Statistics

🎯 Key Points

• Empirical relation: Mode = 3·Median − 2·Mean - lets you find any ONE of the three if the other two are known, without recomputing from raw data
• Step-deviation method (Mean = A+h·Σfu/Σf) avoids large-number arithmetic for grouped data - same final answer as the direct method, just easier computation
• Standard deviation is ALWAYS non-negative and uses every data point (unlike range, which only uses the two extremes) - the most reliable single measure of spread
• Coefficient of Variation (CV=σ/x̄×100%) compares variability across datasets with DIFFERENT means/units - lower CV means more consistent, even if the raw σ values aren't directly comparable

Statistics & Data Analysis

Statistics is the science of collecting, organizing, analyzing, and interpreting numerical data.

Measures of Central Tendency

• Mean (Arithmetic Mean): x̄ = (Σfx) / (Σf); for grouped data use midpoints
• Median: Middle value of ordered data; for grouped: Median = l + [(n/2 - F)/f] × h
• Mode: Most frequent value; for grouped: Mode = l + [f₁-f₀ / 2f₁-f₀-f₂] × h
• Relation: Mode = 3 Median - 2 Mean (empirical)

Measures of Dispersion

• Range: Maximum - Minimum
• Variance: σ² = Σf(x-x̄)² / Σf
• Standard deviation: σ = √variance
• Coefficient of variation: CV = (σ/x̄) × 100%; used to compare variability
• Mean deviation: Σ|x - central value| / n
• Quartiles: Q₁ (25th percentile), Q₃ (75th percentile), IQR = Q₃ - Q₁

Types of Data Representation

• Bar graph, pie chart, histogram (no gaps), frequency polygon, ogive (cumulative frequency curve)
• Stem-and-leaf plot, box plot (five-number summary)

Correlation and Regression

• Correlation coefficient r: -1 ≤ r ≤ 1; r = 1 perfect positive, r = -1 perfect negative, r = 0 no correlation
• Line of best fit (regression line): y = a + bx where b = Σ(x-x̄)(y-ȳ)/Σ(x-x̄)²

Normal Distribution

• Bell-shaped, symmetric about mean
• 68-95-99.7 rule: 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ

Worked Example: Mean by the Step-Deviation Method

For grouped data, the step-deviation (shortcut) method avoids large numbers by using u = (x - A)/h, where A is an assumed mean and h is the class width:

Mean = A + h x (Σfu / Σf)

Consider class intervals 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 15, 16, 6 (N = 50) and midpoints 5, 15, 25, 35, 45. Taking A = 25 and h = 10, the deviations u are -2, -1, 0, 1, 2.

Σfu = 5(-2) + 8(-1) + 15(0) + 16(1) + 6(2) = -10 - 8 + 0 + 16 + 12 = 10.

Mean = 25 + 10 x (10/50) = 25 + 2 = 27. This matches the direct method (Σfx/N), but with much smaller numbers to handle.

Worked Example: Variance by the Shortcut Method

Using the same data, variance by the step-deviation method is σ² = h² x [(Σfu²/N) - (Σfu/N)²].

Σfu² = 5(4) + 8(1) + 15(0) + 16(1) + 6(4) = 20 + 8 + 0 + 16 + 24 = 68.

σ² = 100 x [(68/50) - (10/50)²] = 100 x [1.36 - 0.04] = 100 x 1.32 = 132.

Standard deviation σ = sqrt(132) ≈ 11.49. This is exactly what the direct method Σf(x - xbar)²/N would also give, confirming the shortcut formula.

Variance and SD by the Direct Method (Ungrouped Data)

• Variance: σ² = (Σ(x - xbar)²) / n
• Standard deviation: σ = sqrt(σ²)
• Shortcut formula avoiding the mean subtraction step: σ² = (Σx²/n) - (Σx/n)²
• Tip: Standard deviation is always non-negative, and it is the most reliable measure of dispersion since it uses every data point, unlike range which only uses the extremes.

Using Coefficient of Variation to Compare Two Datasets

CV = (σ/xbar) x 100% lets you compare the relative variability of two datasets even when their means are very different (for example, comparing the consistency of marks in two subjects with different maximum scores). The dataset with the lower CV is considered more consistent or less variable.

Mean Deviation about the Mean and Median

Mean deviation measures dispersion as the average of the absolute distances of the observations from a central value, so it is never negative.

• About the mean: M.D.(x̄) = Σfᵢ|xᵢ - x̄| / N (for ungrouped data all fᵢ = 1, so it is Σ|xᵢ - x̄|/n).
• About the median: M.D.(M) = Σfᵢ|xᵢ - M| / N.
• The mean deviation is least when taken about the median - no other point gives a smaller average absolute deviation.
• Coefficient of mean deviation = M.D. ÷ (the central value used), useful for comparing dispersion across datasets.

