The mean of 100 observations was calculated as 49. It was later discovered that three observations taken as 40, 20, 50 were actually 60, 70, 80 respectively. The correct mean is

Correct — option (3), 50.

Step 1 — recover the total behind the wrong mean.
Wrong total = mean × number of observations = 49 × 100 = 4,900

Step 2 — take out the values as they were wrongly recorded.
Wrong values = 40 + 20 + 50 = 110
4,900 − 110 = 4,790

Step 3 — put in the true values.
True values = 60 + 70 + 80 = 210
4,790 + 210 = 5,000

Step 4 — divide by the number of observations, which is still 100.
Correct mean = 5,000 ÷ 100 = 50

Quicker route: the total rises by 210 − 110 = 100, so the mean rises by 100 ÷ 100 = 1, from 49 to 50.

The idea to remember: a mean is corrected through its total: rebuild the sum, fix the values, then divide again.

Why the others are wrong

• (1)
  48 — 48 is what comes out if the correction is applied in the wrong direction: 4,900 − 100 = 4,800, and 4,800 ÷ 100 = 48.

  The true values (60, 70, 80) are larger than the recorded ones (40, 20, 50), so the total and the mean must go up, not down.

• (2)
  49.33 — A mean of 49.33 over 100 observations means a total of 4,933, which is a rise of 33 in the total.

  The actual rise is 210 − 110 = 100. Spread over the same 100 observations, that lifts the mean by exactly 1, to 50.

• (4)
  49.5 — A mean of 49.5 over 100 observations means a total of 4,950, which is a rise of 50 in the total.

  The corrected values add 100 to the total, not 50 (210 − 110 = 100), so the mean rises by a full 1, from 49 to 50.

The arithmetic mean is the sum of the observations divided by their number. Turned around, sum = mean × number of observations, which is how a total is recovered from a reported mean.

When wrong values are replaced by correct ones, the sum changes by (sum of correct values − sum of wrong values). If the number of observations does not change, the mean changes by that amount ÷ the number of observations.

The mean depends on every observation. Adding k to each value raises the mean by k; multiplying each value by k multiplies the mean by k.

Key facts

• Arithmetic mean = sum of observations ÷ number of observations, so sum = mean × number of observations.
• Replacing wrong values with correct ones changes the sum by (sum of correct values − sum of wrong values).
• With the number of observations unchanged, the mean changes by (change in sum) ÷ (number of observations).
• Adding k to every observation raises the mean by k; multiplying every observation by k multiplies the mean by k.
• For positive values that are not all equal, arithmetic mean > geometric mean > harmonic mean.

Common traps

• Adding the change in the total straight to the mean, 49 + 100 = 149. The change in the total must first be divided by the 100 observations.
• Subtracting the net change. The true values 60, 70 and 80 are larger than the recorded 40, 20 and 50, so the mean goes up.
• Changing the count after the correction. The values were replaced, not added or removed, so the division is still by 100.

UnlockIAS will link similar questions from RAS Pre 2021 and 2013 here once those papers are published on this site.

• practice — not a real PYQ

  The mean of 50 observations is 36. It was later found that an observation 48 was wrongly recorded as 23. The correct mean is

  • (a)35.5
  • (b)36.5
  • (c)61
  • (d)37.5
  Answer(2) — Total = 36 × 50 = 1,800; corrected total = 1,800 − 23 + 48 = 1,825; mean = 1,825 ÷ 50 = 36.5.

  Option (1) applies the change in the wrong direction; option (3) adds the change of 25 straight to the mean; option (4) fits no step.

• practice — not a real PYQ

  The mean of 20 numbers is 15 and the mean of another 30 numbers is 20. The mean of all 50 numbers is

  • (a)17.5
  • (b)18
  • (c)17
  • (d)35
  Answer(2) — Totals are 20 × 15 = 300 and 30 × 20 = 600, so the mean = 900 ÷ 50 = 18. Option (1) averages the two means and ignores the group sizes; option (4) adds the means; option (3) fits no step.