EconomicsCBSE Class 11 · Measures of Central Tendency

Arithmetic Mean: Direct, Assumed Mean, Step Deviation and Weighted Methods

Updated 17 October 2026 · Fact-checked

The arithmetic mean is the sum of observations divided by their number. For grouped data the direct method is mean = Σfm / N; with an assumed mean A, mean = A + Σfd / N; with step deviation, mean = A + (Σfd' / N) x h. The weighted mean is ΣwX / Σw. All methods give the same value.

Notice that all three methods use the same data and land on the same answer, 27.8; the assumed-mean shortcuts only make the arithmetic smaller.

Illustration drawn for QuestionBuddy.

Understand Arithmetic Mean: Direct, Assumed and Weighted

Arithmetic mean (AM). The most commonly used average: the sum of all values divided by their number. It is affected by extreme values and is a good summary of a roughly symmetric distribution.

Individual series. Mean = ΣX ÷ n. Example: 12, 15, 18, 20, 35 → ΣX = 100, n = 5, mean = 20.

Discrete series. Mean = Σfx ÷ N, where N = Σf.

Continuous series. Take the mid-point (m) of each class.

Direct method: Mean = Σfm ÷ N.
Assumed mean method: choose an assumed mean A (usually a mid-point near the centre); d = m − A; Mean = A + Σfd ÷ N.
Step deviation method: d′ = (m − A) ÷ h (h = common class width); Mean = A + (Σfd′ ÷ N) × h.

Worked data. Classes 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 15, 12, 10 (N = 50). Mid-points: 5, 15, 25, 35, 45. Direct: Σfm = 25 + 120 + 375 + 420 + 450 = 1390, so mean = 1390 ÷ 50 = 27.8. Assumed mean A = 25: d = −20, −10, 0, 10, 20; Σfd = −100 − 80 + 0 + 120 + 200 = 140; mean = 25 + 140 ÷ 50 = 27.8. Step deviation (h = 10): d′ = −2, −1, 0, 1, 2; Σfd′ = −10 − 8 + 0 + 12 + 20 = 14; mean = 25 + (14 ÷ 50) × 10 = 27.8.

Weighted arithmetic mean. When items have different importance: Mean = ΣwX ÷ Σw. Example: marks 70, 80, 60 with weights 3, 2, 1: (210 + 160 + 60) ÷ 6 = 430 ÷ 6 = 71.67. Used in index numbers.

Properties. Sum of deviations from the mean is zero; if every value is increased by k, the mean rises by k; if every value is multiplied by k, the mean is multiplied by k; combined mean of two groups = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).

Finding a missing frequency. Use mean = Σfx ÷ Σf and solve for the unknown.

Merits and limits. Simple, uses all values, algebraically treatable; but sensitive to extreme values and cannot be found for open-ended classes without assumptions.

Flow chart: choose method (direct, assumed mean, step deviation) → compute → check with another method.

Arithmetic Mean: Direct, Assumed and Weighted: key facts

Direct: Mean = Sum(f m) / N; Continuous series
Assumed mean: Mean = A + Sum(f d) / N | d = m - A; Shortcut
Step deviation: Mean = A + (Sum(f d') / N) x h | d' = (m - A) / h; Common width h
Weighted mean: Sum(w X) / Sum(w); Different importance

How to answer Arithmetic Mean: Direct, Assumed and Weighted questions

Questions give a series and ask for the mean by a named method, or ask for a missing value.

1Write mid-points and total frequency.
2Pick A near the centre and compute d or d'.
3Apply the formula and keep units.
4Check by another method or by estimating from the data.
5For missing frequency, set up an equation and solve.

Arithmetic Mean: Direct, Assumed and Weighted in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Weighted marks

Suppose a school gives test 1 weight 3, test 2 weight 2 and the final weight 1, with scores 70, 80 and 60.

Find the weighted average.

Try it first, then show the solutionSolution

Multiply each score by its weight: 210, 160, 60.
Sum = 430 and total weight = 6.
Mean = 430 ÷ 6 = 71.67.

Answer: Weighted mean = 71.67.

Memory hook: Heavier items pull more.

Tricks to remember Arithmetic Mean: Direct, Assumed and Weighted

Short memory aids. Say each one aloud once, then use it on a question the same day.

A plus average deviation
Mean = A + average of d.
Use it when: Assumed mean
Step down, step up
Divide by h for d', multiply by h at the end.
Use it when: Step deviation
Weights first
Multiply by weight, add, divide by total weight.
Use it when: Weighted mean

Arithmetic Mean: Direct, Assumed and Weighted: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Individual series · 2 marks · Very short answerBoard level

Find the mean of 12, 15, 18, 20, 35. If each value is increased by 5, what is the new mean?

Try it first, then show the solutionSolution

Mean = 100 / 5 = 20 (1 mark).
New mean = 25 (1 mark).

Answer: 20; 25.

Example 2: Direct method · 4 marks · Short answerBoard level

Compute the mean by the direct method: classes 0-10, 10-20, 20-30, 30-40, 40-50 with f = 5, 8, 15, 12, 10.

