Class 11 Statistics Notes · GSEB

Arithmetic Mean

Arithmetic Mean — master the direct, short-cut and step-deviation methods for raw and grouped data, with mid-values and open-end traps. GSEB Class 11 Statistics notes with formulas, solved sums and a practice lab.

Last updated: 5 Oct 2026

Notes

Arithmetic Mean — Definition and Formulas

Arithmetic Mean (x̄)

The arithmetic mean is the sum of all observations divided by the total number of observations, denoted x̄ (read “x-bar”). It is the most commonly used average.

Raw data — direct

\bar{x} = \frac{\Sigma x}{n}

Discrete frequency — direct

\bar{x} = \frac{\Sigma fx}{n}

Short-cut — assumed mean A

\bar{x} = A + \frac{\Sigma d}{n}

Step-deviation — A + common factor c

\bar{x} = A + \frac{\Sigma fd}{n} \times c

Three computation modes, one answer: direct (Σx/n, Σfx/n), short-cut with an assumed mean A (d = x − A), and step-deviation with a common factor c (d = (x − A)/c). The choice of A and c only changes the size of the numbers you add — never the mean.

The mean is the default average because it is rigidly defined and based on all observations — two people working the same data always get the same x̄.

Money reads better than raw floats

Rupee answers are written with Indian grouping: ₹6,425 (not ₹6425). Non-money answers get 2 decimals — the StatsProblemLab checks your answer in exactly this format.

Words marked with a dotted underline carry the question they usually appear in — tap them to reveal it.

Mid-value Rule for Continuous Classes

A class like 2000–3000 cannot enter the addition as a range — every observation of the class is represented by Mid-value = (lower + upper)/2.

Class 0–5 with f = 7 → mid-value 2.5 stands for all 7 observations. This one rule is what lets the mean work on continuous tables, and the same mid-value becomes x in every method table.

Mid-value

x = \frac{L + U}{2}

L = lower limit, U = upper limit. For the class 0–5: x = (0 + 5) / 2 = 2.5.

Worked Walkthrough — Illustration 7

One full example first, so the labs below have something to imitate: continuous income classes, solved with the step-deviation method.

Solved Example

Problem

Illustration 7 — Find the mean monthly income of 80 families from the classes 2000–3000, 3000–4000, … , 8000–9000.

Solution

x̄ = ₹6,425 per family per month

The same data set is loaded in the StatsProblemLab below as a preset — walk it through the four checks yourself and every method will agree on ₹6,425.

Interactive Lab — Mean as a Balance Point

Before you compute anything, feel the defining property: the mean is the point that balances the data, and only the mean makes Σ(x − x̄) = 0.

Mean Balance Lab — daily chai cups sold (5 days)

Drag any dot along the line — the emerald fulcrum slides to the new balance point every time.

Live mean of the five days

x̄ = (5 + 8 + 10 + 12 + 15) / 5 = 50 / 5 = 10

Move one dot and every number above changes together — the fulcrum is just the arithmetic mean, drawn.

Why it matters: Σ(x − x̄) = 0 is the defining property of the mean — sum of observations = mean × number of observations (here 50 = 10 × 5). No other point on this line zeroes this sum, which is why the mean is the balance point of the whole data set. Drag one dot out to 20 and watch how far the fulcrum is pulled — that pull is exactly what “outliers disturb the mean” means.

Stats Problem Lab — Now You Drive

The chapter's one question generator — the Median, Mode and Combined Mean pages all link back here for practice. Generate a GSEB-level problem (or paste your own data), pick any method, and pass the four checks: enter n → fill the table → substitute → answer. The collapsed “Other methods” panel proves all three methods land on the same mean.

Stats Problem Lab — Bus travel time (min) — discrete frequency

New problem of the chosen type

Generate keeps the same steps: enter n → fill the table → substitute → answer. Book examples reproduce the illustration answers exactly.

Method

Change A or c and the table regenerates — the final mean never changes. That is the point of the exercise.

x = 110, 113, 120, 122, 126 · f = 7, 17, 11, 10, 5

Direct method · formula x̄ = Σfx / n

Step 1 — find n = Σf, the total of the frequency column.

n = Σf

Traps and Quick Tricks

The zero-sum identity — Σ(x − 9) = 0 ⇒ x̄ = 9

Σ(x − x̄) = 0 holds only for the mean. So any “Σ(x − k) = 0” in a question is really telling you the mean: k must be x̄. Nothing else zeroes that sum.

Trap — one value was recorded wrong

Mean 55 kg of 20 persons → Σx = 1100 kg. Later found: 45 was recorded, the person actually weighs 54.

✗ Wrong way

keep Σx = 1100 → x̄ = 1100/20 = 55.00 kg

Still carries the wrong 45.

✓ Right way

Σx = 1100 − 45 + 54 = 1109 → x̄ = 1109/20 = 55.45 kg

Subtract the wrong, add the correct.

