the whole set without changing its spread. Call the old mean m.

New mean: m − 7
New deviation: (Xᵢ − 7) − (m − 7) = Xᵢ − m, the old deviation
Same deviations → same variance → same standard deviation
SD = 5 → option (a)

Why the others are wrong

• (b)
  0 — SD is 0 only when every value is the same. The values X₁ − 7, …, Xₙ − 7 differ from one another exactly as much as the originals did.
• (c)
  10 — Doubling every value would double the SD to 10. Subtracting a constant is a shift, not a scaling, so the SD stays 5.
• (d)
  7 — 7 is the shift, not the spread. Subtracting 7 lowers the mean by 7, and the distances of the values from the mean, which the SD measures, do not change.

The standard deviation measures how far the values sit from their own mean. It does not care where the set sits on the number line.

Adding or subtracting a constant moves every value and the mean together, so each deviation Xᵢ − m is unchanged and so is the SD.

Multiplying by a constant k stretches every deviation by k, so the SD is multiplied by |k| and the variance by k².

Key facts

• SD(X + c) = SD(X) for any constant c.
• SD(kX) = |k| × SD(X), and the variance becomes k² times as large.
• Adding a constant c to every value adds c to the mean, the median and the mode.
• The SD of a set is 0 only when all its values are equal.

Common traps

• Subtracting the constant from the SD as well as from the mean.
• Treating a shift like a scaling and changing the SD.

No directly related past PYQ was found.