Measures of Central Tendency

2.1 Core concepts

• A measure of central tendency summarises a whole set of data into a single representative value so that comparisons (e.g., Baiju's landholding versus other farmers of Balapur) become possible (NCERT §1, pp. 58–59).
• The three commonly used averages are Arithmetic Mean, Median and Mode; Geometric Mean and Harmonic Mean exist but are not (NCERT §1, p. 59).
• Arithmetic Mean (A.M.) is defined as the sum of values of all observations divided by the number of observations — X̄ = ΣX ÷ N (NCERT §2, p. 59).
• For ungrouped data the A.M. can be found by the Direct Method (X̄ = ΣX/N) or, when figures are large, by the Assumed Mean Method: X̄ = A + Σd ÷ N, where d = X − A and A is any value taken as the assumed mean (NCERT §2, pp. 60–61).
• The Step-Deviation Method further simplifies arithmetic by dividing each deviation by a common factor c: X̄ = A + (Σd′/N) × c, where d′ = (X − A)/c (NCERT §2, p. 61).
• For grouped discrete data the direct formula is X̄ = ΣfX ÷ Σf; the assumed-mean version is X̄ = A + Σfd/Σf and the step-deviation version is X̄ = A + (Σfd′/Σf) × c (NCERT §2, p. 62).
• For continuous series, the same formulas are used after replacing each class by its mid-value m; classes may be exclusive, inclusive or of unequal size and the procedure is the same (NCERT §2, p. 63).
• Two key properties of A.M. (NCERT §2, p. 63):
• (i) the algebraic sum of deviations of items from the A.M. is always zero: Σ(X − X̄) = 0.
• (ii) the A.M. is affected by extreme values — any very large or very small value can pull it up or down.
• Weighted Arithmetic Mean assigns weights W₁, W₂, … to items by their importance: X̄ = (ΣWᵢXᵢ) ÷ ΣWᵢ, useful when prices need to be weighted by budget shares (NCERT §2, pp. 63–64).
• Median is the positional middle value: it divides the ordered data into two equal halves and is unaffected by the size of extreme values, only by their position (NCERT §3, p. 64).
• For individual series the position of the median is the [(N+1)/2]ᵗʰ item after arranging the data in order; if N is even, the median is the mean of the two middle observations (NCERT §3, pp. 64–65).
• In a discrete series the (N+1)/2 ᵗʰ item is located through the cumulative frequency column; the corresponding variable value is the median (NCERT §3, p. 65).
• In a continuous series the median class is located by the N/2 ᵗʰ item (not (N+1)/2), and the median is interpolated by Median = L + [(N/2 − c.f.)/f] × h, where L = lower limit of median class, c.f. = cumulative frequency of the class preceding it, f = frequency of the median class, h = class width (NCERT §3, p. 66).
• Quartiles divide the data into four equal parts: Q₁ has 25% items below it, Q₂ is the median, Q₃ has 75% below it. For ordered series Q₁ = [(N+1)/4]ᵗʰ item and Q₃ = [3(N+1)/4]ᵗʰ item (NCERT §4, pp. 67–68).
• Percentiles divide the data into 100 equal parts (P₁ … P₉₉); P₅₀ is the median. Scoring "82 percentile" means 18% of candidates are above you (NCERT §4, p. 67).
• Mode is the value occurring most frequently in the data, denoted Mo. A series can be unimodal, bimodal, multimodal, or have no mode if every value appears the same number of times (NCERT §5, p. 68).
• For continuous series, the modal class is the class with the largest frequency, and Mode = L + [D₁/(D₁ + D₂)] × h, where D₁ = |f₁ − f₀| (modal − preceding) and D₂ = |f₁ − f₂| (modal − succeeding), h = class width; class intervals must be equal and exclusive (NCERT §5, p. 69).
• Relative position: the three averages obey Me > Mi > Mo or Me < Mi < Mo (suffixes in alphabetical order); the median always lies between the arithmetic mean and the mode (NCERT §6, p. 70). For a symmetric distribution, Me = Mi = Mo.
• Conclusion: A.M. is simple and uses all observations but is distorted by extremes; median is better for such data and for open-ended distributions; mode is best for qualitative data and is easily found graphically (NCERT §7, p. 70).
• Baiju's-landholding example: whether Baiju with 7 acres is a "small", "medium" or "large" farmer of Balapur depends on comparison with an average landholding of the village (NCERT §1, p. 58). A one-number summary is needed before any qualitative judgement.
