Statistics

ICSE Class 10 Mathematics • Chapter 14

1. Basic Terminology

Data: Collection of facts, figures, or observations.

Raw Data: Data in its original, unorganized form.

Grouped Data: Data organized into class intervals with frequencies.

Class Interval: A range of values (e.g., 10-20, 20-30).

Class Mark (Mid-value): $x = \frac{\text{Upper Limit + Lower Limit}}{2}$

Class Size (h): Upper Limit − Lower Limit

Frequency (f): Number of observations in a class.

Cumulative Frequency (cf): Running total of frequencies.

Types of Class Intervals

Exclusive (Continuous): 10-20, 20-30, 30-40 (Upper limit of one = Lower limit of next)

Inclusive (Discontinuous): 10-19, 20-29, 30-39 (Gaps between classes)

Conversion: Subtract 0.5 from lower limit, add 0.5 to upper limit.

Example: 10-19 becomes 9.5-19.5

2. Measures of Central Tendency

Mean (Arithmetic Average): Sum of all values ÷ Number of values
Median: Middle value when data is arranged in ascending/descending order
Mode: Value that occurs most frequently

3. Mean of Grouped Data

There are three methods to calculate mean from grouped data:

Method 1: Direct Method

$\bar{x} = \frac{\sum fx}{\sum f}$

Where: $f$ = frequency, $x$ = class mark (mid-value)

Steps:

Find class mark $x$ for each class: $x = \frac{L + U}{2}$
Multiply frequency with class mark: $fx$
Find $\sum f$ (total frequency) and $\sum fx$
Apply formula: $\bar{x} = \frac{\sum fx}{\sum f}$

Method 2: Short-Cut (Assumed Mean) Method

$\bar{x} = A + \frac{\sum fd}{\sum f}$

Where: $A$ = assumed mean, $d = x - A$ (deviation from assumed mean)

Steps:

Find class marks $x$
Choose assumed mean $A$ (usually class mark of middle class or class with highest frequency)
Find deviation: $d = x - A$
Calculate $fd$ for each class
Apply formula: $\bar{x} = A + \frac{\sum fd}{\sum f}$

Method 3: Step-Deviation Method (Most Efficient)

$\bar{x} = A + \frac{\sum ft}{\sum f} \times h$

Where: $t = \frac{x - A}{h} = \frac{d}{h}$, $h$ = class size

Steps:

Find class marks $x$ and choose assumed mean $A$
Find deviation: $d = x - A$
Find step-deviation: $t = \frac{d}{h}$ (where $h$ = class size)
Calculate $ft$ for each class
Apply formula: $\bar{x} = A + \frac{\sum ft}{\sum f} \times h$

When to use which method:

Direct Method: When class marks are small integers
Short-cut Method: When class marks are large numbers
Step-deviation Method: When class size is uniform (BEST for exams)

Solved Example: Finding Mean

Example 1: Find the mean of the following distribution using step-deviation method:

Marks	0-10	10-20	20-30	30-40	40-50
Students	5	10	25	30	20

Solution:

Marks	f	x	A = 25	d = x − A	t = d/h	ft
0-10	5	5		−20	−2	−10
10-20	10	15		−10	−1	−10
20-30	25	25	← A	0	0	0
30-40	30	35		10	1	30
40-50	20	45		20	2	40
Total	90					50

$\bar{x} = A + \frac{\sum ft}{\sum f} \times h = 25 + \frac{50}{90} \times 10 = 25 + 5.56 = \mathbf{30.56}$

4. Median of Grouped Data

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

Where:

$l$ = Lower limit of median class
$n$ = Total frequency ($\sum f$)
$cf$ = Cumulative frequency of class before median class
$f$ = Frequency of median class
$h$ = Class size

Steps to Find Median:

Prepare cumulative frequency column
Find $\frac{n}{2}$ (where $n = \sum f$)
Locate Median Class: The class whose cumulative frequency is just greater than or equal to $\frac{n}{2}$
Identify $l$, $cf$, $f$, $h$ for the median class
Apply the formula

Example 2: Find the median for:

Class	0-10	10-20	20-30	30-40	40-50
Frequency	5	15	30	8	2

Solution:

Class	f	cf
0-10	5	5
10-20	15	20
20-30	30	50 ← Median class
30-40	8	58
40-50	2	60

$n = 60$, $\frac{n}{2} = 30$

Median class = 20-30 (cf = 50 is first to exceed 30)

$l = 20$, $cf = 20$, $f = 30$, $h = 10$

Median $= 20 + \frac{30 - 20}{30} \times 10 = 20 + \frac{10}{30} \times 10 = 20 + 3.33 = \mathbf{23.33}$

5. Mode of Grouped Data

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

Where:

$l$ = Lower limit of modal class
$f_1$ = Frequency of modal class
$f_0$ = Frequency of class before modal class
$f_2$ = Frequency of class after modal class
$h$ = Class size

Modal Class: The class with the highest frequency.

