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Lesson 3 of 13

Tales by Dots and Lines · Lesson 3 of 13

Shifting and Scaling Data

“Explain how the mean changes when every observation is adjusted in the same way.”

Learning Objectives

• Predict the mean after adding or subtracting a constant from every value. • Predict the mean after multiplying or dividing every value by a common factor. • Explain the rules using totals and dot plots. • Apply the rules to measurement corrections and unit conversions.

Suppose every child is measured while wearing shoes that add exactly 1 cm to their height. You are given the mean of the recorded heights. Must you correct every height separately before finding the mean without shoes? No. The same correction affects every observation, so it also affects the mean in a predictable way.

This lesson studies whole-dataset changes. The number of observations stays fixed while each value is shifted or scaled. A shift adds a fixed amount; a scaling multiplies by a fixed factor. Keeping those two actions distinct makes many calculations quicker and easier to check.

Relatively Unchanged!

Consider 2, 4, and 9, whose mean is 5. Add 3 to every value to obtain 5, 7, and 12. The new total is 24 and the new mean is 8. The mean has increased by exactly 3, just as each observation did.

On a dot plot, every dot moves 3 units to the right. Its distance from the new mean is the same as its old distance from the old mean. The whole pattern moves together, so the balance point moves with it.

Before the shiftBefore the shift012345678910111213Mean = 5
Before the shift— The mean is 5; use the same scale in the next diagram.
After adding 3 to every valueAfter adding 3 to every value012345678910111213Mean = 8
After adding 3 to every value— Every dot and the mean move 3 units to the right.
Adding a constantLaTeX
Here m is the original mean and the same amount c is added to every value.
Correcting recorded heights

Problem
The mean recorded height of a group wearing identical 1 cm soles is 150.2 cm. Find the mean height without that added sole height.

  1. 1.Each recorded height includes an extra 1 cm.
  2. 2.Subtracting 1 cm from every observation subtracts 1 cm from the mean.
  3. 3.The corrected mean is 150.2 − 1 = 149.2 cm.
  4. 4.We do not need the number of children because the same correction applies to every observation.

Why the Shift Rule Works

If there are n observations, adding c to each one adds n × c to their total. When we divide the new total by n, the extra n × c becomes c. Subtracting a constant works in exactly the same way, with a decrease instead of an increase.

Notice that this rule says every value. If only one value increases by c, the total increases only by c, so the mean increases by c ÷ n. If all n values increase by c, the total increase is n times as large. This difference is useful when correcting a single recording error.

One change compared with many changes

Problem
Four delivery times have mean 18 minutes. Compare adding 4 minutes to every time with adding 4 minutes to just one time.

  1. 1.The old total is 4 × 18 = 72 minutes.
  2. 2.Adding 4 to every time adds 4 × 4 = 16 to the total. The new mean is 88 ÷ 4 = 22 minutes.
  3. 3.Adding 4 to only one time adds 4 to the total. The new mean is 76 ÷ 4 = 19 minutes.
  4. 4.The two adjustments produce different means because they change different numbers of observations.

Multiplying Every Value

Now double 2, 4, and 9 to obtain 4, 8, and 18. The total doubles from 15 to 30, while the count stays 3. The mean therefore doubles from 5 to 10. More generally, multiplying every value by the same factor multiplies their sum, and hence their mean, by that factor.

This rule helps when converting units. A length in metres becomes 100 times as large numerically when expressed in centimetres. The mean expressed in centimetres is also 100 times the numerical mean expressed in metres. The physical lengths have not changed; only the unit has changed.

Scaling the meanLaTeX
Dividing each observation by a nonzero number divides the mean by that number.
Converting lengths

Problem
The mean length of several pieces of rope is 1.25 m. What is the mean in centimetres?

  1. 1.Each length is multiplied by 100 when metres are converted to centimetres.
  2. 2.The mean undergoes the same conversion: 1.25 × 100 = 125.
  3. 3.The mean length is 125 cm.
  4. 4.The answer includes the new unit; 125 without a unit would not fully describe the measurement.
Dividing quantities equally

Problem
Three containers hold 6, 10, and 14 litres. Half the contents of each container is used. Find the mean amount remaining.

  1. 1.The original total is 30 L, so the original mean is 10 L.
  2. 2.Each remaining amount is half the original: 3, 5, and 7 L.
  3. 3.The mean is also halved: 10 ÷ 2 = 5 L.
  4. 4.Check directly: (3 + 5 + 7) ÷ 3 = 5 L.

Combining a Shift and a Scale

Some situations apply two changes in sequence. Follow their order carefully. Doubling each value and then adding 3 is different from first adding 3 and then doubling. You can apply the same sequence to the mean, because each stage affects every observation uniformly.

Two stages in the right order

Problem
A dataset has mean 7. Every value is multiplied by 4 and then reduced by 5. Find the final mean.

  1. 1.After multiplication by 4, the mean is 7 × 4 = 28.
  2. 2.Subtracting 5 from each transformed value subtracts 5 from that mean.
  3. 3.The final mean is 28 − 5 = 23.
  4. 4.Doing the operations in reverse order would give (7 − 5) × 4 = 8, which answers a different question.

Make a Transformation Table

Choose the small dataset 3, 6, and 9. Use it to explain the rules to a partner.

  1. Find its mean, then list the data after adding 2 to every value.
  2. List the data after multiplying each original value by 3.
  3. List the data after multiplying each original value by 3 and then adding 2.
  4. Compare the new means with 6 + 2, 6 × 3, and 6 × 3 + 2.

The resulting means are 8, 18, and 20. Checking a short dataset makes the rule visible; the explanation using totals shows why it works for any dataset.

Watch out

An average does not remove units. Also, a rule for changing every value cannot be used when only one observation changes or when a new observation is inserted.

Quiz

Quick check

Every value increases by 6. An old mean of 12 becomes what?

Quick check

Every value is divided by 5. The old mean is 30. What is the new mean?

Quick check

Ten values have mean 8. Only one value increases by 10. What is the new mean?

Quick check

The mean is 4. Every value is tripled and then increased by 2. What is the new mean?

Quick check

Which action leaves the number of observations unchanged?

Adding 6 to every observation adds 6 to the mean: 12 + 6 = 18.

Practice Problems

Practice Problems
  1. Six temperatures have mean 24°C. Every sensor reading is 2°C too high. Find the corrected mean.
  2. The mean length is 85 cm. Express the mean in metres.
  3. The values 4, 7, and 10 are each reduced by 3. Find the old and new means.
  4. A dataset has mean 9. First add 5 to every value, then double every result. Find the new mean.
  5. Eight values have mean 20. One value was recorded 8 too low. Correct the mean.
  6. Explain why multiplying every value by 3 triples the mean.

Step 1: Subtract 2°C from every reading. Step 2: The mean decreases by 2°C, giving 22°C.

Key Takeaways

Key Takeaways

• Adding or subtracting a constant from every value shifts the mean by that constant. • Multiplying or dividing every value scales the mean by the same factor. • A change to one observation affects the mean by that change divided by the count. • For several uniform transformations, keep the operation order and units clear.

Previous · Lesson 2

How Adding or Removing Data Changes the Mean

Next · Lesson 4

The Median When Data Changes