• Why three averages, not one: each captures a different facet of "centre" — A.M. measures arithmetic balance, median the positional middle, mode the most typical value. NCERT explicitly says "no one measure is ideal in all situations", which is the basis of the "suitability" CUET MCQs (NCERT §1, p. 59).
• Direct method illustration (NCERT p. 60): marks of 5 students 40, 50, 55, 78, 58. X̄ = (40+50+55+78+58)/5 = 281/5 = 56.2. The mean of 56.2 is not itself an observation — a typical feature of arithmetic means.
• Assumed-mean rationale: when raw values are large or unwieldy (incomes in thousands, prices in lakhs), subtracting an assumed mean A reduces the arithmetic burden. The crucial identity is X̄ = A + (mean of deviations) — true for any choice of A, with computational ease maximised when A lies near the centre (NCERT §2, p. 61).
• Step-deviation rationale: when all deviations share a common factor c (typical when class widths are equal), dividing by c shrinks the numbers further. The end formula X̄ = A + (Σfd′/Σf) × c must therefore be re-multiplied by c — a step students frequently forget, giving the wrong answer (NCERT §2, p. 61).
• Continuous-series A.M. example (Table 5.3): marks-classes 0–10, 10–20, …, 60–70 with class-marks 5, 15, …, 65 and frequencies summing to 100; using direct method Σfm = 3014, so X̄ = 3014/100 = 30.14. The same answer emerges from the step-deviation method with A = 35, c = 10 (NCERT §2, p. 63) — a useful cross-check for CUET MCQs.
• Property Σ(X − X̄) = 0 — quick proof: Σ(X − X̄) = ΣX − NX̄ = ΣX − N(ΣX/N) = 0. This identity is why deviations from the mean cannot be summed to get a measure of dispersion; one must square them or take absolutes (motivating kest106) (NCERT §2, p. 63).
• Weighted A.M. example: a student scores 60, 70, 80 in three subjects with weights 1, 2, 3 (credit hours). Weighted mean = (60×1 + 70×2 + 80×3)/(1+2+3) = (60+140+240)/6 = 440/6 ≈ 73.3, whereas simple mean is 70. The two differ whenever items have unequal importance (NCERT §2, pp. 63–64).
• Median advantage in open-ended classes: in an income distribution with the top class "₹1,00,000 and above", the A.M. cannot be computed without assuming an upper limit for the open class, but the median is computable so long as the median class is below the open class — making median the preferred summary for income, wealth and other open-ended variables (NCERT §3, p. 70, conclusion).
• Quartile coefficient and box plots (implicit): Q₃ − Q₁ is the inter-quartile range that becomes the basis of the quartile deviation measure in kest106; the box plot drawn around (Q₁, Q₂, Q₃) is a graphical summary that combines the median with dispersion in one picture (NCERT §4, pp. 67–68).
• Mode practical use: mode is the measure used in fashion-merchandising (which shoe size to stock most), in public transport planning (most common commute distance) and in survey design (most common response option) — all qualitative or quasi-qualitative settings where averaging numerically makes little sense (NCERT §5, p. 68).
• Bimodal distribution diagnostic: if a histogram shows two peaks, the population is likely a mixture of two sub-populations (e.g., heights of men + women plotted together) — a single mode would conceal this fact, illustrating why summary statistics must be paired with visualisation (NCERT §5, p. 68).
• Empirical relation in words: in a positively skewed distribution (long right tail, like income), mean > median > mode; in a negatively skewed distribution (long left tail, like age at retirement), mean < median < mode. The median is always sandwiched, which is the operational meaning of "median lies between mean and mode" (NCERT §6, p. 70).
• Choice algorithm (NCERT §7 paraphrased): (i) data qualitative → mode; (ii) data has open classes or extreme values → median; (iii) data symmetric, no extremes, every observation should count → mean; (iv) items have unequal importance → weighted mean. CUET often poses this as a "which measure is suitable in situation X" item.

