How to Perform in PSLE | Learner’s Guide Vol 0682 | Mathematics: When One Data Value Changes, the Average Changes by That Amount Divided by the Number of Values

CHAPTER 1 OF 12 · Orient · Back to contents

1. Why one changed value affects everyone’s equal share

An average is an equal-share value. If five numbers have total 90, their average is 18 because 90 shared across five gives 18. When one data value rises by 10 and the number of values stays five, the total rises by 10. The new equal share therefore rises by 10÷5=2, not by 10.

This relationship produces a powerful but conditional rule: average change = total change ÷ number of values. The condition matters. It works when the count stays fixed. If a value is added or removed, both the total and denominator change, so the total must be rebuilt.

The rule is not a trick to memorise in isolation. It follows directly from average = total ÷ count. Understanding the derivation lets you handle replacement, correction, several changes, reverse questions and decimal data.

• Confirm the count remains fixed.
• Find the signed net change in total.
• Divide the net change by the count.
• Apply the average change to the old average.
• Keep units and direction.
• Rebuild totals when count changes.

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CHAPTER 2 OF 12 · Model · Back to contents

2. Derive the change rule

Let n be the number of values, T the old total and A the old average. Then A=T÷n. If the total changes by d while n stays fixed, the new average is (T+d)÷n = T÷n + d÷n = A+d÷n. This is the change rule.

For a replacement, d = new value − old value. Replacing 40 with 52 creates d=12, not 52. For several corrections, add signed changes before dividing.

For a reverse question, multiply the required average change by the count. If an average of 12 values must rise by 3, total must rise by 36. If only one entry changes, that entry must rise by 36.

Direction provides an estimate. A positive total change raises the average; a negative change lowers it; net zero leaves it unchanged. The average change’s magnitude is smaller than the single total change when more than one value shares it.

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CHAPTER 3 OF 12 · Model · Back to contents

3. Totals, tables and balance models

A balance model makes the equal-share mechanism visible. Imagine the total distributed evenly into n boxes. Adding d units to the total and redistributing gives d÷n extra units in each box. This is why a single corrected value can affect the average without changing every original value.

A four-row audit table is efficient: old count, old total, net total change, new average. For fixed count, the new count repeats the old count. For count-changing questions, add a separate new-count row and rebuild.

Use two methods for verification. The shortcut finds average change directly. The full method calculates old total, adjusts it and divides. Agreement between them is strong evidence; disagreement reveals a sign, replacement or count error.

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CHAPTER 4 OF 12 · Work examples · Back to contents

4. Worked foundations

Worked case 0: one value increases

Question. The average of 5 numbers is 18. One number increases by 10 while the other four stay unchanged. Find the new average.

Tempting error. Increasing the average by 10 to get 28. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. The total increases by 10. Spread across 5 values, the average increases by 10÷5=2. New average = 20.

Mechanism. Old total is 90; new total is 100; 100÷5 confirms 20. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Change the increase to 15 and predict the new average before calculating. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

Worked case 1: one value decreases

Question. The average mass of 8 parcels is 6 kg. One parcel’s recorded mass is reduced by 4 kg. What is the corrected average?

Tempting error. Subtracting 4 kg directly from 6 kg. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. Total decreases by 4 kg across 8 parcels, so average decreases by 4÷8=0.5 kg. Corrected average = 5.5 kg.

Mechanism. The equal-share effect of one correction is diluted by the number of values. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Reconstruct old and new totals to verify units. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

Worked case 2: replacement with a larger value

Question. Six scores have average 72. A score of 65 is replaced by 83. Find the new average.

Tempting error. Adding 83 to the old total without removing 65. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. Net total change = 83−65=18. Average change = 18÷6=3. New average = 75.

Mechanism. Replacement requires a net change, not the full incoming value. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Reverse the replacement and explain the sign change. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

Worked case 3: correction of a data entry

Question. The average of 10 values was calculated as 24, but 17 was entered as 71. Find the correct average.

