Statistics & Measures of Dispersion

Concept Core

Measures of central tendency and dispersion — the complete statistics framework.

Measures of Central Tendency

Measure	Formula	Best For
Mean (x̄)	Σfx / Σf (or Σx/n for raw data)	Symmetric data with no outliers
Median	Middle value; (n+1)/2 th term when sorted	Skewed data, outliers present
Mode	Most frequently occurring value	Categorical data, most common item

Empirical relation: Mode ≈ 3×Median − 2×Mean (for moderately skewed data)

Variance & Standard Deviation

Variance (σ²) = Σf(x−x̄)² / N = Σfx²/N − (x̄)² Standard Deviation (σ) = √(Variance) SD (shortcut) = (1/N)√(NΣfx² − (Σfx)²)

Variance measures average squared deviation from mean. SD is in the same units as data (variance is squared units).

Coefficient of Variation (CV)

CV = (σ/x̄) × 100 %

CV is a dimensionless measure of relative spread — used to compare variability between datasets with different means or units.

More consistent data → smaller CV. Less consistent → larger CV.

Mean Deviation

MD about mean = Σf|x − x̄| / N MD about median = Σf|x − median| / N

MD about median is minimum (among all central values). MD about mean ≤ SD (always). SD ≥ MD for any dataset.

Properties of Variance & SD

Effect of adding constant k: Mean increases by k; Variance, SD unchanged (shifting doesn't change spread).

Effect of multiplying by k: Mean × k; Variance × k²; SD × |k| (scaling changes spread).

Variance of (aX + b) = a² × Var(X). SD of (aX + b) = |a| × SD(X).

Range & Quartile Deviation

Range: Maximum − Minimum value. Simple but sensitive to outliers.

Quartile Deviation (QD) = (Q₃ − Q₁)/2

Q₁ = lower quartile (25th percentile), Q₃ = upper quartile (75th percentile). IQR = Q₃ − Q₁.

Formula Vault

Statistics formulas for EAPCET — all in one place.

Arithmetic Mean

x̄ = Σfx / N

N = total frequency = Σf

Median (ungrouped)

Middle value after sorting

n odd: (n+1)/2 th; n even: avg of n/2 and n/2+1

Variance (direct)

σ² = Σf(x−x̄)²/N

Average squared deviation

Variance (shortcut)

σ² = Σfx²/N − x̄²

Faster for computation

Standard Deviation

σ = √(σ²)

Same units as data

Coefficient of Variation

CV = (σ/x̄) × 100%

For comparing consistency

Mean Deviation (mean)

MD = Σf|x−x̄|/N

Always less than or equal to σ

Variance of (aX+b)

Var(aX+b) = a²·Var(X)

Constant b doesn't affect variance

Worked Examples

5 problems — mean, variance, SD, CV, and a property trap.

EasyFind mean and variance of: 2, 4, 6, 8, 10▾

Find the mean and variance of 2, 4, 6, 8, 10.

Mean x̄ = (2+4+6+8+10)/5 = 30/5 = 6

Variance = Σ(x−x̄)²/n = [(2−6)²+(4−6)²+(6−6)²+(8−6)²+(10−6)²]/5

= [16+4+0+4+16]/5 = 40/5 = 8

SD = √8 = 2√2

✓ Mean = 6, Variance = 8, SD = 2√2

EasyFind variance using shortcut formula for data 1, 3, 5, 7▾

Find variance of 1, 3, 5, 7 using the shortcut formula.

n=4; Σx = 16; x̄ = 4; Σx² = 1+9+25+49 = 84

σ² = Σx²/n − x̄² = 84/4 − 16 = 21 − 16 = 5

✓ Variance = 5

MediumTwo series: Series A has CV=25%, x̄=40; Series B has CV=35%, x̄=60. Which is more consistent?▾

Compare consistency: Series A: CV=25%, x̄=40. Series B: CV=35%, x̄=60.

Lower CV → more consistent data.

CV_A = 25% < CV_B = 35%

Series A is more consistent (less relative spread around its mean)

✓ Series A is more consistent (lower CV = 25%)

EAPCET LevelIf each observation is multiplied by 3, find new SD if original SD = 4▾

A dataset has SD = 4. Each observation is multiplied by 3. Find the new standard deviation.

Multiplying each observation by k: new SD = |k| × old SD

New SD = 3 × 4 = 12

New Variance = 9 × 16 = 144

✓ New SD = 12

Trap QuestionAdding 5 to every observation increases variance by 25 — True or False?▾

Each observation is increased by 5. A student claims variance increases by 5² = 25. Evaluate.

Adding a constant k to every observation: mean increases by k.

But variance = Σ(x−x̄)²/n. If all x and x̄ both increase by k, the differences (x−x̄) are unchanged.

Therefore variance is unchanged. SD is also unchanged.

Only multiplication (scaling) changes variance. Adding a constant (shifting) never changes spread.

✓ False — adding a constant shifts data but does not change variance or SD

Mistake DNA

3 statistics errors from EAPCET distractor analysis.

➕

Adding Constant Changes Variance

Shifting all data points by a constant does NOT change variance. The deviations from mean are identical.

❌ Wrong

Add 10 to every value: new variance = old + 10² ✗ (adding constant doesn't change spread)

✓ Correct

Variance unchanged ✓ Mean increases by 10 ✓ SD unchanged ✓ Only scaling changes variance

Variance measures spread around the mean. If all values and the mean shift by the same amount, all (x−x̄) values are unchanged. Hence Σ(x−x̄)² is unchanged.

📊

Using Variance Formula Without Squaring Deviations

MD = Σ|x−x̄|/n (absolute values). Variance = Σ(x−x̄)²/n (squared deviations). These are different.

❌ Wrong

Variance = Σ|x−x̄|/n ✗ (absolute value — that's Mean Deviation!)

✓ Correct

Variance = Σ(x−x̄)²/n ✓ SD = √Variance ✓ MD uses |·|; Variance uses ²

Mean Deviation uses absolute values. Variance uses squares. Squaring makes variance sensitive to outliers and ensures mathematical tractability (differentiable, additive).

🔢

Coefficient of Variation: Forgetting to Multiply by 100

CV = (σ/x̄) × 100%. The × 100 converts the ratio to a percentage. Forgetting it gives a decimal, not a percentage.

❌ Wrong

CV = σ/x̄ = 4/20 = 0.2 ✗ (not a percentage form)

✓ Correct

CV = (σ/x̄) × 100 = 20% ✓ Expressed as percentage ✓

CV is always expressed as a percentage, which makes comparison between different datasets meaningful. Without × 100, the value appears misleadingly small.

Chapter Intelligence

Statistics is a direct formula-application chapter — practise the variance shortcut.

EAPCET Weightage (2019–2024)

Variance and SD

Mean, median, mode

Coefficient of variation

Properties: add/multiply constant

Mean deviation

High-Yield PYQ Patterns

Variance from raw dataSD using shortcut formulaCompare CV of two seriesEffect of multiplying by k on SDFind mean when variance givenEmpirical relation Mode-Median-Mean

Exam Strategy

Variance shortcut: σ² = Σx²/n − x̄². This is faster than computing each (x−x̄)² separately. Memorise and use it.
Multiplying each observation by k: new SD = k × old SD; new Variance = k² × old Variance.
Adding constant k: Mean increases by k; Variance and SD are UNCHANGED.
CV = (σ/x̄)×100%. Lower CV = more consistent. This appears as "which team/batch is more consistent?" questions.