Statistics

Introduction

Statistics is the science of collecting, organising, analysing, and interpreting data. It is widely used in economics, business, sciences, and daily life. ICSE Class 9 covers descriptive statistics fundamentals.

Types of Data

Primary Data

Collected directly from the source (surveys, experiments, questionnaires) for a specific purpose.

Secondary Data

Already collected by someone else and used for a different purpose (government records, published reports).

Data Classification

Type	Description	Example
Qualitative	Non-numerical attributes	Hair colour, gender, city
Quantitative	Numerical measurements	Height, weight, marks
Discrete	Countable values (integers)	Number of students, cars
Continuous	Any value in a range	Temperature, weight

Frequency Distribution

Ungrouped Frequency Distribution

Listing each observation with its frequency.

Example: Marks of 20 students: 5, 6, 7, 5, 8, 6, 7, 5, 9, 6, 7, 8, 5, 6, 7, 7, 8, 6, 5, 7

Marks	Tally Marks	Frequency
5
6
7
8
9
Total		20

Grouped Frequency Distribution

Continuous data is grouped into class intervals.

Key Terms:

Class Interval: Range of values (e.g., 10-20)
Class Size: Upper limit - Lower limit
Class Mark: Midpoint = (Upper limit + Lower limit)/2
Frequency: Number of observations in each class

<ICSEExample title="Create a Grouped Frequency Table"> Marks of 30 students (out of 100): 45, 67, 89, 34, 56, 78, 92, 43, 55, 77, 88, 23, 65, 71, 82, 49, 58, 69, 73, 85, 95, 38, 62, 74, 81, 52, 63, 79, 87, 41

Create a grouped frequency table with classes 20-30, 30-40, etc. <Solution>

Class	Tally	Frequency
20-30
30-40
40-50
50-60
60-70
70-80
80-90
90-100
Total		30

</Solution> </ICSEExample>

Graphical Representation

Histogram

A histogram consists of adjacent rectangles for continuous data. The width represents the class interval and height represents the frequency.

Rules:

Class intervals on x-axis
Frequency on y-axis
Bars touch each other (no gaps)
Area of each bar is proportional to frequency

Frequency Polygon

A frequency polygon is formed by joining the midpoints of the top of histogram bars with straight lines.

Method:

Find the class mark (midpoint) of each class
Plot class mark vs frequency
Join the points with straight lines
Extend to x-axis at both ends (add imaginary classes with 0 frequency)

Measures of Central Tendency

Mean (Average)

Arithmetic Mean = Sum of all observations / Number of observations

For ungrouped data: Mean = x̄ = (x1 + x2 + ... + xn)/n

Median

The median is the middle value when data is arranged in ascending or descending order.

Steps:

Arrange data in ascending order
If n is odd: Median = value at position (n+1)/2
If n is even: Median = average of values at positions n/2 and n/2 + 1

<ICSEExample title="Find Median (Odd)"> Find the median: 15, 12, 18, 20, 25, 10, 8 <Solution> Arrange: 8, 10, 12, 15, 18, 20, 25 n = 7 (odd) Median = value at position (7+1)/2 = 4th value Median = 15 </Solution> </ICSEExample> <ICSEExample title="Find Median (Even)"> Find the median: 5, 8, 3, 7, 10, 6 <Solution> Arrange: 3, 5, 6, 7, 8, 10 n = 6 (even) Median = (3rd value + 4th value)/2 = (6 + 7)/2 = 6.5 </Solution> </ICSEExample>

Mode

The mode is the value that occurs most frequently in a data set.

<ICSEExample title="Find Mode"> Find the mode: 2, 3, 5, 3, 4, 5, 3, 6, 7, 3, 8 <Solution> Frequency: 2(1), 3(4), 4(1), 5(2), 6(1), 7(1), 8(1) Mode = 3 (occurs 4 times, highest frequency) </Solution> </ICSEExample>

Common Mistakes With Fixes

Mistake	Correction
Using gaps in histogram	Histogram bars touch; gaps are for bar graphs
Confusing class mark with class limits	Class mark = (upper + lower)/2
Not arranging data before finding median	Data MUST be sorted before finding median
Confusing mode with mean	Mode is most frequent, mean is average

ICSE Exam Focus

Topic	Marks (approx.)	Frequency
Mean, median, mode calculation	4-5 marks	Very common
Histogram and frequency polygon	4 marks	Common
Grouped frequency distribution	3-4 marks	Very common
Identifying data types	2 marks	Occasionally asked

Self-Test

Q1: Find the mean, median, and mode of: 4, 7, 2, 4, 9, 4, 6, 7, 4, 8

Q2: Draw a histogram for: Classes 0-10(5), 10-20(8), 20-30(12), 30-40(7), 40-50(3)

Q3: Find the median: 15, 22, 18, 30, 25, 19, 28, 21, 16, 24

Q4: What is the class mark of the interval 35-45?

Q5: The mean of 5 numbers is 18. If one number 20 is removed, find the new mean.

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