Lesson 32 of 100 | Quantitative Aptitude / Averages / औसत
Arithmetic Average of a Group
Learning outcome
Calculate an arithmetic mean, recover a total or missing value, and distinguish an average of zero from an undefined average.
Meaning and method
The arithmetic average, or mean, is the value each member would have if the group's total were shared equally. It summarises the group; it does not claim that every member actually has that value.
For a nonempty group:
Mean = total of all values ÷ count.
Total = count × mean.
The second identity reverses the first: multiplying the equal share by the number of shares restores the total. Include every observation, even repeated or zero values, in the count. The count must be a positive integer.
The mean lies between the smallest and largest values, including the endpoints. An equal share cannot be below every original value or above every original value. However, the mean need not be a listed value or an integer.
If the total and a nonzero mean are known, count = total ÷ mean. For exact data, a fractional count is inconsistent with a group of individual items. Do not divide by the mean when it is zero.
A zero total can give a mean of zero when the count is positive. An empty group has count zero, so its mean is undefined, not zero.
Worked examples
Example 1 — Find an equal share. In five practice sessions, a learner solves 14, 18, 16, 22 and 20 questions. Total = 14 + 18 + 16 + 22 + 20 = 90. Mean = 90 ÷ 5 = 18 questions per session. Check: 5 × 18 = 90.
Example 2 — Recover a missing value. Six boxes contain an average of 17 packets. Five contain 14, 16, 19, 18 and 15 packets. Required total = 6 × 17 = 102. Known total = 14 + 16 + 19 + 18 + 15 = 82. The sixth box contains 102 - 82 = 20 packets.
Example 3 — Recover the count. Parcels have total mass 234 kg and mean mass 19 1/2 kg. Since 19 1/2 = 39/2, count = 234 ÷ (39/2) = 234 × 2 ÷ 39 = 12 parcels. The result is a positive integer, as required.
Common mistakes
Do not divide by the number of distinct values instead of the number of observations. Keep measurement units consistent. An average describes equal redistribution, not the most frequent value.
Practice questions
- Find the mean of 11, 15, 17 and 21.
- Five notebooks cost an average of ₹38. Four cost ₹31, ₹36, ₹42 and ₹39. Find the fifth price.
- Ropes have total length 126 m and mean length 10 1/2 m. How many ropes are there?
- Find the mean of -7, 0 and 7. Would an empty group also have mean zero?
Worked answers
- Total = 11 + 15 + 17 + 21 = 64. There are 4 values, so mean = 64 ÷ 4 = 16.
- Total cost = 5 × 38 = ₹190. Known cost = 31 + 36 + 42 + 39 = ₹148. The fifth price is 190 - 148 = ₹42.
- Count = 126 ÷ (21/2) = 126 × 2 ÷ 21 = 12 ropes. Here 10 1/2 = 21/2.
- The total is (-7) + 0 + 7 = 0. Dividing by the positive count gives 0 ÷ 3 = 0. An empty group has no positive count; its mean would require division by zero and is undefined.