Tales by Dots and Lines · Lesson 2 of 13
How Adding or Removing Data Changes the Mean
“Predict and explain changes to a mean when observations enter or leave a dataset.”
• Predict how inserting or removing a value affects the mean. • Calculate a new mean using the old total. • Choose pairs or groups that leave a mean unchanged. • Distinguish adding an observation from changing every observation.
A reading group has recorded its daily reading times, and the mean is 20 minutes. A new record of 35 minutes arrives. You can already predict that the mean will rise: the new time brings more than the old fair share. But the mean will not jump straight to 35, because the earlier observations still matter.
In this lesson, first make a prediction and then calculate to check it. Thinking about excesses and shortages often reveals the direction of change before you know every original value. The old mean and the old count are enough to recover the old total.
Adding One Value
Suppose there are n observations with mean m. Their total is n × m. Adding a new observation x changes both the total and the count: the total becomes n × m + x, and the count becomes n + 1. Forgetting the new count is a common source of wrong answers.
If x is greater than m, the extra observation supplies more than one old fair share, so the new mean rises. If x is less than m, the new mean falls. If x equals m, the new observation supplies exactly one fair share and the mean stays unchanged.
| Inserted value | Effect on the mean |
|---|---|
| Greater than the old mean | Increases |
| Equal to the old mean | Unchanged |
| Less than the old mean | Decreases |
Problem
Four reading sessions have mean 20 minutes. A fifth session lasts 35 minutes. Find the new mean.
- 1.Predict an increase because 35 is greater than 20.
- 2.The old total is 4 × 20 = 80 minutes.
- 3.The new total is 80 + 35 = 115 minutes, and the new count is 5.
- 4.The new mean is 115 ÷ 5 = 23 minutes. The extra 15 minutes are shared across all 5 observations.
Removing One Value
Removal reverses the excess-and-shortage idea. Removing a value above the old mean removes an excess, so the remaining mean falls. Removing a value below the old mean removes a shortage, so the remaining mean rises. Removing a value equal to the mean leaves the mean unchanged, provided at least one observation remains.
Only a value actually present in the dataset can be removed. Also, an empty dataset has no mean: we cannot divide a total by zero. These conditions matter when explaining a rule in words.
Problem
The values are 8, 10, 12, and 14. Remove 8 and find the new mean.
- 1.The original total is 44, so the original mean is 44 ÷ 4 = 11.
- 2.Because 8 is below 11, predict that removing it will increase the mean.
- 3.The remaining total is 44 − 8 = 36, and there are 3 observations.
- 4.The new mean is 36 ÷ 3 = 12, agreeing with the prediction.
Problem
Five measurements have mean 18 cm. One of the measurements is 18 cm and is removed. Find the new mean.
- 1.Recover the old total: 5 × 18 = 90 cm.
- 2.Subtract the removed value: 90 − 18 = 72 cm.
- 3.There are now 4 measurements, so the new mean is 72 ÷ 4 = 18 cm.
- 4.Removing one full fair share preserves the mean.
Unchanging Mean!
You can add more than one value without changing a mean. The inserted group must have the same mean as the original dataset. For two new observations, their total must be twice the old mean. For three, their total must be three times the old mean.
The inserted values need not be identical. For a mean of 10, the pair 7 and 13 works because its shortage of 3 balances its excess of 3. The triple 6, 9, and 15 also works: its two shortages total 5, matching the excess of 5. Many different groups can satisfy the same condition.
Problem
A dataset has mean 12. Add three different values, two below 12 and one above 12, without changing the mean.
- 1.The three added values must total 3 × 12 = 36.
- 2.Choose two below 12, such as 8 and 10. Their total is 18.
- 3.The third must be 36 − 18 = 18, which is above 12.
- 4.Check the balance: shortages are 4 + 2 = 6; the excess is 6. Thus 8, 10, and 18 work.
Comparing Two Groups
When several observations are added together, compare their group mean with the old mean. A group whose mean is higher raises the combined mean; a group whose mean is lower lowers it. One high value alone does not settle the question if other inserted values are low.
The size of the original dataset controls how far the mean changes. Adding 30 to two observations with mean 10 gives a larger shift than adding 30 to twenty observations with mean 10. In the second case, the extra amount is shared among more observations.
Design a Group That Balances
Use 15 as a target mean and create several possible additions.
- Write a pair with one value below 15 and one above it.
- Write a triple with two values above 15 and one below it.
- Check each group total before deciding that the mean is preserved.
The pair 11 and 19 totals 30. The triple 8, 17, and 20 totals 45. Both have mean 15, although their individual values are quite different.
Adding one new observation changes the count. Adding the same amount to every existing observation leaves the count unchanged. These are different operations, so read the question carefully.
Quiz
A dataset has mean 9. A new value 14 is inserted. What happens?
Which pair preserves a mean of 8?
A value below the old mean is removed, leaving some observations. What happens?
Six values have mean 5. After inserting 12, what is the new mean?
Three added values will preserve mean 11. What must their total be?
The inserted value is above the old mean, so it supplies an excess and increases the mean.
Practice Problems
- Five values have mean 14. Insert 20. Find the new mean.
- Four values have mean 9. Remove one value equal to 15. Find the remaining mean.
- A pair 7 and x is inserted into data with mean 13. Find x if the mean is unchanged.
- Construct three positive values with mean 10, two above 10 and one below 10.
- Can inserting one value above the old mean and one below it still raise the mean?
- Six values have mean 10. An existing pair 6 and 14 is removed. What is the new mean?
Step 1: The old total is 5 × 14 = 70. Step 2: The new total is 90 and the new count is 6. Step 3: The new mean is 15, which is greater than 14.
Key Takeaways
• Insert an above-mean value to raise the mean, or a below-mean value to lower it. • Removing a value reverses these directions. • Use the old mean and count to recover the old total. • A group preserves the mean exactly when its own mean matches the old mean.