Board exam focus — Statistics (CBSE & HBSE)

Measures of dispersion — range, mean deviation, variance, standard deviation, and coefficient of variation. CBSE asks 5–6 marks; HBSE 4–5 marks. Formula-heavy chapter with frequent tabular computations.

Measures of Dispersion: Range & Mean Deviation

What Is Dispersion?

Dispersion measures how spread out the observations are from a central value. Two data sets can have the same mean but very different spreads.

Range

Range = Maximum − Minimum.

The simplest dispersion measure; depends only on two values and ignores everything in between. Sensitive to outliers.

Coefficient of Range: (Max − Min)/(Max + Min).

Mean Deviation (MD)

The arithmetic mean of the absolute deviations of all observations from a central value (mean or median).

MD about Mean (ungrouped data)

MD = (1/n) Σ |xᵢ − x̄|.

MD about Median (ungrouped data)

MD = (1/n) Σ |xᵢ − M|, where M is the median.

MD for Grouped Data (Discrete Frequency)

MD = (1/N) Σ fᵢ |xᵢ − x̄|, where N = Σ fᵢ.

MD for Continuous Frequency Distribution

Replace each xᵢ by the class mid-point and use the same formula.

Coefficient of Mean Deviation

MD / (central value used).

Why Use Median for MD?

The mean deviation about the median is always least (smaller than MD about any other value). This is a useful theoretical property.

Computation Strategy

1. Find the mean (or median) of the data.
2. Compute |xᵢ − x̄| for each observation.
3. Multiply by frequency (if grouped).
4. Sum and divide by N.

CBSE Tip: Always tabulate intermediate columns (xᵢ, fᵢ, xᵢ − x̄, |xᵢ − x̄|, fᵢ|xᵢ − x̄|) — examiners give partial credit for the table even if final answer is wrong.

Variance & Standard Deviation

Variance (σ²)

The arithmetic mean of the squared deviations from the mean.

Ungrouped Data

σ² = (1/n) Σ (xᵢ − x̄)²

Discrete Frequency Distribution

σ² = (1/N) Σ fᵢ (xᵢ − x̄)², where N = Σ fᵢ.

Continuous Frequency Distribution

Use class mid-points for xᵢ.

Computational (Shortcut) Formula

Expanding (xᵢ − x̄)² and simplifying: σ² = (Σ fᵢxᵢ²)/N − [(Σ fᵢxᵢ)/N]² = (Σ fᵢxᵢ²)/N − x̄²

This avoids computing each (xᵢ − x̄)² individually.

Standard Deviation (σ)

σ = √(variance) = √σ².

Same units as the original data — easier to interpret than variance.

Properties

• σ ≥ 0; σ = 0 iff all observations are equal.
• Adding a constant to all observations does not change σ (or σ²).
• Multiplying all observations by a constant c changes σ to |c| · σ (and σ² to c² · σ²).

Step-Deviation Method

When mid-points are large, use uᵢ = (xᵢ − A)/h where A is the assumed mean and h is the class width. σ² = h² · [(Σ fᵢuᵢ²)/N − ((Σ fᵢuᵢ)/N)²]. Reduces arithmetic substantially.

Variance of Combined Data (Optional)

For two groups with n₁, n₂ observations: Combined variance involves means and variances of each group plus the squared difference of group means.

HBSE Tip: Use the shortcut formula (Σx²/n − x̄²) unless the data is grouped with unusually small mid-points; it's significantly faster.

Coefficient of Variation & Comparing Distributions

Coefficient of Variation (CV)

CV = (σ / x̄) × 100%.

• Dimensionless — allows comparison of variability between datasets with different units or means.
• Smaller CV ⇒ more consistent / less variable data.

When to Use CV

Use CV when:

• Comparing two or more sets with different means or units.
• Determining which dataset is more consistent.

Example: Comparing Two Cricket Teams

Team A: mean = 50, σ = 5 ⇒ CV = 10%. Team B: mean = 60, σ = 12 ⇒ CV = 20%.

Team A is more consistent (lower CV), even though Team B has a higher mean.

Variance / SD for Frequency Distribution (Full Recipe)

1. Construct columns: xᵢ (mid-points if continuous), fᵢ, fᵢxᵢ, fᵢxᵢ².
2. Compute N = Σ fᵢ, Σ fᵢxᵢ, Σ fᵢxᵢ².
3. Mean x̄ = (Σ fᵢxᵢ)/N.
4. σ² = (Σ fᵢxᵢ²)/N − x̄².
5. σ = √σ².
6. CV = (σ/x̄) × 100.

Relationship Among Dispersion Measures

For any data set:

• Range ≥ MD ≥ σ (approximate inequality; depends on data).
• σ²(uniform data) = 0 only when all values are identical.

Variance of Standard Distributions

For first n natural numbers 1, 2, …, n:

• Mean = (n + 1)/2.
• Variance = (n² − 1)/12.

For an AP a, a + d, …, a + (n − 1)d:

• Variance = d² (n² − 1)/12.

Quartile Deviation (Optional Note)

Q.D. = (Q₃ − Q₁)/2 — used in classes 9-10 for related dispersion.

CBSE Tip: Always present the comparison clearly: state "Team A is more consistent because its CV is smaller" — labelled conclusions earn the final mark.

Frequently asked questions

Are these Statistics notes free?

Yes — the Statistics notes for Mathematics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Statistics notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Statistics chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Statistics.