Real data often need a change of units: minutes to hours, Celsius to Fahrenheit, or a teacher adding 5 bonus points. You do not need to recompute everything. Shifting and scaling change the summaries in predictable ways, and z-scores are one example.
Key idea
If every value changes by $y = a + bx$: centers and positions (mean, median, quartiles) become $a + b(\text{old})$. Spreads (standard deviation, IQR, range) are multiplied by $|b|$ and are not affected by adding $a$.
Why it works
Spread is about distances between values. Adding $a$ to every value slides the whole data set along the number line: every distance stays the same, so the spread stays the same, while every center slides by $a$. Multiplying by $b$ stretches every distance by $|b|$: the deviation $bx - b\bar{x} = b(x - \bar{x})$, so the standard deviation becomes $|b|s$ (never negative).
Worked example: Adding a constant
Scores have mean 72, median 74, standard deviation 9 and IQR 12. Everyone gets 5 bonus points. Find the new summaries.
- Centers move by 5: mean $77$, median $79$.
- Spreads do not change: standard deviation $9$, IQR $12$.
Answer: Mean 77, median 79, SD 9, IQR 12.
Worked example: Changing units
Heights have mean 65 inches and SD 3 inches. Convert to centimeters ($1 \text{ in} = 2.54 \text{ cm}$).
- Every value is multiplied by 2.54.
- Mean: $65 \times 2.54 = 165.1$ cm.
- SD: $3 \times 2.54 = 7.62$ cm (stretched by the same factor).
Answer: Mean 165.1 cm, SD 7.62 cm.
Worked example: Both at once
Temperatures have mean $20^\circ$C and SD $5^\circ$C. Use $F = 1.8C + 32$.
- Mean: $1.8(20) + 32 = 68^\circ$F.
- SD: only the multiplier matters, $1.8 \times 5 = 9^\circ$F.
Answer: Mean $68^\circ$F, SD $9^\circ$F.
Common mistake
Adding the constant to the standard deviation too ("the SD goes from 9 to 14"). Adding 5 to every score moves every score, so nobody gets farther from anybody else. Check: did the change move the data, stretch it, or both? Only stretching changes spread.
Standardizing is a shift and a scale: subtract $\mu$ (the mean becomes 0), then divide by $\sigma$ (the SD becomes 1). That is why z-scores always have mean 0 and standard deviation 1.