Average Based on Misread Numbers: Corrected Average Formula

Average Based on Misread Numbers questions involve correcting an average after one or more values were recorded incorrectly. The correction depends on the difference between the correct and misread values divided by the total number of observations. Use the wrong average to find the wrong total, apply the value correction, and then calculate the corrected average.

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What Is Average Based on Misread Numbers?

Average based on misread numbers is the average calculated using one or more incorrect values instead of their correct values. The corrected average is found by adding the net correction to the reported average.

If there are n observations and a value x was misread as y, the reported total contains y instead of x. Replace y with x by changing the total by x − y. Therefore, corrected average = reported average + (x − y)/n. If x is less than y, the corrected average decreases; if x is greater than y, it increases.

Average Based on Misread Numbers Formula & Tricks

Important Formulas

Single misread number
Corrected average = Wrong average + (Correct value − Misread value) / n

Here, n is the total number of observations. The correction is based on the difference between the correct and misread values.

Using totals
Corrected total = n × Wrong average − Misread value + Correct value

First calculate the total based on the wrong average, remove the misread value, and add the correct value.

Multiple misread numbers
Corrected average = Wrong average + Σ(Correct value − Misread value) / n

Add the corrections for all misread values and divide their net correction by the total number of observations.

Finding a misread value
Correct value = Misread value + n × (Corrected average − Wrong average)

This rearranged formula is used when the corrected average and the misread average are known.

Quick Tricks

Check the direction of correction

If the correct value is smaller than the misread value, subtract from the wrong average. If the correct value is larger, add to it.

Example: A value 48 was read as 84 in a set of 9 numbers. Correction = (48 − 84)/9 = −4, so the corrected average is 4 less than the wrong average.
Correct the total before dividing

When signs seem confusing, calculate the wrong total first, replace each misread value, and divide the corrected total by n.

Example: For 6 numbers with wrong average 20, a value 14 was read as 41. Wrong total = 120; corrected total = 120 − 41 + 14 = 93; corrected average = 93/6 = 15.5.
Combine all errors into one net correction

For several misread values, add every correct-minus-misread difference before dividing by the number of observations.

Example: If errors are 25 read as 30 and 18 read as 12, net correction = (25 − 30) + (18 − 12) = 1. For 10 values, the average increases by 0.1.

Average Based on Misread Numbers Concepts

Correction for One Misread Value

For one misread value, add the difference between the correct value and the misread value divided by the number of observations.

Let the reported average be A, the correct value be x, the misread value be y, and the number of observations be n. The wrong total is nA. After replacement, the corrected total is nA − y + x, so the corrected average is A + (x − y)/n.

Example: The average of 8 numbers was reported as 24 because 17 was recorded as 71. Corrected average = 24 + (17 − 71)/8 = 24 − 6.75 = 17.25.

Correction Through Wrong and Correct Totals

The total based on the wrong average is found by multiplying the wrong average by the number of observations.

Use Wrong total = n × Wrong average. Remove the misread number and insert the correct number. This method gives Correct total = Wrong total − Misread value + Correct value, followed by Correct average = Correct total/n.

Example: For 5 numbers, the wrong average is 32, and 46 was entered instead of 16. Wrong total = 5 × 32 = 160. Correct total = 160 − 46 + 16 = 130, so the corrected average is 26.

Multiple Misread Numbers

When several values are misread, apply all replacements to the total before dividing by the same number of observations.

For each replacement, calculate correct value − misread value. Add these differences to the wrong average after dividing their sum by n. The number of observations remains unchanged when values are only corrected, not added or removed.

Example: For 10 numbers, the wrong average is 30. Values 25 and 40 were recorded as 35 and 32. Net correction = (25 − 35) + (40 − 32) = 2. Corrected average = 30 + 2/10 = 30.2.

Finding the Misread or Correct Value

The correction formula can be rearranged to find an unknown correct or misread value.

If the correct average is known, Correct value − Misread value = n × (Corrected average − Wrong average). Thus, Correct value = Misread value + n × (Corrected average − Wrong average). The corresponding misread value is Misread value = Correct value − n × (Corrected average − Wrong average).

Example: For 12 observations, the wrong average is 18 and the corrected average is 20. If the misread value was 9, the correct value = 9 + 12 × (20 − 18) = 33.

Average Based on Misread Numbers Video Lessons

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Average Based on Misread Numbers

Learn how to calculate the correct average when one or more numbers are misread, using the difference between the incorrect and actual values to adjust the result.

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Practice Average Based on Misread Numbers Questions

Practise published questions related to this topic.

1Least Common Multiple of A and B is 15. Least Common Multiple of C and D is 25. What is the Least Common Multiple of A, B, C and D?→ 2Find the single equivalent discount (rounded off to two decimal places) for successive discounts of 16%, 29% and 4%.→ 3A, B and C start at the same time in the same direction to run around a circular stadium. A completes a round in 180 seconds, B in 270 seconds and C in 360 seconds. All of them start from the same point. After how many minutes will they again be together at the starting point?→ 4What is the smallest value of a in the number a3724, so that the number formed is divisible by 3?→ 5When an article is sold for a discount of 40 percent, then there is a profit 50 percent. If the cost price of article is Rs. 1000, then what is the marked price?→ 6The 7-digit number 77A444B is divisible by 24. What is the maximum value of (A + B)?→ 7A dealer purchased an AC for ₹1,848. After allowing a discount of 58% on its marked price, he still gains 63%. Find the marked price of the AC.→ 8Find the value of √(((169)² × (2.56)² × (0.25)²)/((13)⁴ × (1.6)⁴ × (0.5)²))→ 9A trader marked the price of an item 40% above the cost price. He allowed two successive discounts of 20% and 25% to a customer. If he incurred a loss of ₹224, at what price did he sell the item to that customer?→ 10A trader purchased goods for Rs. 1200 and sold them for Rs. 1440. What is the profit percentage?→

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Quick Revision Notes

Average Based on Misread Numbers: Quick Revision

Use the net change in the total to correct an average after replacing misread values.

  • Wrong total = Number of observations × Wrong average.
  • For one replacement, corrected average = Wrong average + (Correct value − Misread value)/n.
  • For several replacements, add all correct-minus-misread differences and divide the sum by n.
  • If the correct value is smaller than the misread value, the average decreases.
  • The number of observations remains unchanged when an existing value is corrected.
  • When the total is not given, calculate it from the reported average before making corrections.

Average Based on Misread Numbers FAQs

What is the corrected average formula when one number is misread?

Corrected average = Wrong average + (Correct value − Misread value)/n, where n is the number of observations.

A value 12 was read as 42 in a set of 6 numbers. How much does the average change?

Change in average = (12 − 42)/6 = −30/6 = −5. The corrected average is 5 less than the reported average.

The average of 7 numbers is 18 because 25 was used instead of 11. What is the corrected average?

Corrected average = 18 + (11 − 25)/7 = 18 − 2 = 16.

How are multiple misread numbers corrected?

Use Corrected average = Wrong average + Σ(Correct value − Misread value)/n. Add the correction for every replaced value before dividing by n.

Does the denominator change when a number is corrected?

No. If an existing observation is replaced by its correct value, the number of observations remains the same, so the denominator remains n.

The wrong average is 25 for 10 values, and the corrected average is 27. If the misread value was 15, what was the correct value?

Correct value = 15 + 10 × (27 − 25) = 35.

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