Statistics Class 9 ICSE — Maths Chapter 18
Statistics Class 9 ICSE, Chapter 18 of the Mathematics book, is about turning raw numbers into readable tables and graphs. The chapter defines data and variables, builds frequency tables using tally marks, and ends with histograms and frequency polygons. Each section adds one tool you will use in the exercise and in the exams. Work through the sections in order, and the exercise becomes routine.
Download Statistics Class 9 ICSE Mathematics Chapter 18 PDF
Download the Statistics Class 9 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 12 pages, 8 sections, 13 questions.
What is inside Chapter 18
12 pages · 8 sections · 6 worked examples · 13 exercise questions
| Section | Title |
|---|---|
| 18.1 | INTRODUCTION |
| 18.2 | VARIABLE |
| 18.3 | TABULATION OF DATA |
| 18.4 | FREQUENCY |
| 18.5 | FREQUENCY DISTRIBUTION |
| 18.6 | TYPES OF FREQUENCY DISTRIBUTIONS |
| 18.7 | CLASS INTERVALS AND CLASS LIMITS |
| 18.10 | GRAPHICAL REPRESENTATION OF CONTINUOUS FREQUENCY DISTRIBUTION |
- EXERCISE 18(A): 13 questions
18.1 INTRODUCTION
Statistics is used in two different senses, and the distinction matters. In the singular sense, it names the whole subject: the branch of knowledge that deals with methods for collecting, analysing and interpreting data. In the plural sense, it refers to the numerical data itself, gathered in a systematic way with a definite object in view.
A set of data is a collection of numerical facts gathered with a definite object. Suppose you record the heights of five students as 150 cm, 158 cm, 162 cm, 155 cm and 149 cm. Those five numbers together form a set of data, because they were collected with a clear purpose and can be analysed as one group.
Why does the chapter begin here? Every tool that follows, from frequency to the histogram, works on a set of data. Once you know what data means, the rest of this ICSE chapter becomes a sequence of steps you apply to it.
18.2 VARIABLE
A variable is a quantity that can vary from one individual to another. Height, weight and age are variables, because each one changes from person to person. The chapter separates variables into two types.
A continuous variable can take any numerical value within a certain range. Wages, heights of children and rainfall records are continuous, because between any two values you can always find another. A discrete variable is incapable of taking all possible values. The number of children in a family is discrete: you cannot have 2.5 children.
The same data can also be described by how it is arranged. Raw or ungrouped data is data in the form it was collected. Arrayed data is that same data arranged in ascending or descending order. Consider six test scores: 18, 7, 25, 12, 7, 20. As written, this is raw data. Arranged in ascending order it becomes 7, 7, 12, 18, 20, 25, which is arrayed data.
18.3 TABULATION OF DATA
Tabulation is the representation of data in the form of a table, with rows and columns arranged so that totals and comparisons are visible at a glance. A good table lets a reader see the whole picture without reading every number.
Take three shops and the items they sell. Shop P sells 4 pens and 6 notebooks. Shop Q sells 5 pens and 3 notebooks. Shop R sells 2 pens and 7 notebooks. The table below sets these facts in order.
| Shop | Pens | Notebooks | Total |
|---|---|---|---|
| P | 4 | 6 | 10 |
| Q | 5 | 3 | 8 |
| R | 2 | 7 | 9 |
| Total | 11 | 16 | 27 |
The Total row and the Total column are the point of tabulation. In one glance you read that the three shops together sold 11 pens and 16 notebooks, or 27 items in all. This is exactly what the chapter means by tabulation: arranging data so its meaning surfaces quickly.
18.4 FREQUENCY
Frequency is the number that tells how many times a particular value appears in a given set of data. It is the single building block for every table in the rest of the chapter.
Consider the list: 4, 4, 7, 4, 9, 7. The value 4 appears three times, so the frequency of 4 is 3. The value 7 appears twice, so the frequency of 7 is 2. The value 9 appears once, so its frequency is 1. If a value never appears, its frequency is zero.
Why does frequency matter? A frequency table converts a messy list into a clear count. Instead of scanning the list, you read one number per value, and that number is what later tables and graphs use as their heights.
18.5 FREQUENCY DISTRIBUTION
A frequency distribution is a tabular arrangement of data showing their corresponding frequencies. The table that carries it is called a frequency distribution table. There are two kinds.
An ungrouped frequency distribution lists each distinct value with its count. The tally mark method builds it step by step. Write the values in ascending order in the first column. For each occurrence, mark a short vertical stroke in the tally column.
When four strokes stand for one value, draw the fifth stroke across the first four — this makes a bundle of five. Then count the strokes and write the number in the frequency column. Take the ten values: 5, 7, 5, 9, 7, 5, 8, 7, 5, 6. The value 5 gets four strokes, so its frequency is 4. The value 7 gets three strokes, frequency 3. Values 6, 8 and 9 each get one stroke, frequency 1.
A grouped frequency distribution collects values into class intervals. Suppose marks are grouped as 10-20, 20-30, 30-40. A mark x in the class 10-20 satisfies 10 \le x \lt 20. This inequality notation is the rule for placing boundary values. A mark of 30 belongs in 30-40, not in 20-30, because it equals the lower limit of the higher class.
18.6 TYPES OF FREQUENCY DISTRIBUTIONS
Frequency distributions come in two types. In an inclusive distribution, the upper limit of one class does not coincide with the lower limit of the next. If a mark x belongs to the class 1-10, it means 1 \le x \le 10, with both ends included. In an exclusive distribution, the upper limit of one class equals the lower limit of the next.
