Board exam focus — Statistics (CBSE & HBSE)

Statistics involves measures of central tendency for grouped data. CBSE tests all three methods for mean and ogive-based median and mode questions. HBSE focuses on direct, assumed mean, and step deviation methods.

Mean of Grouped Data

Mean — Three Methods

Method 1: Direct Method $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ where xᵢ = class mark (midpoint) = (lower + upper)/2

Method 2: Assumed Mean Method (Short-cut) $$\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$$ where a = assumed mean, dᵢ = xᵢ - a

Method 3: Step Deviation Method $$\bar{x} = a + \left(\frac{\sum f_i u_i}{\sum f_i}\right) \times h$$ where uᵢ = (xᵢ - a)/h, h = class width

When to Use Which Method?

Steps for Mean Calculation

1. Find class marks xᵢ = (lower limit + upper limit)/2
2. Multiply: fᵢ × xᵢ (for direct method)
3. Find Σfᵢxᵢ and Σfᵢ
4. Mean = Σfᵢxᵢ / Σfᵢ

Table Format

CBSE Exam: Step deviation method saves time when class width is constant. But direct method is also fully correct.

Mode and Median of Grouped Data

Mode of Grouped Data

The modal class is the class with the highest frequency.

$$\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$

where:

• l = lower limit of modal class
• f₁ = frequency of modal class
• f₀ = frequency of class preceding modal class
• f₂ = frequency of class following modal class
• h = class width

Median of Grouped Data

First, find the median class: the class containing the (n/2)th observation.

$$\text{Median} = l + \frac{\frac{n}{2} - cf}{f} \times h$$

where:

• l = lower limit of median class
• n = total frequency (Σfᵢ)
• cf = cumulative frequency of class preceding median class
• f = frequency of median class
• h = class width

Steps for Median

1. Find n = Σfᵢ
2. Calculate n/2
3. Build cumulative frequency column
4. Identify median class (first class whose cf ≥ n/2)
5. Apply formula

Empirical Relationship

$$\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}$$

Use when 2 of 3 are known to find the third!

HBSE Exam: The modal class formula needs careful substitution of f₀, f₁, f₂. Always identify which is 'preceding' and which is 'following'.

Cumulative Frequency and Ogive

Cumulative Frequency

Cumulative Frequency (cf): Running total of frequencies up to a certain class.

Ogive (Cumulative Frequency Curve)

Less Than Ogive: Plot (upper class boundary, cumulative frequency) and join with smooth curve.

More Than Ogive: Plot (lower class boundary, n - cumulative frequency up to previous class) and join.

Finding Median from Ogive

1. Draw 'Less Than' ogive
2. Mark n/2 on y-axis
3. Draw horizontal line to curve, then vertical to x-axis
4. The x-axis value = Median

OR: Draw both 'Less Than' and 'More Than' ogives on same graph. The x-coordinate of their intersection = Median.

Quartiles from Ogive

• Q₁ (Lower Quartile): at n/4 on y-axis
• Q₃ (Upper Quartile): at 3n/4 on y-axis
• Interquartile Range = Q₃ - Q₁

CBSE Exam: Ogive-based questions ask you to draw the graph and read median/quartiles. Practice this carefully as it is 4-5 marks.

Frequently asked questions

Are these Statistics notes free?

Yes — the Statistics notes for Mathematics (Class 10) 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.