Statistics Class 9 Maths Revision Notes

Statistics is one of the most practical and scoring chapters in Class 9 Mathematics. These revision notes are designed to help you quickly revisit every key concept from understanding how data is collected and classified, to drawing histograms and calculating mean, median, and mode.

Whether you are preparing for your school exams or strengthening your foundation for higher classes, this guide covers all the essential topics with clear definitions, important formulas, and fully solved examples.

Learnr data collection, frequency distributions, graphical representations, and measures of central tendency with solved examples and exam-ready explanations.

Introduction to Statistics

Statistics is a branch of mathematics concerned with the collection, organisation, analysis, and interpretation of numerical data. Its use in India dates back to ancient times, from population records to agricultural yield tracking.

In everyday life, expressing facts numerically gives precision. Rather than saying "India has a large population," it is more meaningful to state that India's population exceeded one billion according to the 2001 census.

Types of Data

Primary Data

Data collected directly by the investigator for the first time — through surveys, interviews, or experiments. Example: A researcher recording workers' monthly incomes in a factory.

Secondary Data

Data already collected by someone else — available in published reports, government records, or academic papers. Must be used carefully, verifying source reliability.

Note:

Primary data is more reliable and specific to your research question. Secondary data saves time but may introduce bias or outdated information.

Classification of Data

Raw data (also called Crude Data) is data in its original, unorganised form. For example, marks of 20 students recorded randomly. Classification organises this data into meaningful groups.

Four Bases of Classification

Important Definitions

Variate

A numerical quantity whose value varies from observation to observation, typically denoted x.

• Discrete variate: Takes fixed, countable values (e.g., number of teachers: 30, 35, 40).
• Continuous variate: Takes any value within a range (e.g., heights in groups 150–160 cm, 160–170 cm…).

Range

Range = Maximum value − Minimum value

Class Interval (h or i)

Class Interval (h) = Range ÷ Number of Classes

Class Mark (Mid Value)

Class Mark = (Lower Limit + Upper Limit) ÷ 2

Solved Example — Finding Class Limits from Mid Values

Q: The mid values of a distribution are 54, 64, 74, 84, and 94. Find the class interval and class limits.

Step 1: Class interval h = 64 − 54 = 10

Step 2: Class limits = (mid value − h/2) to (mid value + h/2)

1st class: 54 − 5 to 54 + 5 → 49 – 59

2nd class: 64 − 5 to 64 + 5 → 59 – 69

3rd class: 74 − 5 to 74 + 5 → 69 – 79

4th class: 84 − 5 to 84 + 5 → 79 – 89

5th class: 94 − 5 to 94 + 5 → 89 – 99

Frequency Distribution

The frequency of a value is how many times it appears in a dataset. A frequency distribution table organises data to show each value alongside its frequency, making patterns visible at a glance.

Types of Frequency Distribution

1. Individual Frequency Distribution

Each data point is listed separately. Frequency of each variable is 1. Used for small datasets, e.g., marks of 10 students.

2. Discrete Frequency Distribution (Ungrouped)

Used when a variable takes specific, fixed values. Similar values are grouped under one entry with a frequency count.

3. Continuous Frequency Distribution (Grouped)

Used when the variate is continuous (can take any value). Data is placed into class intervals such as 0–5, 5–10, 10–15, etc.

Important Note

In continuous distributions, classes must be continuous. Gaps between classes (e.g., 0–5, 6–10) leave values like 5.5 unclassified. Fix this by using the exclusive series method where the upper limit of one class equals the lower limit of the next.

Exclusive vs Inclusive Series

Exclusive Series

Upper limit of one class=Lower limit of next class. The upper limit value is not counted in that class it goes to the next. E.g., 10–20, 20–30, 30–40.

Inclusive Series

Both limits are included in the same class. E.g., 10–19, 20–29, 30–39. To convert to exclusive, add/subtract half the difference between consecutive limits.

Cumulative Frequency

Cumulative frequency is the running total of frequencies. It tells you how many observations fall at or below a certain value.

This table can be read as less than (e.g., fewer than 10 students aged less than 10 years) or more than (e.g., 178 students aged 5 and above).

Graphical Representation of Data

Graphs transform tables of numbers into visual stories. Class 9 Statistics covers six key graphical forms:

1. Bar Graphs
2. Histograms
3. Frequency Polygons
4. Frequency Curves
5. Cumulative Frequency Curves (Ogives)
6. Pie Diagrams

Bar Graphs

Bar graphs use separate rectangular bars to represent categorical data. Each bar's height is proportional to its value. There are gaps between bars.

Histograms

A histogram looks like a bar graph but represents continuous grouped data.

• No gaps between bars (unlike bar graphs).
• The width of each bar equals the class interval.
• The area of each rectangle is proportional to its class frequency.
• When class intervals are unequal, heights are redefined using:

Redefined frequency = (h ÷ class interval) × original frequency, where h is the smallest class interval.

