Transforming data: effect on center and spread

Three quick questions from earlier lessons.

Everyone in a class grows 2 cm over the summer. What happens to the standard deviation of their heights?

Shifts move the center; scalings stretch center and spread.

For new values

a + bx

: center (mean, median) becomes

a + b \times

center; spread (SD, IQR, range) becomes

|b| \times

spread. Adding

a

does not change the spread.

Scores: mean 60, SD 8. New score $= 1.5 \times$ old $+ 10$ .

New mean

= 1.5 \times 60 + 10 = 100

Apply to the center.

New SD

= 1.5 \times 8 = 12

Only the factor affects spread.

Your turn: mean 20, SD 4, new $= 3x - 5$ . The new SD.

12

|3| × 4.

A set of test scores has standard deviation 14. The teacher adds 14 points to every score. What is the new standard deviation?
A set of test scores has standard deviation 8. The teacher adds 12 points to every score. What is the new standard deviation?

Lengths have standard deviation 7 cm. Each length is multiplied by 3. What is the new standard deviation, in the new units?
Lengths have standard deviation 10 cm. Each length is multiplied by 3. What is the new standard deviation, in the new units?

Daily temperatures have mean 11 °C. With $F = \tfrac{9}{5}C + 32$ , what is the mean in °F?
Daily temperatures have mean 29 °C. With $F = \tfrac{9}{5}C + 32$ , what is the mean in °F?

Three questions to check you have got it.

Lengths have standard deviation 7 cm. Each length is multiplied by 3. What is the new standard deviation, in the new units?
Lengths have standard deviation 10 cm. Each length is multiplied by 3. What is the new standard deviation, in the new units?
Lengths have standard deviation 2 cm. Each length is multiplied by 2. What is the new standard deviation, in the new units?

Transforming data

$a + bx$ : center → $a + b\cdot$ center; spread → $|b|\cdot$ spread.

Watch out: adding a constant never changes the spread.

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