Tempting error. Subtracting 71 or adding 17 without comparing them. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. The recorded total is too high by 71−17=54. Correct average = 24−54÷10 = 18.6.

Mechanism. A transposition error changes total by the difference between wrong and correct entries. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Verify through totals: recorded 240, correct 186. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

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CHAPTER 5 OF 12 · Work examples · Back to contents

5. Replacement and correction cases

Worked case 5: two values change

Question. The average of 7 numbers is 30. One value rises by 8 and another falls by 1. Find the new average.

Tempting error. Treating the changes separately without combining signs. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. Net total change = +8−1=+7. Average rises by 7÷7=1. New average = 31.

Mechanism. Multiple fixed-count changes combine into one signed total change. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Change the fall to 15 and predict direction first. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

Worked case 6: average unchanged

Question. In a set of 9 values, one rises by 12 and another falls by 12. What happens to the average?

Tempting error. Assuming any changed data must change the average. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. Net total change is 0, count remains 9, so the average is unchanged.

Mechanism. Average depends on total and count, not on whether individual positions changed. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Give two different changes with net zero. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

Worked case 7: find the required correction

Question. The average of 12 readings must increase from 15 to 17 by changing only one reading. By how much must that reading increase?

Tempting error. Increasing it by 2 because the average needs to rise by 2. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. Required total increase = 12×2=24. The one reading must increase by 24.

Mechanism. Reverse the rule: total change = average change × count. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Check by old total 180 and new total 204. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

Worked case 8: unknown replacement

Question. Five numbers have average 14. Replacing 9 with x raises the average to 17. Find x.

Tempting error. Setting x=12 because the average rose by 3. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. Total must rise by 5×3=15. Since x replaces 9, x−9=15, so x=24.

Mechanism. Average change reveals net total change; replacement equation identifies the new value. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Solve again through old and new totals. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

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CHAPTER 6 OF 12 · Work examples · Back to contents

6. Reverse and comparison cases

Worked case 9: fractional change

Question. The mean of 4 measurements is 7.5 cm. One measurement increases by 1.2 cm. Find the new mean.

Tempting error. Avoiding the shortcut because decimals are present. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. Mean increase = 1.2÷4=0.3 cm. New mean = 7.8 cm.

Mechanism. The change rule is structural and works with decimals when count is fixed. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Rebuild totals 30.0 and 31.2. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

Worked case 10: count changes: new member joins

Question. Five pupils average 60 marks. A sixth pupil scores 90. Can we add (90÷5) to the average?

Tempting error. Using the fixed-count change rule even though a value was added. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. No. Old total = 300; new total = 390; new count = 6; average = 65.

Mechanism. The short rule assumes a fixed count. Adding a value changes both total and denominator. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Compare replacing one score with 90, which keeps the count fixed. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

Worked case 11: count changes: value removed

Question. Eight readings average 12. One reading of 5 is removed. Find the new average.

Tempting error. Subtracting 5÷8 from 12. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. Old total = 96. Remove 5 to get 91, then divide by new count 7: 13.

Mechanism. Removal changes both total and count, so rebuild rather than use total change divided by the old count. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Explain why the average rises even though total falls. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

Worked case 12: comparison of two data sets

Question. Set A has 4 values and one rises by 8. Set B has 8 values and one rises by 8. Which average changes more?

Tempting error. Saying both averages rise equally because the same value change occurred. This usually happens when the learner changes the average directly without first identifying how the total changed.

Controlled solution. A rises by 8÷4=2; B rises by 8÷8=1. Set A changes more.

Mechanism. The same total change is spread across fewer values in Set A. For a fixed number of values, average = total ÷ count. Therefore a total change is shared across the same count, so average change = total change ÷ count.

Transfer. Generalise how count affects sensitivity of the average. Solve the variation once through total reconstruction and once through the change rule, then compare the reasoning.

Audit. Check whether the number of values stayed fixed. If the count changed, the short rule cannot be applied without rebuilding the total.

