Chapter 13: Statistics - Board Exam Notes

Exam Weightage & Blueprint

Total: ~7-8 Marks

Statistics is a calculation-heavy but high-scoring chapter. The Board exam typically features one long-answer question (often involving finding missing frequencies) and shorter questions on formulas or empirical relationships.

Question Type	Marks	Frequency	Focus Topic
MCQ	1	High	Formulas, Empirical Relation, Mean of simple data
Short Answer	2 or 3	Medium	Finding Mean (Direct method), Mode
Long Answer	4 or 5	Very High	Missing Frequency (Mean/Median), Step-Deviation Method

1. Mean of Grouped Data ($\bar{x}$)

There are three methods to calculate the mean. The result is the same for all, but some are faster for larger numbers.

A. Direct Method

$$ \bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i} $$

Use when: Values of class mark ($x_i$) and frequency ($f_i$) are small.

B. Assumed Mean Method

$$ \bar{x} = a + \frac{\Sigma f_i d_i}{\Sigma f_i} $$

Where $a$ is assumed mean (middle of $x_i$), and $d_i = x_i - a$.

C. Step-Deviation Method (Best for large numbers)

$$ \bar{x} = a + \left(\frac{\Sigma f_i u_i}{\Sigma f_i}\right) \times h $$

Where $u_i = \frac{x_i - a}{h}$, and $h$ is the class size.

Tip: Always double-check your $\Sigma f_i u_i$ sum. One negative sign error can ruin the whole calculation!

2. Mode of Grouped Data

The mode is the value inside the modal class (the class interval with the maximum frequency).

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

Formula Breakdown:

$l$: Lower limit of modal class
$h$: Class size
$f_1$: Frequency of modal class (Highest frequency)
$f_0$: Frequency of class preceding modal class
$f_2$: Frequency of class succeeding modal class

3. Median of Grouped Data

The median divides the distribution into two equal halves. It is found using Cumulative Frequency (cf).

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

Critical Steps to find Median Class: 1. Calculate Cumulative Frequency ($cf$) for all classes. 2. Find $n = \Sigma f_i$. Calculate $n/2$. 3. Locate the class whose cumulative frequency is just greater than $n/2$. This is the Median Class.
Warning: In the formula, use the $cf$ of the class preceding the median class, but use the $f$ of the median class itself!

⚡ Empirical Relationship (Important for 1 Mark) ⚡
$$ 3 \text{ Median} = \text{Mode} + 2 \text{ Mean} $$

Important PYQs & Solved Examples

🔥 PYQ Type 1: Missing Frequencies (Median Given)

Problem: The median of the following data is 525. Find values of $x$ and $y$, if total frequency is 100.

Class	Freq ($f$)	Cum. Freq ($cf$)
0-100	2	2
100-200	5	7
200-300	x	7+x
300-400	12	19+x
400-500	17	36+x
500-600	20	56+x
600-700	y	56+x+y
700-800	9	65+x+y
800-900	7	72+x+y
900-1000	4	76+x+y

Step 1: Form Equation 1 1 Mark

Total frequency $n = 100$. From table, last $cf = 76 + x + y$.

$\Rightarrow 76 + x + y = 100 \Rightarrow x + y = 24$ ... (i)

Step 2: Identify Median Class 0.5 Mark

Given Median = 525. This lies in class 500-600.

$l = 500, f = 20, cf = 36 + x, h = 100$.

Step 3: Apply Formula 1.5 Marks

$525 = 500 + \left(\frac{50 - (36+x)}{20}\right) \times 100$

$25 = (14 - x) \times 5$

$5 = 14 - x \Rightarrow x = 9$.

Step 4: Solve for y 1 Mark

From (i): $9 + y = 24 \Rightarrow y = 15$.

Answer: $x = 9, y = 15$.

🔥 PYQ Type 2: Missing Frequency in Mean (3 Marks)

Question: The mean pocket allowance is ₹18. Find the missing frequency $f$ in the distribution.

Strategy: Use the Direct Method. Construct a table with $x_i$ (class mark) and $f_i x_i$. Then solve equation: $\frac{\Sigma f_i x_i}{\Sigma f_i} = 18$.

🔥 PYQ Type 3: Discontinuous Class Intervals (4 Marks)

Scenario: Classes given as 118-126, 127-135... (Example 13.3 Q4).

Solution Step: Convert to continuous intervals by subtracting 0.5 from lower limit and adding 0.5 to upper limit. New classes: 117.5-126.5, 126.5-135.5...

Exam Strategy & Mistake Bank

Mistake Bank 🚨

Continuity Check: Always check if class intervals are continuous (e.g., 0-10, 10-20). If not (e.g., 1-10, 11-20), fix them first!

Step-Deviation Error: Students often forget to multiply by $h$ at the very end of the Step-Deviation calculation.

Wrong CF Selection: In the Median formula, $cf$ is the cumulative frequency of the preceding class, NOT the median class itself. This is the most common error.

Scoring Tips 🏆

Table Clarity: Draw neat tables with a ruler. Misaligned columns lead to calculation errors.

Sanity Check: Your calculated Mean, Median, or Mode MUST lie within the boundaries of their respective classes. If you get a Mode of 65 for a modal class 40-50, recheck your math immediately!

Empirical Formula: If a question asks for all three measures, calculate Mean and Median, then use $3 \text{ Median} = \text{Mode} + 2 \text{ Mean}$ to find Mode quickly (unless specified otherwise).