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Part A · Statistics for Economics CBSE Code 030 · NCERT-aligned Variables defined inline
Formula Sheet · Class 11 Economics

Class 11 Economics — Statistics Formulas at a Glance

Every rule and identity for the Statistics for Economics half. Class-interval helpers, the three averages, all four dispersion measures, Karl Pearson and Spearman correlation, the four standard price-index formulas plus CPI and IIP. Part B (Indian Economic Development) is descriptive — no formulas there.

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On this page
  1. Ch 3 · Organisation of Data
  2. Ch 5 · Measures of Central Tendency
  3. Ch 6 · Measures of Dispersion
  4. Ch 7 · Correlation
  5. Ch 8 · Index Numbers
  6. Chs 1, 2, 4, 9 — descriptive, no formulas

Ch 3 Organisation of Data

Themes: class limits and class mark, exclusive vs inclusive series, class size, relative and cumulative frequency. The arithmetic helpers that make every later chapter work on grouped data.

3.1 Class limits, class mark, class size

Class mark (mid-value)
m = (lower limit + upper limit) / 2
The single value taken as representative of an interval — used as x in grouped-data mean / SD formulas.
Class size (class width)
h = upper limit − lower limit
In an exclusive series of equal width h, all intervals share the same value.
Inclusive → exclusive adjustment
d = (lower of next class − upper of this class) / 2 new lower = lower − d new upper = upper + d
Eliminates the gap between successive inclusive classes (e.g. 0–9, 10–19) so they become continuous (−0.5–9.5, 9.5–19.5).

3.2 Frequency & density

Relative frequency
rf = f / N (often expressed as % = 100 × f / N)
f = frequency of the class · N = Σf = total frequency.
Cumulative frequency (less-than ogive)
CFi = f1 + f2 + … + fi
Running total of frequencies; used to locate the median and quartiles graphically (ogive) and via formula.
Frequency density (unequal classes)
density = f / h
For unequal-width histograms — the bar HEIGHT becomes density (not frequency) so areas stay proportional to f.

Ch 5 Measures of Central Tendency

Themes: the three averages — arithmetic mean (direct, assumed-mean, step-deviation), median, mode — plus the empirical relation and combined mean. Picking the right average for the data type is half the chapter.

5.1 Arithmetic mean

Mean — ungrouped (raw data)
x̄ = Σx / N
x̄ = mean · Σx = sum of all observations · N = number of observations.
Mean — discrete series / grouped (direct)
x̄ = Σ f·x / Σ f
x = observation (discrete) or class mark (grouped) · f = frequency · Σf = N.
Mean — assumed-mean (short-cut)
x̄ = A + (Σ f·d / Σ f) where d = x − A
A = assumed mean (any value, often a central class mark) · d = deviation from A. Saves arithmetic when x is large.
Mean — step-deviation
x̄ = A + (Σ f·d' / Σ f) × h where d' = (x − A) / h
Equal-width classes only. h = class size. Reduces deviations to small integers, easiest to add.
Weighted arithmetic mean
x̄w = Σ w·x / Σ w
w = weight assigned to observation x (importance / population / value). Reduces to simple mean when all w are equal.
Combined mean (two groups)
x̄12 = (N1·x̄1 + N2·x̄2) / (N1 + N2)
Extends to k groups: numerator = Σ Ni·x̄i, denominator = Σ Ni.

5.2 Median (positional average)

Median — ungrouped, N odd
Median = value at position (N + 1) / 2
Sort data ascending first. For N even, median = average of the two middle values, i.e. positions N/2 and (N/2 + 1).
Median — grouped data
Median = L + [(N/2 − CF) / f] × h
L = lower limit of median class · N/2 = halfway cumulative frequency · CF = cumulative frequency of class BEFORE the median class · f = frequency of median class · h = class size.
Quartiles (grouped)
Qk = L + [(k·N/4 − CF) / f] × h for k = 1, 2, 3
Q2 = median. Q1, Q3 feed the inter-quartile range and Quartile Deviation in Ch 6.