Trap — unequal class lengths: c = HCF, not class length

For classes 0–10, 10–20, 20–30, 30–50, 50–70 the lengths differ (10, 10, 10, 20, 20). Step-deviation still works — take c as the HCF of the deviations |x − A| (= 5 when A = 25), never as “the class length”. Ill-formed c is the fastest way to lose a sum.

Trick — two numbers, one missing

Mean of two numbers = (sum of the two numbers) / 2, so the other number = 2 × mean − known number. Example: mean 5, one number 6 → other = 2 × 5 − 6 = 4.

Change of Origin and Scale

If every x is turned into y = bx + a (multiply by b, then add a), the mean transforms by exactly the same recipe — ȳ = b·x̄ + a.

Example: x̄ = 12 and y = 4x − 7 → ȳ = 4(12) − 7 = 41. Nothing else needs recalculating.

The short-cut and step-deviation methods are exactly this idea in disguise: subtracting A is a change of origin, dividing by c is a change of scale — and the formula above is why the final mean comes out identical whatever A and c you choose.

y = bx + a

\bar{y} = b\bar{x} + a

Works forward and backward: ȳ known, find x̄ → x̄ = (ȳ − a) / b.

When the Mean Cannot Be Computed

Open-ended classes stop the mean

Mean cannot be computed for open-ended classes — classes like “below 100” or “200 and above” have no upper (or lower) limit, so no mid-value exists, so x has nothing to add. Unequal class lengths, by contrast, are perfectly fine (c = HCF).

Worked Example Gallery

One solved example per sum type — expand any row for the full table, substitution and answer.

Worked Example Gallery — one example per sum type

9 problems

Raw data — direct method

Full question

The following data show the number of scooters repaired daily at a garage. Find the mean number of scooters repaired per day.

7, 13, 4, 8, 6, 9, 10, 4

Σx = 7 + 13 + 4 + 8 + 6 + 9 + 10 + 4 = 61

x̄ = Σx / n = 61 / 8

x̄ = 7.63

Raw data: every number counts once, so n = how many numbers you were given.

Raw data — short-cut with assumed mean

Full question

The rainfall (in mm.) at 10 different places of a district was recorded as :

126, 110, 91, 115, 112, 80, 101, 93, 97, 113

Find mean rainfall.

d = x − A for each reading; the deviations add to Σd = 38

x̄ = A + Σd / n = 100 + 38 / 10 = 100 + 3.8

x̄ = 103.8 mm

A is picked near the middle of the data so the d column stays small and easy to add.

Correcting a wrongly recorded value

Full question

The mean weight of a group of 20 persons was found to be 55 kg. Later, it was discovered that one of them reported her weight as 45 kg which was actually 54 kg. Find the correct mean of their weights.

Old Σx = 55 × 20 = 1100 kg (carries the wrong 45)

Correct Σx = 1100 − 45 + 54 = 1109

x̄ = 1109 / 20

x̄ = 55.45 kg

Subtract the wrong value, add the correct one — never recompute the whole sum from scratch.

Discrete frequency — direct method

Full question

The following table shows the number of children per family in a certain area. Find the mean number of children per family.

No. of children	0	1	2	3	4	5
No. of families	4	8	23	8	6	3

x	f	fx = x × f
0	4	0
1	8	8
2	23	46
3	8	24
4	6	24
5	3	15
Σ	52	117

n = Σf = 52 · Σfx = 117

x̄ = Σfx / n = 117 / 52

x̄ = 2.25 children per family

Discrete — step-deviation, c = 1

Full question

The time (in minutes) taken by a bus to travel between two towns on different days is shown in the following table.

Time (min.)	110	113	120	122	126
No. of days	7	17	11	10	5

Find the mean travel time.

d = (x − A) / c = (x − 120) / 1 → the deviations ARE the d column

x̄ = A + (Σfd / n) × c = 120 + (−139 / 50) × 1 = 120 − 2.78

x̄ = 117.22 minutes

This is the default problem loaded in the StatsProblemLab — every method agrees on 117.22.

Discrete — step-deviation, c = 2

Full question

The price of an item changes from shop to shop. The following data are available.

Find the mean price.

Price (₹)	206	212	218	220	224	230
No. of shops	5	8	9	14	3	1

c = HCF of |x − A| = 2 (prices jump in twos around the middle)

x̄ = A + (Σfd / n) × c = 220 + (−65 / 40) × 2 = 220 − 3.25

x̄ = ₹216.75

Continuous classes — mid-values + step-deviation

Full question

The following data represent monthly income (in ₹) of workers in a factory. Find their mean income.

