Class 10 Statistics Notes

Exam Weightage & Blueprint

Total: ~7-8 Marks

This chapter falls under Unit VII: Statistics & Probability (11 marks total). As per the latest syllabus, focus is on: Mean, median and mode of grouped data (bimodal situation to be avoided). Note: Ogives are excluded from the current syllabus.

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).

📝 More Solved Board Questions

Q4. In a distribution, the Mean and Mode are 26.4 and 27.2 respectively. Find the Median. 2 Marks

Sol. Using Empirical Relation: $3 \text{ Median} = \text{Mode} + 2 \text{ Mean}$

$3 \text{ Median} = 27.2 + 2(26.4)$

$3 \text{ Median} = 27.2 + 52.8 = 80.0$

Median = $80/3 = 26.67$ (approx).

Answer: 26.67

🎯 Board Pattern (2018–2025): 1-mark or 2-mark questions often test the empirical relation. Make sure you don't confuse the coefficients (it's 3-Median, not 3-Mean).

📋 Board Revision Checklist

✅ Class Mark ($x_i$) = (Upper Limit + Lower Limit) / 2
✅ Mean (Direct): $\Sigma f_ix_i / \Sigma f_i$
✅ Mean (Assumed): $a + \Sigma f_id_i / \Sigma f_i$
✅ Mean (Step Dev): $a + h \times (\Sigma f_iu_i / \Sigma f_i)$
✅ Modal Class = Class with highest frequency
✅ Median Class = Class where $cf \geq n/2$
✅ Empirical Relation: $3 \text{ Median} = \text{Mode} + 2 \text{ Mean}$
✅ Syllabus Check: Ogive/Cumulative Frequency graphs removed

💡 Exam Tip:
Always write the formula before starting the table or calculation. It ensures you get "step marks" even if you make a calculation error later.

Concept Mastery Quiz 🎯

Test your readiness for the board exam.

1. Which of the following cannot be determined graphically?

2. If the mean of the data is 27 and median is 33, then the mode is:

3. The class mark of the class 100-150 is:

4. In the formula for Mean by Assumed Mean Method, $d_i$ is:

5. The sum of the frequencies is denoted by:

Chapter 13: Statistics

Overview