5.3 Mode & empirical relation

Mode — grouped data
Mode = L + [(f1 − f0) / (2·f1 − f0 − f2)] × h
L = lower limit of modal class · f1 = frequency of modal class · f0 = frequency of PRE-modal class · f2 = frequency of POST-modal class · h = class size.
Empirical relation (moderately skewed data)
Mode ≈ 3·Median − 2·Mean
Use only when mode is hard to compute directly. Holds for unimodal, moderately asymmetric distributions.
For symmetric data — Mean = Median = Mode.

Ch 6 Measures of Dispersion

Themes: all four absolute measures (range, QD, MD, SD) plus their relative cousins, the assumed-mean and step-deviation short-cuts for SD, and Lorenz curve concept. CV makes two unlike-unit distributions comparable.

6.1 Range & Quartile Deviation

Range
R = L − S
L = largest value · S = smallest value. Simplest dispersion measure; sensitive to extremes.
Coefficient of Range
Coeff. of Range = (L − S) / (L + S)
Unit-free version of range, used to compare two distributions with different magnitudes.
Inter-Quartile Range & Quartile Deviation
IQR = Q3 − Q1 QD = (Q3 − Q1) / 2 (also called semi-interquartile range)
QD ignores the extreme 25% on each side — robust to outliers.
Coefficient of QD
Coeff. of QD = (Q3 − Q1) / (Q3 + Q1)
Unit-free analogue of QD for cross-distribution comparison.

6.2 Mean Deviation

Mean Deviation from the MEAN — ungrouped
MDx̄ = Σ |x − x̄| / N
Average of absolute deviations from the arithmetic mean.
Mean Deviation from the MEDIAN — ungrouped
MDM = Σ |x − Median| / N
MD is MINIMUM when taken from the median — a key NCERT result.
Mean Deviation — grouped
MDA = Σ f·|x − A| / Σ f where A = mean or median, x = class mark
Drop the absolute-value bars and you get signed deviations; their sum from the mean is exactly zero (the very property that motivates MD and SD).
Coefficient of Mean Deviation
Coeff. of MD = MDA / A (A = mean or median, whichever was used)
Unit-free MD for comparison.

6.3 Variance & Standard Deviation

Variance & SD — ungrouped (direct)
σ² = Σ (x − x̄)² / N σ = √[ Σ (x − x̄)² / N ]
σ² = variance · σ = standard deviation. Squaring removes signs and penalises larger deviations more than MD does.
SD — short-cut (mean-square minus square-of-mean)
σ = √[ Σx² / N − (Σx / N)² ]
Useful when x̄ is not a whole number. Equivalent to the direct formula algebraically.
SD — grouped data (direct)
σ = √[ Σ f·(x − x̄)² / Σ f ]
x = class mark · f = frequency · Σf = N.
SD — assumed-mean short-cut
σ = √[ Σ f·d² / N − (Σ f·d / N)² ] where d = x − A
Use whenever direct deviations are messy. Result is independent of A — it is invariant under shift.
SD — step-deviation short-cut
σ = h × √[ Σ f·d'² / N − (Σ f·d' / N)² ] where d' = (x − A) / h
Equal-width classes only. The extra factor h at the front compensates for shrinking deviations by h.
Combined SD (two groups)
σ12 = √{ [N1(σ1² + d1²) + N2(σ2² + d2²)] / (N1 + N2) } where d1 = x̄1 − x̄12, d2 = x̄2 − x̄12
Requires the combined mean x̄12 first (see Ch 5).

6.4 Coefficient of Variation & Lorenz

Coefficient of Variation
CV = (σ / x̄) × 100 (expressed as %)
Unit-free measure — the only way to compare variability across distributions with different units or different means. Smaller CV = more consistent / stable.
Lorenz curve — equality line
plot cumulative % of variable (income / wealth) against cumulative % of population
Greater area between the Lorenz curve and the 45° equality line = greater inequality. NCERT treats it visually (no Gini coefficient at Class 11).

Ch 7 Correlation

Themes: the two NCERT-prescribed coefficients (Karl Pearson's product-moment r and Spearman's rank ρ) plus the standard ranges and interpretation. Scatter diagram and r both live in the range −1 ≤ r ≤ 1.