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Direct-method worked example for ungrouped A.M. (marks 40, 50, 55, 78, 58 → X̄ = 56.2), p. 60.
• Table 5.1 — computation of A.M. by assumed-mean method for weekly family income; A = 850, Σd = 2,660, N = 10, X̄ = ₹1,116, p. 61.
• Table 5.2 — discrete-series A.M. by direct method (plots of 100/200/300 sq m; X̄ = 126.92 sq m), p. 62.
• Table 5.3 — continuous-series A.M. for student marks using both direct (X̄ = Σfm/Σf = 30.14) and step-deviation methods (A = 35, c = 10), p. 63.
• Table 5.5 — discrete-series median via cumulative frequency (median income = ₹30), p. 66.
• Table 5.6 — continuous-series median, daily wages of factory workers (median = ₹35.83), p. 67.
• Unimodal vs Bimodal distribution sketch illustrating Mode, p. 68.
• Table 5.7 — conversion of a "less-than" cumulative frequency table into an ordinary frequency table to find Mode (Mo = ₹27,273), p. 69.
• Average decision flow: identify data type (qualitative / quantitative) → check for extremes or open-ended classes → choose A.M., median or mode → compute using direct or assumed-mean method.
• Full worked A.M. (continuous, step-deviation) using NCERT Table 5.3-style data: classes 0–10, 10–20, 20–30, 30–40, 40–50, 50–60, 60–70 with frequencies 5, 12, 15, 25, 18, 15, 10 (Σf = 100). Take A = 35, c = 10, so d′ = (m − 35)/10 = −3, −2, −1, 0, 1, 2, 3. Then fd′ = −15, −24, −15, 0, 18, 30, 30 = 24. X̄ = 35 + (24/100) × 10 = 35 + 2.4 = 37.4. Same data via direct method: Σfm = 5×5 + 12×15 + 15×25 + 25×35 + 18×45 + 15×55 + 10×65 = 25 + 180 + 375 + 875 + 810 + 825 + 650 = 3740 → X̄ = 3740/100 = 37.4 ✓.
• Full worked median (continuous) for the same data: cumulative frequencies are 5, 17, 32, 57, 75, 90, 100; N/2 = 50, which falls in the class 30–40 (since cumulative 32 < 50 ≤ 57). L = 30, c.f. = 32, f = 25, h = 10. Median = 30 + ((50 − 32)/25) × 10 = 30 + (18/25) × 10 = 30 + 7.2 = 37.2.
• Full worked mode (continuous) for the same data: highest frequency = 25 in class 30–40, so L = 30, f₁ = 25, f₀ = 15, f₂ = 18, h = 10. D₁ = |25 − 15| = 10, D₂ = |25 − 18| = 7. Mode = 30 + (10/(10+7)) × 10 = 30 + 5.88 = 35.88. The three averages are X̄ ≈ 37.4, Me ≈ 37.2, Mo ≈ 35.88 — consistent with the empirical ordering Mean > Median > Mode for a slightly positively-skewed distribution, and the rough rule Mode ≈ 3 Median − 2 Mean = 3(37.2) − 2(37.4) = 111.6 − 74.8 = 36.8 (close to 35.88, confirming the approximation).
• Quartile worked example: ordered series 12, 18, 22, 25, 28, 32, 35, 38, 41, 44, 46 (N = 11). Q₁ position = (11+1)/4 = 3, so Q₁ = 22 (3rd value); Q₂ position = (11+1)/2 = 6, so median = 32; Q₃ position = 3(11+1)/4 = 9, so Q₃ = 41. Inter-quartile range Q₃ − Q₁ = 41 − 22 = 19 — the dispersion summary that kest106 will build on.

2.5 Key formulas

2.4 Common confusions / NTA trap points

• For continuous-series median use N/2; for individual/discrete series use (N+1)/2. NTA loves swapping these.
• Σ of deviations from A.M. is zero — but this is NOT true for deviations taken from the median.
• A.M. is affected by extreme values; median and mode are not — the median depends only on position, not magnitude of extremes.
• For Mode in continuous series, class intervals must be equal and exclusive; convert if given inclusive/unequal.
• The empirical relation is Me > Mi > Mo or Me < Mi < Mo — the median is the middle one of the three. "Median equals mean" holds only for symmetric data.
• Mode in a "less than" cumulative frequency table cannot be read directly — convert to an ordinary frequency distribution first.
• Weighted A.M. is different from the simple A.M.; do not interchange the formulas.
• Q₂ is the median, not Q₁ or Q₃.
• P₅₀ is the median; "82 percentile" means 82% are below, 18% above.
• Geometric and harmonic means are not developed in this chapter — distractors that ask "best average for ratios" should be flagged as outside-syllabus for kest105.
• The mode formula has D₁ in the numerator, not D₂.
• The step-deviation correction multiplies the deviation sum by c, the common divisor — forgetting c is a frequent error.