Example 3: Find the mode for the data in Example 1.

Modal class = 30-40 (highest frequency = 30)

$l = 30$, $f_1 = 30$, $f_0 = 25$, $f_2 = 20$, $h = 10$

Mode $= 30 + \frac{30 - 25}{2(30) - 25 - 20} \times 10$

$= 30 + \frac{5}{60 - 45} \times 10 = 30 + \frac{5}{15} \times 10 = 30 + 3.33 = \mathbf{33.33}$

6. Empirical Relationship

For moderately asymmetric distribution:

Mode = 3 × Median − 2 × Mean

or equivalently: Mean − Mode = 3(Mean − Median)

7. Graphical Representation

A. Histogram

A bar graph with no gaps between bars for continuous data.

X-axis: Class intervals
Y-axis: Frequency
Width of bar = Class size
Area of bar ∝ Frequency

[Diagram: Histogram with bars for different class intervals, showing frequency on Y-axis]

Finding Mode from Histogram

Identify the tallest bar (modal class)
Draw diagonal lines from top corners of modal bar to top corners of adjacent bars
Where diagonals intersect, drop a perpendicular to x-axis
The x-coordinate is the mode

B. Frequency Polygon

A line graph formed by joining mid-points of tops of histogram bars.

Can be drawn with or without histogram
Starts and ends at x-axis (using class marks of imaginary classes before and after)

[Diagram: Frequency polygon overlaid on histogram, showing connected mid-points]

C. Ogive (Cumulative Frequency Curve)

Less than Ogive (Rising Curve):

Plot: Upper class limit (x) vs Cumulative frequency (y)
Starts from origin or near it
Curve rises from left to right

More than Ogive (Falling Curve):

Plot: Lower class limit (x) vs Cumulative frequency from above (y)
Curve falls from left to right

[Diagram: Both Less than and More than Ogive curves, intersecting at the median point]

Finding Median from Ogive

Method 1 (Using one Ogive):

Locate $\frac{n}{2}$ on y-axis
Draw horizontal line to meet the ogive
From intersection, drop perpendicular to x-axis
The x-coordinate = Median

Method 2 (Using both Ogives):

The x-coordinate of intersection of Less than and More than Ogive = Median

8. Important Points to Remember

Mean is affected by extreme values; Median and Mode are not.
For symmetrical distribution: Mean = Median = Mode
Class mark is always used for calculations, not class limits.
Cumulative frequency is always a running total (increases or stays same, never decreases).
In step-deviation, $t$ values are always integers (positive, zero, or negative).

Quick Reference: Formulas at a Glance

Measure	Formula
Mean (Direct)	$\bar{x} = \frac{\sum fx}{\sum f}$
Mean (Short-cut)	$\bar{x} = A + \frac{\sum fd}{\sum f}$
Mean (Step-deviation)	$\bar{x} = A + \frac{\sum ft}{\sum f} \times h$
Median	$l + \frac{\frac{n}{2} - cf}{f} \times h$
Mode	$l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$
Relationship	Mode = 3 Median − 2 Mean

Exam Practice Questions (PYQ Trends)

PYQ: 2023

BOARD The following distribution gives the daily income of 50 workers:

Income (₹)	100-120	120-140	140-160	160-180	180-200
Workers	12	14	8	6	10

Convert to 'more than type' distribution and draw its ogive.

PYQ: 2022

BOARD The mean of the following frequency distribution is 50. Find the missing frequencies $f_1$ and $f_2$:

Class	0-20	20-40	40-60	60-80	80-100	Total
Freq	17	$f_1$	32	$f_2$	19	120

Additional Practice

HOTS Draw both 'less than' and 'more than' ogives for the following distribution. Hence find the median.

Class	30-40	40-50	50-60	60-70	70-80
Freq	14	6	10	8	12