Try it first, then show the solutionSolution

Mid-points 5, 15, 25, 35, 45; N = 50 (1 mark).
fm = 25, 120, 375, 420, 450 (1 mark).
Sum fm = 1390 (1 mark).
Mean = 1390 / 50 = 27.8 (1 mark).

Answer: 27.8.

Example 3: Step deviation · 4 marks · Short answerHarder board level

Repeat the above data by the step deviation method with A = 25 and h = 10.

Try it first, then show the solutionSolution

d' = -2, -1, 0, 1, 2 (1 mark).
fd' = -10, -8, 0, 12, 20, sum = 14 (1 mark).
Mean = 25 + (14 / 50) x 10 (1 mark).
= 27.8, matching the direct method (1 mark).

Answer: 27.8.

Example 4: Missing frequency · 4 marks · Short answerHarder board level

The values 10, 20, 30, 40, 50 have frequencies 4, 6, x, 5, 3 and the mean is 29. Find x.

Try it first, then show the solutionSolution

Sum fx = 40 + 120 + 30x + 200 + 150 = 510 + 30x (1 mark).
N = 18 + x (1 mark).
(510 + 30x) / (18 + x) = 29 gives 510 + 30x = 522 + 29x (1 mark).
x = 12; check: Sum fx = 870, N = 30, mean = 29 (1 mark).

Answer: x = 12.

Example 5: Combined and corrected means · 5 marks · Long answerHOTS / competency

(a) 30 students average 60 marks and 20 students 70: find the combined mean. (b) A mean of 10 items was found as 50, but 60 was read for 80 and 54 for 45. Find the correct mean.

Try it first, then show the solutionSolution

(a) Total = 30 x 60 + 20 x 70 = 1800 + 1400 = 3200 (1 mark).
Combined mean = 3200 / 50 = 64 (1 mark).
(b) Wrong sum = 10 x 50 = 500 (1 mark).
Correct sum = 500 - 60 - 54 + 80 + 45 = 511 (1 mark).
Correct mean = 511 / 10 = 51.1 (1 mark).

Answer: 64; 51.1.

Where students lose marks in Arithmetic Mean: Direct, Assumed and Weighted

The slips examiners see most often, and what to write instead.

Mistake: Mean with mid-points omitted
Used class limits.
Wrong working
Used the upper limit as x.
Right working
Use mid-points m.
Do this: Mid-point.
Usually costs: 1 mark.
Mistake: Forgetting to multiply by h
Step deviation.
Wrong working
Mean = A + Sum fd'/N.
Right working
Mean = A + (Sum fd'/N) x h.
Do this: Multiply by h.
Usually costs: 1 mark.
Mistake: Simple average of two means
Sizes ignored.
Wrong working
Combined mean = (40 + 30)/2.
Right working
Weight by sizes.
Do this: Weighted.
Usually costs: 1 mark.

Exam tips

Always show N and mid-points.
Check one method against another.
Write the unit.
State properties such as zero sum of deviations.

Practice questions on Arithmetic Mean: Direct, Assumed and Weighted

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If every observation of a series is multiplied by 3, the arithmetic mean
(a) stays the same (b) is multiplied by 3 (c) rises by 3 (d) becomes zero
Hint
Scaling.
Answer
(b).
Q2 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The mean is affected by extreme values.
Reason (R): It uses every observation in its calculation.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
All values included.
Answer
(a).
Q3 · 2 marks · Very short answerBoard level
Find the weighted mean of prices 20, 30, 50 with weights 5, 3, 2.
Hint
Sum wX / sum w.
Answer
(100 + 90 + 100) / 10 = 290 / 10 = 29.
Q4 · 4 marks · Case-basedHarder board level
Read and answer.
Suppose five workers earn Rs 100, 120, 130, 150 and 400 per day.
(a) Find the mean. (1)
(b) Remove the highest wage and find the new mean. (1)
(c) Comment on the effect of the extreme value. (2)
Hint
Outlier.
Answer
(a) 900 / 5 = 180. (b) 500 / 4 = 125. (c) The extreme value Rs 400 raises the mean by Rs 55, showing the mean is sensitive to extreme values.
Q5 · 3 marks · Short answerHOTS / competency
Suppose the mean of 20 items is 40 and the mean of 30 other items is 30. Find the mean of all 50, and explain why it lies nearer to 30 than to 40.
Hint
Weighted by sizes.
Answer
Combined mean = (20 x 40 + 30 x 30) / 50 = (800 + 900) / 50 = 34. It is closer to 30 because the group with mean 30 has more items, so it carries more weight.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Mean formula (individual)?Show answer
Sum / n.
2.Assumed mean formula?Show answer
A + Sum fd / N.
3.Weighted mean?Show answer
Sum wX / Sum w.
4.Sum of deviations from mean?Show answer
Zero.
5.Mean affected by?Show answer
Extreme values.
6.d' formula?Show answer
(m - A) / h.

Arithmetic Mean: Direct, Assumed and Weighted: frequently asked questions

What is the arithmetic mean?

Sum of values divided by their number.

Why use the assumed mean method?

It reduces large calculations.

What is a weighted mean?

Mean allowing for different importance of items.

Does step deviation change the answer?

No.

Can the mean be an impossible value?

Yes, e.g. 2.3 children.