Find the Arithmetic Mean of the following weekly wages (in ₹): 12, 18, 24, 30, 36.

Numerical Problems · Q1

Q. Find the Arithmetic Mean of the following weekly wages (in ₹): 12, 18, 24, 30, 36.

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✓ Free question

Concept understanding — Measures of Central Tendency

Measures of central tendency are single values that summarise the centre of a data set. The three mathematical averages are the arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM); the two positional averages are the median and mode.

Each average captures "the typical value" differently — pick the one that suits the data, and use the fixed inequality between AM, GM and HM to sanity-check answers.

How it works

The arithmetic mean adds and divides; the geometric mean multiplies and takes a root (ideal for ratios and growth rates); the harmonic mean handles rates like speed. The median is the middle value in order, and the mode is the most frequent value — both unaffected by extreme outliers.

For two positive numbers a neat identity ties the three mathematical means together, so any two of them determine the third.

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Key properties

$\text{AM}\ge\text{GM}\ge\text{HM}$ for any set of positive values, with equality only when all values are equal.
The identity $\text{GM}^{2}=\text{AM}\times\text{HM}$ holds exactly for two observations.
Median and mode ignore extreme values; the arithmetic mean does not.

Quick example

Given $\text{AM}=64$ and $\text{HM}=16$ , find the GM.

Use $\text{GM}=\sqrt{\text{AM}\times\text{HM}}$ .
$\text{GM}=\sqrt{64\times16}=\sqrt{1024}$ .
$\sqrt{1024}=32$ .
Check the ordering: $64\ge32\ge16$ holds. Result: GM $=32$ .

Watch out

Applying $\text{GM}^{2}=\text{AM}\times\text{HM}$ to more than two observations — it is exact only for two.
Using the arithmetic mean for average speed or average rates, where the harmonic mean is correct.
Forgetting the mean is dragged toward outliers, while median/mode are not.

Tip

Memorise $\text{AM}\ge\text{GM}\ge\text{HM}$ — it instantly flags an impossible answer (a GM larger than the AM). For "average speed over equal distances" questions, reach for the harmonic mean.

Using the direct method for ungrouped data:

\bar{X} = \frac{\sum X}{n}

Working: $\sum X = 12+18+24+30+36 = 120$ , and $n = 5$ .

\bar{X} = \frac{120}{5} = 24

Independent re-check: grouping the additions differently, $(12+36)+(18+30)+24 = 48+48+24=120$ — the sum agrees, confirming $\bar X = 24$ .

✓Final answer

Arithmetic Mean = ₹24.

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2025Set MARCH1 markMCQ
$Q. The correct relationship among A.M., G.M., and H.M., is :$
(a) $H.M \geq G.M \geq A.M$
(b) $A.M < G.M < H.M$
(c) $A.M \geq G.M \geq H.M$
(d) $G.M \geq A.M \geq H.M$
›Reveal solutionSolution

The standard inequality among the three means is $A.M \ge G.M \ge H.M$ .

For any set of positive real numbers, the arithmetic mean (A.M.), geometric mean (G.M.) and harmonic mean (H.M.) always satisfy

$A.M \ge G.M \ge H.M,$

with equality if and only if all the observations are equal.

For two positive numbers $a,b$ this is easy to see: $A.M=\dfrac{a+b}{2}$ , $G.M=\sqrt{ab}$ , $H.M=\dfrac{2ab}{a+b}$ , and one can verify $\dfrac{a+b}{2}\ge\sqrt{ab}\ge\dfrac{2ab}{a+b}$ . A useful related identity is $G.M=\sqrt{A.M\times H.M}$ .

This relationship among the averages is a key result in the measures-of-central-tendency section of the TN HSC Class-11 Business Statistics syllabus.

✓Final answer
$A.M \ge G.M \ge H.M$
CBSE 2023Set MARCH1 markMCQ
$Q. Which of the following represents Median ?$
(a) $Q_3$
(b) $Q_1$
(c) $D_2$
(d) $Q_2$
›Reveal solutionSolution
The median is the value below which half the data lie, which is exactly the second quartile $Q_2$ , so the answer is option (d).

Quartiles divide an ordered data set into four equal parts:

$Q_1$ (first quartile) cuts off the lowest 25%,

$Q_2$ (second quartile) cuts off the lowest 50%,

$Q_3$ (third quartile) cuts off the lowest 75%.

The median is the middle value that splits the data into two equal halves — that is precisely the point with 50% below it, so

$\text{Median} = Q_2.$

(Equivalently, Median $= D_5 = P_{50}$ .)

✓Final answer
$Q_2$

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Q.Find the Arithmetic Mean of the following weekly wages (in ₹): 12, 18, 24, 30, 36.

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