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CHAPTER 7 OF 12 · Repair errors · Back to contents

7. Misconceptions and repairs

0. Add the full value change to the average

Why it fails. The change affects the total, which is then shared over all values. Repair the reasoning by writing old total, total change, new total and count before dividing.

Diagnostic prompt. How many values share the total change? Ask for an estimate: should the average change by the full amount, a fraction of it, or in the opposite direction?

1. Divide by the changed value

Why it fails. The denominator is the number of data values, not the size of the changed entry. Repair the reasoning by writing old total, total change, new total and count before dividing.

Diagnostic prompt. What remains fixed in average = total ÷ count? Ask for an estimate: should the average change by the full amount, a fraction of it, or in the opposite direction?

2. Use the incoming replacement value as the change

Why it fails. A replacement removes the old value and adds the new, so use their difference. Repair the reasoning by writing old total, total change, new total and count before dividing.

Diagnostic prompt. What is new minus old? Ask for an estimate: should the average change by the full amount, a fraction of it, or in the opposite direction?

3. Ignore signs when several values change

Why it fails. Increases and decreases combine as signed total changes. Repair the reasoning by writing old total, total change, new total and count before dividing.

Diagnostic prompt. What is the net change before division? Ask for an estimate: should the average change by the full amount, a fraction of it, or in the opposite direction?

4. Use the shortcut when a value is added

Why it fails. The rule requires a fixed count; a new member changes the denominator. Repair the reasoning by writing old total, total change, new total and count before dividing.

Diagnostic prompt. Did the number of values stay fixed? Ask for an estimate: should the average change by the full amount, a fraction of it, or in the opposite direction?

5. Use the shortcut when a value is removed

Why it fails. Removal changes both total and count. Repair the reasoning by writing old total, total change, new total and count before dividing.

Diagnostic prompt. What are the new total and new count? Ask for an estimate: should the average change by the full amount, a fraction of it, or in the opposite direction?

6. Assume changed data means changed average

Why it fails. Offsets can leave total unchanged even when individual values move. Repair the reasoning by writing old total, total change, new total and count before dividing.

Diagnostic prompt. What is the net total change? Ask for an estimate: should the average change by the full amount, a fraction of it, or in the opposite direction?

7. Forget units

Why it fails. Average retains the unit of the data, while total may represent unit-sum across entries. Repair the reasoning by writing old total, total change, new total and count before dividing.

Diagnostic prompt. What unit should the final average carry? Ask for an estimate: should the average change by the full amount, a fraction of it, or in the opposite direction?

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CHAPTER 8 OF 12 · Examination control · Back to contents

8. Examination control

Use the CNT routine: Count fixed? Net total change? Total change divided by count. If the count is not fixed, abandon the shortcut and rebuild total and denominator.

• Circle words such as replaced, corrected, added or removed.
• Write new minus old for a replacement.
• Combine increases and decreases with signs.
• Estimate direction before division.
• Attach units to the average.
• Verify with total reconstruction when time allows.
• Stop after count, total and final average agree.

Stop rule. Never apply the short rule merely because a question mentions average and change. It is authorised only after the count check passes.

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CHAPTER 9 OF 12 · Practice · Back to contents

9. Independent practice I

Practice 0: Average of 6 values is 20. One value rises by 12.

Explained answer. Average rises by 12÷6=2, so new average is 22. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Verify using old and new totals. Return later and verify by reconstructing a possible data set or its total.

Practice 1: Average of 9 values is 14. One falls by 18.

Explained answer. Average falls by 18÷9=2, so new average is 12. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Create a possible data set and alter one value. Return later and verify by reconstructing a possible data set or its total.

Practice 2: Eight scores average 50. Replace 42 with 58.

Explained answer. Net increase 16; average increases 16÷8=2; new average 52. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Reverse the replacement. Return later and verify by reconstructing a possible data set or its total.

Practice 3: Ten readings average 31. A wrong 46 should be 26.