A mark x in the class 10-20 means 10 \le x \lt 20, with the upper end excluded.
| Distribution type | Class limits | Meaning of x |
|---|---|---|
| Inclusive | 1-10, 11-20 | 1 \le x \le 10 |
| Exclusive | 10-20, 20-30 | 10 \le x \lt 20 |
The common mistake happens at a boundary. In an exclusive distribution, a value exactly on the boundary, such as 30, is counted in the class where it is the lower limit, never in the previous class. A mark of 40 belongs to 40-50, not 30-40. The inequality 30 \le x \lt 40 makes that placement unambiguous.
18.7 CLASS INTERVALS AND CLASS LIMITS
In an exclusive distribution, the pair 10-20 is a class interval, bounded by two numbers called class limits. The smaller number, 10, is the lower class limit. The larger number, 20, is the upper class limit. The same naming applies to every class.
Inclusive classes must be converted to exclusive form before graphing. The adjustment factor makes this possible. Find the difference between the upper limit of one class and the lower limit of the next, then divide by 2. Subtract this factor from every lower limit and add it to every upper limit.
Take the classes 5-14, 15-24, 25-34. The difference between 15 and 14 is 1, and half of that is 0.5. Subtract 0.5 from each lower limit and add 0.5 to each upper limit, giving 4.5-14.5, 14.5-24.5 and 24.5-34.5. These adjusted limits are the actual or true class limits, also called class boundaries.
Two formulas follow from the actual limits. Class size is the difference between the actual upper and lower limits: 14.5 - 4.5 = 10. Class mark is the value midway between them:
\text{Class-mark} = \frac{\text{Upper limit} + \text{Lower limit}}{2} = \frac{4.5 + 14.5}{2} = 9.5
Cumulative frequency is the sum of the frequencies of all classes up to and including a given class. For classes 10-20, 20-30, 30-40 with frequencies 4, 9 and 5, the cumulative frequencies are 4, then 13, then 18. Written as a ‘less than’ table, this reads: less than 20 is 4, less than 30 is 13, less than 40 is 18.
18.10 GRAPHICAL REPRESENTATION OF CONTINUOUS FREQUENCY DISTRIBUTION
A histogram draws rectangles with class intervals as bases and frequencies as heights. Before drawing, convert the data into exclusive form if it is inclusive. Mark class intervals on the x-axis and frequencies on the y-axis; the two scales need not match. If the x-axis does not start at zero, show a kink or zig-zag near the origin.
A frequency polygon joins the mid-points of adjacent class intervals with straight line segments. To close the figure, join the ends to the mid-points of the classes just before the first and just after the last, where the frequency is zero. There are two ways to draw it.
Using a histogram, mark the mid-point of the top of each rectangle, then join those points in order. Without a histogram, compute each class mark first:
\text{Class-mark} = \frac{\text{Upper limit} + \text{Lower limit}}{2}
Take the classes 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 3, 8, 12, 7, 5. The class marks are 5, 15, 25, 35, 45. Plot the points (5, 3), (15, 8), (25, 12), (35, 7), (45, 5), join them with straight lines, and close the ends at zero frequency. The same frequency tables you build here feed directly into the next chapter on mean and median of ungrouped data.
What to practise from Statistics Class 9 ICSE
The chapter carries one exercise block, EXERCISE 18(A), with 13 questions. The questions test the skills section by section:
- classifying continuous and discrete variables
- building grouped frequency tables from raw marks
- filling blanks on limits and class marks
- converting classes to actual limits and boundaries
- reading class boundaries from a given table
- constructing cumulative frequency tables
- converting a cumulative table back to a frequency table
- making frequency polygons from distributions
These same skills are the ones the graphical section of the chapter builds on, so working through the full exercise prepares you for both halves of the chapter. The scope of the exercise follows the official CISCE syllabus for Class 9 mathematics, which you can verify on the CISCE website.
Statistics Class 9 ICSE frequently asked questions
What is the difference between inclusive and exclusive class intervals in statistics class 9 icse?
In an inclusive distribution the upper limit of one class does not match the lower limit of the next, and both ends are included, so 1 \le x \le 10. In an exclusive distribution the upper limit of one class equals the lower limit of the next, and the upper end is excluded, so 10 \le x \lt 20.
How do I convert inclusive class intervals into exclusive form?
Find the difference between the upper limit of one class and the lower limit of the next, and divide by 2. That gives the adjustment factor. Subtract it from every lower limit and add it to every upper limit.
What is a class mark and how do I calculate it?
The class mark is the value midway between the actual lower and upper limits of a class. Use \frac{\text{Upper limit} + \text{Lower limit}}{2}. For the class 4.5-14.5, the class mark is 9.5.
How do I draw a frequency polygon without using a histogram?
Compute the class mark of each class, plot the points with class marks on the x-axis and frequencies on the y-axis, and join the consecutive points with straight lines. Close the figure by joining the ends to the mid-points at zero frequency just before the first class and just after the last.
How do I decide whether a variable is discrete or continuous?
A continuous variable can take any value within a range, like height or wages. A discrete variable cannot take all possible values, like the number of children in a family.
Reference: ICSE Class 9 Mathematics textbook material, chapter 18: Statistics — CISCE-aligned.