Statistics: Grouped Data, Missing Frequencies & Ogives Class 10

In modern data science, business forecasting, and medical research, decisions are governed by statistical distributions. In CBSE Class 10 Mathematics, Chapter 13 (Statistics) carries between $7$ and $9$ marks. It tests computational precision across the three measures of central tendency—Mean, Mode, and Median—along with the celebrated two missing frequencies ($x$ and $y$) problem and the graphical determination of the median via Ogives.

In this master guide, we review the exact formulas, algebraic shortcuts, and presentation templates needed to secure full marks.

What You Will Learn

• Comparison of the three methods to calculate the Mean: Direct, Assumed Mean, and Step-Deviation
• The Mode Formula: Identifying the modal class and the $0 - 1 - 2$ frequency subscripts
• The Median Formula and constructing cumulative frequency ($cf$) tables
• Complete 5-mark solution to the Missing Frequencies ($x$ and $y$) Problem
• Karl Pearson's Empirical Relationship: $3\,\text{Median} = \text{Mode} + 2\,\text{Mean}$
• Ogives (Cumulative Frequency Curves): Less than and More than ogives
• Graphical determination of the median ($N/2$ projection and double-ogive intersection)

1. Master Formula Compendium for Grouped Data

    1. MEAN (Step-Deviation):    x̄ = a + [ (Σ fi ui) / (Σ fi) ] × h      where ui = (xi - a) / h
    2. MODE:                     Mode = l + [ (f1 - f0) / (2f1 - f0 - f2) ] × h
    3. MEDIAN:                   Median = l + [ (N/2 - cf) / f ] × h     (cf is PRECEDING class!)
    4. EMPIRICAL RELATION:       3 Median = Mode + 2 Mean

2. Solving the 5-Mark Missing Frequencies Problem ($x$ and $y$)

This problem appears in Section D of CBSE board exams almost every year:

Problem Statement:

The median of the following frequency distribution of $60$ observations is $28.5$. Find the values of $x$ and $y$:

Step-by-Step Solution:

Step 1: Construct the Cumulative Frequency ($cf$) Table:

Step 2: Establish Equation 1 from Total Frequency ($N = 60$)

The sum of all frequencies must equal $60$: $45 + x + y = 60$ $x + y = 60 - 45 \implies \mathbf{x + y = 15} \quad \text{--- (Equation 1)}$

Step 3: Identify the Median Class from the Given Median ($28.5$)

• The given median is $28.5$.
• $28.5$ lies in the class interval $20 - 30$.
• Therefore, the Median Class is $20 - 30$!

Step 4: Extract All Parameters:

• Lower limit of median class: $l = \mathbf{20}$
• Class size: $h = 30 - 20 = \mathbf{10}$
• Frequency of median class: $f = \mathbf{20}$
• Half of total frequency: $\frac{N}{2} = \frac{60}{2} = \mathbf{30}$
• Cumulative frequency of preceding class: $cf = \mathbf{5 + x}$

Step 5: Substitute into the Median Formula:

$\text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h$ $28.5 = 20 + \left( \frac{30 - (5 + x)}{20} \right) \times 10$ $28.5 - 20 = \frac{30 - 5 - x}{2}$ $8.5 = \frac{25 - x}{2}$ $8.5 \times 2 = 25 - x \implies 17 = 25 - x$ $\mathbf{x = 25 - 17 = 8}$

Step 6: Solve for $y$ using Equation 1:

$x + y = 15 \implies 8 + y = 15 \implies \mathbf{y = 15 - 8 = 7}$

Step 7: State the Final Answer:

<u>The missing frequencies are $\mathbf{x = 8}$ and $\mathbf{y = 7}$.</u>

3. Graphical Determination of the Median (Ogives)

An Ogive is a smooth cumulative frequency curve:

1. Less Than Ogive: Plot Upper Class Limits on the $x$-axis against Less than cumulative frequencies on the $y$-axis (an ascending S-curve).
2. More Than Ogive: Plot Lower Class Limits on the $x$-axis against More than cumulative frequencies on the $y$-axis (a descending S-curve).

    Cumulative                       Intersection of Both Ogives
    Frequency (cf)
         ^          \                             /  (Less than ogive)
       N |           \                           /
         |            \                         /
     N/2 + - - - - - - \ - - - - - - - - - - - /
         |              \         P           /
         |               \        *          /
         |                \      / \        /
         | (More than)     \    /   \      /
         |                  \  /     \    /
         +-------------------+--------+--+-------------------------->
         O                            M (MEDIAN on x-axis)

The Two Graphical Methods:

• Method 1 (Single Ogive): On the $y$-axis of a Less Than Ogive, locate $\frac{N}{2}$. Draw a horizontal line to intersect the curve at point $P$. Drop a perpendicular from $P$ to the $x$-axis. The $x$-coordinate gives the Median!
• Method 2 (Double Ogive): The $x$-coordinate of the point of intersection of the Less Than and More Than ogives gives the exact Median!

4. Summary and Examination Tips

Exam Tip: In the missing frequency problem, never search for $N/2 = 30$ in the $cf$ column to find the median class! The median class is determined directly by where the given number ($28.5$) falls (in $20-30$)!

Common Mistake: Forgetting parentheses when subtracting $cf$ in the median formula: $30 - (5 + x) = 30 - 5 - x = 25 - x$. Writing $30 - 5 + x = 25 + x$ ruins the entire solution!