Bar Graph vs Histogram

In a bar graph: bars are separate, width is irrelevant, used for categorical data.

In a histogram: bars are adjacent, width represents class interval, used for continuous data.

Frequency Polygon

A frequency polygon is drawn by connecting the midpoints of the top edges of histogram bars with straight lines. It can also be drawn independently using class marks (midpoints) plotted against frequencies.

Steps to draw a Frequency Polygon (with histogram):

1. Construct the histogram.
2. Mark the midpoint of the top edge of each rectangle.
3. Connect consecutive midpoints with straight lines.
4. Extend the polygon to the x-axis at both ends (using assumed classes with zero frequency).

Solved Example Histogram with Unequal Class Intervals

Q. Weekly wages: 1000–2000 (26 workers), 2000–2500 (30), 2500–3000 (20), 3000–5000 (16), 5000–5500 (1). Draw the histogram.

Smallest interval h = 500

Redefined frequencies:

1000–2000: (500/1000) × 26 = 13

2000–2500: (500/500) × 30 = 30

2500–3000: (500/500) × 20 = 20

3000–5000: (500/2000) × 16 = 4

5000–5500: (500/500) × 1 = 1

Plot class intervals on the x-axis and redefined frequencies on the y-axis to draw the histogram.

Measures of Central Tendency

A measure of central tendency provides a single representative value that summarises an entire dataset. The three main measures studied in Class 9 are Mean, Median, and Mode.

Mean (Arithmetic Mean)

The mean is the sum of all observations divided by the total number of observations. Denoted by x̄ (read as "x-bar").

x̄ = (x₁ + x₂ + x₃ + … + xₙ) ÷ n

For frequency distribution: x̄ = Σ(fᵢxᵢ) ÷ Σfᵢ

Properties of Mean

• Adding a constant a to all values → new mean = x̄ + a
• Subtracting a constant a → new mean = x̄ − a
• Multiplying all values by a → new mean = a · x̄
• Dividing all values by a → new mean = x̄ ÷ a

Combined Mean

x̄ = (n₁x̄₁ + n₂x̄₂ + …) ÷ (n₁ + n₂ + …)

Solved Example — Finding Unknown Value from Mean

Q: If the mean of 6, 4, 7, P, and 10 is 8, find P.

8 = (6 + 4 + 7 + P + 10) ÷ 5

40 = 27 + P

P = 13

Median

The median is the middle value that divides a ranked dataset into two equal halves.

Steps for Ungrouped Data:

1. Arrange data in ascending order.
2. Count total observations (n).
3. If n is odd: Median = value of the ((n+1)/2)ᵗʰ observation.
4. If n is even: Median = average of (n/2)ᵗʰ and (n/2 + 1)ᵗʰ observations.

Q. Find the median of: 37, 31, 42, 43, 46, 25, 39, 45, 32

Arranged: 25, 31, 32, 37, 39, 42, 43, 45, 46

n = 9 (odd) → Median = value of 5th term = 39

Q. Observations 11, 12, 14, 18, x+2, x+4, 30, 32, 35, 41 are in ascending order; median = 24. Find x.

n = 10 (even) → Median = [(5th) + (6th)] ÷ 2

24 = [(x+2) + (x+4)] ÷ 2

48 = 2x + 6

x = 21

Mode

The mode is the value that appears most frequently in a dataset. A dataset may have one mode (unimodal), two modes (bimodal), or more.

Q. Find the mode of: 7, 7, 10, 12, 12, 12, 11, 13, 13, 17

Frequency of 12 = 3 (highest)

Mode = 12

Empirical Relationship: Mean, Median & Mode

Karl Pearson's Empirical Formula

Mode = 3 × Median − 2 × Mean

This formula is useful when one of the three measures is unknown you can estimate it from the other two. It holds well for moderately skewed distributions.

Uses of Each Measure

Mean: Best for numerical data without extreme values. Used in calculating averages: marks, income, output per machine.

Median: Best for qualitative data or skewed distributions. Used in wage analysis and wealth distribution studies.

Mode: Best for identifying the most common value. Used in manufacturing (most common shoe size), fashion, and business forecasting.

06 RangeMeasure of Dispersion

While measures of central tendency describe where data is centred, range describes how spread out the data is. It is the simplest measure of dispersion.

Range = Highest value (h) − Lowest value (ℓ)

Coefficient of Range = (h − ℓ) ÷ (h + ℓ)

For grouped data, range = midpoint of the last class − midpoint of the first class.

Solved Example — Range of Grouped Data

Q: Find the range for classes: 0–5, 5–10, 10–15, 15–20, 20–25.

Midpoint of first class=(0 + 5)/2 = 2.5

Midpoint of last class=(20 + 25)/2 = 22.5

Range = 22.5 − 2.5 = 20