Income (₹)	2000 - 3000	3000 - 4000	4000 - 5000	5000 - 6000	6000 - 7000	7000 - 8000	8000 - 9000
No. of workers	2	3	7	15	25	16	12

Class	x (mid)	f	d = (x − 5500)/1000	fd
2000–3000	2500	2	−3	−6
3000–4000	3500	3	−2	−6
4000–5000	4500	7	−1	−7
5000–6000	5500	15	0	0
6000–7000	6500	25	1	25
7000–8000	7500	16	2	32
8000–9000	8500	12	3	36
Σ		80		74

x̄ = A + (Σfd / n) × c = 5500 + (74 / 80) × 1000 = 5500 + 925

x̄ = ₹6,425

'Less than' cumulative table

Full question

The data about weight (in grams) of mangoes from a tree are as follows. Moreover, the minimum weight for these mangoes is 410 grams. Find the mean weight of mangoes.

Weight of mango (gram)	Below 420	Below 430	Below 440	Below 450	Below 460	Below 470
No. of mangoes	14	34	76	130	165	180

Less than	Cumulative f	Class	f by subtraction
420	14	410–420	14
430	34	420–430	20
440	76	430–440	42
450	130	440–450	54
460	165	450–460	35
470	180	460–470	15

First lower limit = first boundary − class width = 420 − 10 = 410 (from the note: minimum weight 410 g).

Mid-values 415, 425, …, 465 → A = 435, c = HCF = 10, Σf = 180, Σfd = 121

x̄ = 435 + (121 / 180) × 10 = 435 + 6.72

x̄ = 441.72 gm

Unequal class lengths

Full question

The distribution of annual sales tax of different companies in a zone is given below.

Find the mean sales tax of these companies :

Sales tax (thousand ₹)	0 - 10	10 - 20	20 - 30	30 - 50	50 - 70
No. of companies	3	14	32	40	21

Class	x (mid)	f	d = (x − 25)/5	fd
0–10	5	3	−4	−12
10–20	15	14	−2	−28
20–30	25	32	0	0
30–50	40	40	3	120
50–70	60	21	7	147
Σ		110		227

Class lengths are 10, 10, 10, 20, 20 — unequal, so c = HCF of deviations = 5, NOT 10.

x̄ = 25 + (227 / 110) × 5 = 25 + 10.32

x̄ = ₹35.32 thousand

Advantages and Disadvantages of the Mean

Six each — learn them as pairs

Advantages

Rigidly defined and based on all observations — same data, same mean, always.
Easy to understand and quick to calculate with basic arithmetic.
Suitable for further algebraic treatment — Σx = n × x̄ unlocks combined and missing-value sums.
Takes every observation into account; no item is quietly ignored.
Less affected by sampling fluctuation — stable across samples of the same size.
Best device for comparing two or more series of data.

Disadvantages

The mean is unduly affected by too large or too small values
Mean cannot be computed for open-ended classes.
Not suitable for qualitative (ordinal) data like intelligence or beauty ratings.
Averaging becomes impossible if even one observation is missing from the data.
May not correspond to any actual observation — 7.63 children per family exists in no family.
Misleading for skewed data — one extreme value drags it away from the typical value.

Negative example — the rainy-night average

One stormy evening, a few Zomato orders stretched to 90 minutes and a restaurant's displayed average jumped from 28 to 41 minutes — even though most riders still delivered in 25–30. A manager who punished riders on that single number punished them for the weather. That is disadvantage #1 in action: one extreme pulls the mean, and the mean pulls the decision with it.

Quick Reference

Raw: x̄ = Σx / n · Short-cut: x̄ = A + Σd / n
Discrete: x̄ = Σfx / n · Step-deviation: x̄ = A + (Σfd / n) × c
Mid-value = (L + U)/2 — represents its whole class in continuous data.
⭐ Σ(x − x̄) = 0 — only the mean zeroes the deviations.
⭐ ȳ = b·x̄ + a when y = bx + a (change of origin and scale).
Sum of observations = mean × number of observations, so Σx = n × x̄.
Mean of two numbers = (sum of the two numbers) / 2 — the other number = 2 × mean − known number.
Open-ended classes → the mean cannot be computed; unequal classes → fine, c = HCF.

Key Takeaways

x̄ = Σx/n for raw data, Σfx/n for frequency tables; n is always Σf.
Short-cut (x̄ = A + Σd/n) and step-deviation (x̄ = A + (Σfd/n) × c) exist to keep numbers small — A and c never change the answer.
Continuous classes enter the sum through their mid-value x = (L + U)/2; cumulative tables are converted to classes by successive subtraction first.
Σ(x − x̄) = 0 — the mean is the unique balance point of the data, and ȳ = b·x̄ + a handles any y = bx + a transformation.
Corrections: subtract the wrong value and add the right one; unequal class lengths use c = HCF of deviations; open-ended classes kill the mean.
So what? Whether it is your own test average, a Zomato delivery time or an IPL run rate — you now know exactly how that single number was built, and what it quietly hides.