7.1 Karl Pearson's coefficient of correlation

Definition (covariance form)
r = Σ (x − x̄)(y − ȳ) / [ N · σx · σy ]
σx, σy = SDs of x and y. Pure number, dimensionless, lies in [−1, +1].
Direct (deviations from actual means)
r = Σ x'·y' / √( Σ x'² · Σ y'² ) where x' = x − x̄, y' = y − ȳ
Use when x̄ and ȳ are integers. No square roots inside.
Short-cut (deviations from assumed means)
r = [N·Σdx·dy − Σdx·Σdy] / √[ (N·Σdx² − (Σdx)²) · (N·Σdy² − (Σdy)²) ] where dx = x − Ax, dy = y − Ay
Use any convenient values for Ax, Ay; the value of r is unchanged.
Range & interpretation
−1 ≤ r ≤ +1
r = +1 perfect positive · r = −1 perfect negative · r = 0 no LINEAR association (may still be non-linear).
  • |r| ≥ 0.75 — high
  • 0.5 ≤ |r| < 0.75 — moderate
  • |r| < 0.25 — low

7.2 Spearman's rank correlation

Spearman ρ — no tied ranks
ρ = 1 − [ 6·Σd² / (N(N² − 1)) ] where d = Rx − Ry
Rx, Ry = ranks of x and y · d = difference of ranks · N = number of pairs. Same range as r: [−1, +1].
Spearman ρ — with tied ranks
ρ = 1 − [ 6·{Σd² + Σ(m(m² − 1) / 12)} / (N(N² − 1)) ]
m = number of items tied at a particular rank; the Σ runs over each tie-group. Each tied item gets the average of the ranks they would have occupied.
When to choose ρ over r
Use Spearman when (a) data are qualitative / ordinal (beauty, intelligence, preferences), (b) extreme values would distort r, or (c) only ranks — not actual scores — are available.

Ch 8 Index Numbers

Themes: simple and weighted aggregate methods, the four standard weighted price indices (Laspeyres, Paasche, Fisher, Marshall–Edgeworth), plus the two index numbers NCERT covers in detail — CPI and IIP. Base year carries subscript 0, current year subscript 1.

8.1 Simple methods

Simple aggregate price index
P01 = (Σ p1 / Σ p0) × 100
p0 = price in base year · p1 = price in current year. Simplest but ignores quantity/importance — affected by the units chosen for each commodity.
Simple average of price relatives
P01 = ( Σ (p1 / p0) × 100 ) / N
Average of individual price relatives — unit-free, but still treats all commodities equally.

8.2 Weighted aggregate price indices

Laspeyres (base-year weights)
P01L = (Σ p1·q0 / Σ p0·q0) × 100
q0 = base-year quantities. Tends to overestimate the price rise — assumes consumers haven't substituted away from items whose prices rose.
Paasche (current-year weights)
P01P = (Σ p1·q1 / Σ p0·q1) × 100
q1 = current-year quantities. Tends to underestimate the price rise — assumes substitution already happened.
Fisher's ideal index
P01F = √( P01L × P01P )
Geometric mean of Laspeyres and Paasche. Called "ideal" because it satisfies BOTH the time-reversal and factor-reversal tests; balances over- and under-estimation.
Marshall–Edgeworth
P01ME = [Σ p1·(q0 + q1) / Σ p0·(q0 + q1)] × 100
Uses sum of base- and current-year quantities as weights. A practical compromise between Laspeyres and Paasche, simpler to compute than Fisher.

8.3 Consumer Price Index (CPI)

CPI — Aggregate Expenditure Method (= Laspeyres)
CPI = (Σ p1·q0 / Σ p0·q0) × 100
India's CPI(IW), CPI(AL) and CPI(RL) all use this Laspeyres form — base-year quantities serve as the consumption basket.
CPI — Family Budget Method
CPI = Σ (W · R) / Σ W where R = (p1 / p0) × 100, W = p0·q0
R = price relative of each item · W = base-year expenditure (used as weight). Algebraically equivalent to Aggregate Expenditure — just rearranged.

8.4 Index of Industrial Production (IIP)

IIP
IIP = [Σ (Wi · qi1 / qi0) / Σ Wi] × 100
qi0, qi1 = production of industry i in base and current years · Wi = weight (share in base-year industrial output). A quantity index measuring real production growth across mining, manufacturing and electricity.
Inflation rate from CPI
Inflation rate (%) = [(CPI1 − CPI0) / CPI0] × 100
Year-on-year % change in CPI. Used by the RBI as the official inflation reference.
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