Statistics

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Statistics — Key Facts for JEE Main Mean: average of data; for grouped data: ∑f_ix_i/∑f_i Median: middle value when data is sorted; for grouped: L + [h/f](n/2 − c) Mode: most frequent value; for grouped: L + [h/f](f₁ − f₀)/(2f₁ − f₀ − f₂) Variance: σ² = ∑(x_i − x̄)²/n or ∑f_i(x_i − x̄)²/∑f_i for grouped Standard deviation: σ = √variance ⚡ Exam tip: For grouped data mean, remember to multiply frequency by class mark (midpoint), not class boundaries!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Statistics — JEE Main Study Guide

Ungrouped data:

• Mean: x̄ = (∑x_i)/n
• Median: sort data; if n odd, middle; if n even, average of two middle values
• Mode: value with highest frequency

Grouped data:

• Class mark (midpoint): x_i = (upper + lower)/2
• Mean: x̄ = ∑f_i x_i / ∑f_i
• Assumed mean method: x̄ = A + ∑f_i d_i/∑f_i where d_i = x_i − A
• Step deviation method: x̄ = A + h·∑f_i u_i/∑f_i where u_i = (x_i − A)/h

Median of grouped data: Median = L + [h/f](n/2 − c) where L = lower limit of median class, h = class width, f = frequency of median class, c = cumulative frequency before median class

Mode of grouped data: Mode = L + [h/f](f₁ − f₀)/(2f₁ − f₀ − f₂) where L = lower limit of modal class, f₁ = frequency of modal class, f₀ = frequency before modal class, f₂ = frequency after modal class

Variance:

• Population variance: σ² = ∑(x_i − x̄)²/n
• Sample variance: s² = ∑(x_i − x̄)²/(n−1)
• For grouped: σ² = ∑f_i(x_i − x̄)²/∑f_i

Standard deviation: σ = √[∑(x_i − x̄)²/n] = √[∑f_i(x_i − x̄)²/∑f_i]

Combined mean: If two groups have means x̄₁, x̄₂ and sizes n₁, n₂: x̄ = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂)

Coefficient of variation: CV = (σ/x̄) × 100%; used to compare variability of different datasets

Range: Range = Maximum value − Minimum value

Quartiles: Q₁ = value at position (n+1)/4; Q₃ = value at position 3(n+1)/4 Interquartile range = Q₃ − Q₁

• Key formula: Mean = ∑f_i x_i/∑f_i; Variance = ∑f_i(x_i − x̄)²/∑f_i
• Common trap: In assumed mean method, the deviation d_i = x_i − A must be calculated correctly — A is chosen as an assumed mean from class marks
• Exam weight: 1 question per year (4 marks); occasionally combined with probability

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Statistics — Comprehensive JEE Main Notes

Effect of changes on mean and standard deviation:

• Adding constant c to each value: mean increases by c, SD unchanged
• Multiplying each value by constant k: mean multiplies by k, SD multiplies by |k|
• Adding constant c to each value: variance unchanged

Standard deviation shortcuts: σ² = (∑x_i²)/n − (x̄)² = (∑x_i²)/n − (∑x_i/n)² For grouped: σ² = (∑f_i x_i²)/∑f_i − (x̄)²

Mean deviation: MD = ∑|x_i − x̄|/n for ungrouped MD about median = ∑|x_i − M|/n

Quartile deviation: QD = (Q₃ − Q₁)/2 For grouped data, Q₁ and Q₃ can be found using cumulative frequency method

Dispersion measures comparison:

• Range: easiest but most affected by outliers
• Quartile deviation: good for skewed data
• Mean deviation: uses all values but not as good as SD for normal distribution
• Standard deviation: most commonly used; mathematically convenient (involves squared deviations)

Skewness: Pearson coefficient of skewness = 3(x̄ − Mode)/σ Also: 3(Mean − Median)/σ

• Skewness = 0: symmetric distribution
• Positive skewness: tail to the right (mode < median < mean typically)
• Negative skewness: tail to the left (mean < median < mode typically)

Kurtosis: Measure of peakedness of distribution relative to normal distribution Leptokurtic: more peaked than normal (γ₂ > 0) Platykurtic: flatter than normal (γ₂ < 0) Mesokurtic: normal (γ₂ = 0)

Bivariate data: For paired observations (x_i, y_i):

• Covariance: cov(x, y) = ∑(x_i − x̄)(y_i − ȳ)/n
• Correlation coefficient: r = cov(x, y)/(σ_x·σ_y)
• r ranges from −1 to +1

Regression lines: y on x: y − ȳ = r(σ_y/σ_x)(x − x̄) x on y: x − x̄ = r(σ_x/σ_y)(y − ȳ)

Effect of data transformation: If y = a + bx, then ȳ = a + bx̄, σ_y = |b|σ_x

Moment generating: rth moment about mean: μ_r = ∑(x_i − x̄)^r/n For grouped: μ_r = ∑f_i(x_i − x̄)^r/∑f_i

Normalization: z-score = (x − x̄)/σ; this standardizes any distribution to mean 0, SD 1

Weighted mean: x̄_w = ∑w_i x_i/∑w_i where w_i are weights

Harmonic mean: HM = n/(1/x₁ + 1/x₂ + … + 1/x_n); useful for rates and ratios

Geometric mean: GM = (x₁·x₂·…·x_n)^{1/n}; less affected by outliers than arithmetic mean

• Remember: Mean = ∑f_i x_i/∑f_i; Variance = ∑f_i(x_i − x̄)²/∑f_i = (∑f_i x_i²/∑f_i) − x̄²; σ² = (∑x_i²)/n − (x̄)² is the quick calculation method
• Previous years: “Find mean and variance of first n natural numbers” [2023]; “For grouped data: find median class and calculate median” [2024]; “If mean of 10 numbers is 20 and adding 5 more numbers makes mean 25, find original sum” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Statistics: 1 question per year, 4 marks
• Common patterns: calculate mean of grouped data, find mode/median, standard deviation calculation
• Weight: low-medium frequency, easy marks if formulas are known

💡 Pro Tips

• For grouped data mean, always use class marks (midpoints), not class boundaries
• Assumed mean method reduces calculation time — pick A from middle class marks
• Step deviation method further simplifies when class width is constant
• Variance shortcut formula: σ² = (∑x_i²)/n − (x̄)² saves computation time
• Remember: adding same constant to all values doesn’t change SD; multiplying by same constant multiplies SD by that constant

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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