Worked example: for the data 4, 7, 8, 9, 10, 12, 14 (n = 7) the mean is 64/7 ≈ 9.14 and the median is 9. Mean deviation about the median = (|4-9|+|7-9|+|8-9|+|9-9|+|10-9|+|12-9|+|14-9|)/7 = (5+2+1+0+1+3+5)/7 = 17/7 ≈ 2.43.

Range, Quartile Deviation, and Their Coefficients

• Range = Largest value (L) - Smallest value (S); Coefficient of range = (L - S)/(L + S).
• Quartile deviation (semi-interquartile range) = (Q₃ - Q₁)/2; Coefficient of quartile deviation = (Q₃ - Q₁)/(Q₃ + Q₁).
• Range uses only the two extreme values, while quartile deviation ignores the extreme 25% at each end, making it more resistant to outliers than the range.

Choosing Between Mean, Median, and Mode

• The mean uses every observation and is best for symmetric data, but it is pulled toward extreme values (outliers).
• The median depends only on the middle position, so it is the preferred average for skewed data or when outliers are present (e.g. incomes).
• The mode is the only average usable for purely categorical data and identifies the most typical or most frequent value.
• For a symmetric distribution mean = median = mode; for a right-skewed distribution mean > median > mode, and the order reverses for left skew.

🚀 JEE Advanced Edge

Why variance is NOT additive under a linear transformation in the obvious way: If Y=aX+b, then Var(Y)=a²·Var(X) - the additive constant b shifts the mean but doesn't affect spread at all (Var unaffected by b), while the multiplicative constant a scales the variance by a² (not a) because variance involves SQUARED deviations. This "shift doesn't matter, scale squares" rule resolves a huge class of "find the new variance after transformation" problems instantly.

Effect of combining datasets on the combined mean and variance: If two datasets have n₁, n₂ observations with means x̄₁, x̄₂, the combined mean is the WEIGHTED average (n₁x̄₁+n₂x̄₂)/(n₁+n₂), NOT the simple average of the two means - this weighting by sample size is essential whenever group sizes differ and is a common error source in board/JEE-adjacent problems.

Worked problem: A dataset has variance 16. If every value in the dataset is multiplied by 3 and then increased by 5, find the new variance. Approach: Y=3X+5, so Var(Y)=3²·Var(X)=9×16=144. (The "+5" shift has zero effect on variance.)

Measures of Dispersion - Overview

• Dispersion measures how spread out or scattered the data values are about a central value
• Two datasets can share the same mean yet differ greatly in spread, so a measure of central tendency alone is not enough
• The main measures are range, mean deviation, variance, and standard deviation
• Range is the crudest (uses only two values); standard deviation is the most reliable (uses every value)
• A small dispersion means the data are consistent and clustered; a large dispersion means the data are widely scattered
• Dispersion is always a non-negative quantity

Standard Deviation for a Frequency Distribution

• For values xᵢ with frequencies fᵢ and total N = Σfᵢ, the variance is σ² = (1/N) Σ fᵢ(xᵢ − x̄)²
• The standard deviation σ is the positive square root of the variance
• A convenient shortcut form is σ² = (1/N) Σ fᵢxᵢ² − (x̄)², avoiding repeated subtraction of the mean
• For a continuous (grouped) distribution, xᵢ is the mid-point (class mark) of each class interval
• Standard deviation carries the same units as the original data, unlike variance which is in squared units
• Standard deviation is the most widely used measure because it is least affected by sampling fluctuations

Effect of Change of Origin and Scale on Variance

• Adding (or subtracting) a constant to every observation does not change the variance or standard deviation - dispersion is unaffected by a shift of origin
• Multiplying every observation by a constant k multiplies the standard deviation by |k| and the variance by k²
• If yᵢ = a + b·xᵢ, then variance(y) = b²·variance(x) and SD(y) = |b|·SD(x)
• This principle justifies the step-deviation method, where values are shifted and scaled to make arithmetic easier
• Example: if each mark is increased by 5 grace marks, the standard deviation of the class stays exactly the same
• The mean, by contrast, does shift by the constant and scale by the multiplier

Mean Deviation for Grouped Data

• For a frequency distribution, mean deviation about the mean is M.D.(x̄) = (1/N) Σ fᵢ|xᵢ − x̄|
• Mean deviation about the median uses M.D.(M) = (1/N) Σ fᵢ|xᵢ − M|
• For grouped data, xᵢ is taken as the class mark of each interval
• Mean deviation about the median is the least among deviations taken about any point
• Because it uses absolute values rather than squares, mean deviation is simpler but mathematically less tractable than standard deviation
• It gives an average distance of the observations from the chosen centre