Shifting and stretching data

What you will be able to do

• predict the mean, median, standard deviation, variance and IQR after adding a number to every value or multiplying every value;
• use the rules for a change of unit, such as minutes to seconds or degrees Celsius to degrees Fahrenheit;
• standardise a data set to mean 0 and standard deviation 1.

Marks out of 40

A made-up test is marked out of 40. The seven marks are $17, 20, 23, 26, 29, 32$ and $35$. Their mean is 26 and their population standard deviation is 6 marks. The teacher wants percentages, so she multiplies every mark by 2.5. Then she adds a bonus of 5 for a hard question. What happens to the mean and the standard deviation?

Adding a number

Adding 5 to every value slides the whole data set 5 to the right. Every value moves, and so does every centre: the mean, the median and the mode all go up by 5. But the distances between the values do not change. So the range, the IQR, the MAD and the standard deviation stay the same. In the picture, the bonus moves the mean from 65 to 70, and the SD stays 15.

Multiplying by a number

Multiplying every value by 2.5 stretches the data. Every distance between values becomes 2.5 times as large. So the centres and the spreads are all multiplied by 2.5. The mean goes from 26 to 65, and the SD from 6 to 15 marks. The variance, which is a square, is multiplied by $2.5^2 = 6.25$, from 36 to 225.

What if the multiplier is negative, such as $-2$? The data is flipped as well as stretched. The mean is multiplied by $-2$, but the SD is multiplied by $|-2| = 2$. A spread is never negative.

The rules together

For a change $y = a + bx$:

The median follows the same rule as the mean, and the IQR and the range follow the rule of the SD. These rules hold for the sample SD $s$ as well.

Changing units. A temperature in degrees Fahrenheit is $F = 1.8C + 32$. A town's daily highs in spring have mean $15\ ^\circ\text{C}$ and SD $2.5\ ^\circ\text{C}$. In Fahrenheit the mean is $1.8 \times 15 + 32 = 59\ ^\circ\text{F}$, and the SD is $1.8 \times 2.5 = 4.5\ ^\circ\text{F}$. The 32 moves the centre only.

You can meet more of these rules, together with combining two groups, in the Maths course on Comparing and Combining Data.

Standardising

Here is a special change. Subtract the mean from every value, then divide by the standard deviation:

Subtracting the mean moves the centre to 0. Dividing by $\sigma$ makes the spread 1. So the new values always have mean 0 and standard deviation 1, whatever the units were. These new values are called z-scores, and the next lesson puts them to work.

You should find that the slider $a$ moves the amber dots along without changing their spacing, so only the mean changes. The slider $b$ stretches or squeezes them, and with a negative $b$ their order is reversed. The SD after is always $|b|$ times the SD before, about $3.85$.

Practise with new numbers

Try it in Python

The program changes the seven marks to percentages, adds the bonus, and prints the mean and the population SD at each step.

Common mistakes

• Adding to the SD. Adding a number never changes the spread.
• Multiplying the variance by $b$. The variance is multiplied by $b^2$.
• A negative SD. Multiply the SD by $|b|$.
• Changing only some values. These rules need the same change for every value.

Remember these

• Adding $a$: centres go up by $a$; spreads stay the same.
• Multiplying by $b$: centres times $b$; SD, IQR and range times $|b|$; variance times $b^2$.
• $y = a + bx$ gives $\bar{y} = a + b\bar{x}$ and $\sigma_y = |b|\,\sigma_x$.
• Standardising, $z = \dfrac{x - \bar{x}}{\sigma}$, gives mean 0 and SD 1.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A data set has mean 40 and SD 6. Every value is increased by 12. Find the new mean and the new SD.

2. Times have mean 8 minutes and SD 1.5 minutes. Change them to seconds.

3. A data set has variance 9. Every value is multiplied by 4. Find the new variance and the new SD.

4. Every value is changed by $y = 10 - 2x$. The SD of $x$ was 3. What is the SD of $y$?

5. After standardising, what are the mean and the SD of any data set?

References

• OpenStax, Introductory Statistics, Chapter 2 (Descriptive Statistics), section on measures of the spread of the data.