Explained answer. Total decreases 20; average decreases 2; correct average 29. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Show the total reconstruction. Return later and verify by reconstructing a possible data set or its total.

Practice 4: Seven values: one rises 9, one falls 2.

Explained answer. Net +7; average rises 1. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Change the fall to 16 and predict the new direction. Return later and verify by reconstructing a possible data set or its total.

Practice 5: What change to one entry raises the average of 15 values by 4?

Explained answer. Required total and entry increase = 15×4=60. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Explain why the answer exceeds 4. Return later and verify by reconstructing a possible data set or its total.

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CHAPTER 10 OF 12 · Practice · Back to contents

10. Independent practice II

Practice 7: Four numbers average 25. Replace 18 with x to make average 30.

Explained answer. Total rise 4×5=20; x−18=20; x=38. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Solve through totals. Return later and verify by reconstructing a possible data set or its total.

Practice 8: A mean of 11 values should fall by 0.5 after correcting one entry. What total correction is needed?

Explained answer. Total must fall by 11×0.5=5.5. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Choose an old and new entry with that difference. Return later and verify by reconstructing a possible data set or its total.

Practice 9: Six pupils average 70; a seventh scores 84.

Explained answer. Old total 420; new total 504; new average 72. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Explain why fixed-count rule is invalid. Return later and verify by reconstructing a possible data set or its total.

Practice 10: Nine values average 18; a value 10 is removed.

Explained answer. Old total 162; new total 152; new count 8; average 19. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Explain why average rose. Return later and verify by reconstructing a possible data set or its total.

Practice 11: One value rises by 20 in a set of 5 and in a set of 10.

Explained answer. Average changes are 4 and 2 respectively; fewer values produce the larger change. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Write a general statement. Return later and verify by reconstructing a possible data set or its total.

Practice 12: Two entries swap positions. What happens to the average?

Explained answer. Nothing. The total and count are unchanged; order does not affect average. The solution states the count and the total change, not only the final average.

Delayed-transfer variation. Contrast swapping with replacing one value. Return later and verify by reconstructing a possible data set or its total.

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CHAPTER 11 OF 12 · Transfer · Back to contents

11. Delayed transfer and proof

On day one, practise one-value increases and decreases. On day three, mix replacements and data-entry corrections. On day seven, include added or removed values so the learner must decide whether the fixed-count shortcut is allowed.

Prove the change rule algebraically or with a balance drawing, then explain it in words. A learner who can move among equation, model and verbal explanation is less likely to misuse the rule.

Use sensitivity comparisons. The same total change produces a smaller average change in a larger data set because it is shared across more values. This prepares learners to reason before calculating.

Seven-day test. Solve four unseen cases: fixed-count increase, replacement, reverse correction and added member. State whether the shortcut is valid before each solution and verify one case by totals.

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CHAPTER 12 OF 12 · Transfer · Back to contents

12. Family guidance, FAQs and next routes

Guidance for parents and tutors

Ask “Did the count stay the same?” before correcting arithmetic. If yes, ask for net total change. If no, ask for old total and new count. These two prompts reveal the precise branch of reasoning.

Fade support from balance models to audit tables to mental estimates. Do not reward shortcut speed when the learner cannot state its condition. Conditional fluency is safer than fast rule recall.

Frequently asked questions

Why divide the change by the count?

Because the average is the total shared equally across that count. A total change is redistributed across all values.

What if two values change?

Add their signed changes to find the net total change, then divide by the unchanged count.

What if one value is replaced?

Use new minus old as the net total change. Do not add the incoming value without removing the outgoing one.

Can the average stay unchanged when values change?

Yes. If increases and decreases cancel so total remains unchanged and count stays fixed, the average remains unchanged.

Why does removing a low value sometimes raise the average?

Removal lowers total but also lowers the count. The remaining values may have a larger equal-share value, so rebuild both.

How do I reverse the rule?

Total change = average change × count when the count is fixed.

Next routes

Continue at the Mathematics tuition hub or return to the series directory.

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