Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Chapter Introduction
Raw data is like a pile of unsorted books — it contains all the information but is useless without organisation. Organisation of data is the second step in statistical analysis after collection. It involves classifying and arranging data into meaningful groups so that patterns, trends, and relationships become visible. This chapter teaches you how to classify data, construct frequency distributions, and understand the key terms (class intervals, class boundaries, class mark, cumulative frequency) that form the basis for all further statistical work.
The concepts in this chapter are foundational — every subsequent chapter (from Presentation of Data to Correlation) depends on your understanding of frequency distributions, class intervals, and cumulative frequencies.
Key Definitions
| Term | Definition |
|---|---|
| Raw data | Unorganised, unclassified numerical facts collected from a survey or experiment |
| Classification | The process of arranging data into groups or classes according to common characteristics |
| Variable | A characteristic that can take different values — e.g., height, income, marks |
| Frequency | The number of times a particular value or class occurs in the data |
| Frequency distribution | A table showing values or classes and their corresponding frequencies |
| Class interval | The range of values grouped into one class (e.g., 10-20) |
| Class limit | The smallest and largest values that can belong to a class (lower and upper limits) |
| Class boundary | The true limits of a class — adjusted so there is no gap between consecutive classes |
| Class mark | The midpoint of a class interval: (lower limit + upper limit) / 2 |
| Class width (h) | The difference between upper and lower boundaries of a class: h = upper boundary - lower boundary |
| Cumulative frequency | The running total of frequencies, either less than or more than type |
| Bivariate frequency distribution | A frequency distribution that shows frequencies for two variables simultaneously |
| Magnitude of class | The size or width of a class interval |
Detailed Concepts
Classification of Data
Classification groups data according to common characteristics so that it becomes meaningful and analysable.
| Basis of classification | Type | Description | Example |
|---|---|---|---|
| Chronological | Time-based | Data arranged in order of time of occurrence | GDP data from 2000 to 2025; monthly sales figures |
| Geographical | Location-based | Data classified by region or place | Literacy rate by state; per capita income by district |
| Qualitative | Attribute-based | Data classified by qualities or characteristics | Population by religion, gender, occupation, marital status |
| Quantitative | Value-based | Data classified by numerical values into class intervals | Income groups: 0-10,000, 10,001-20,000; marks distribution |
Objectives of classification:
1. To simplify and condense raw data 2. To facilitate comparison 3. To highlight significant features 4. To enable further statistical analysis 5. To arrange data in a logical order
Frequency Distribution
A frequency distribution summarises raw data by showing how often each value or group of values occurs.
Discrete Frequency Distribution
Used when data takes specific, separate values (not continuous). Each distinct value is listed with its frequency.
| Marks | No. of students (frequency) |
|---|---|
| 5 | 3 |
| 10 | 7 |
| 15 | 12 |
| 20 | 8 |
| 25 | 5 |
| Total | 35 |
Continuous Frequency Distribution
Used when data can take any value within a range. Values are grouped into class intervals.
| Class interval (marks) | No. of students (frequency) |
|---|---|
| 0-10 | 4 |
| 10-20 | 8 |
| 20-30 | 12 |
| 30-40 | 6 |
| 40-50 | 5 |
| Total | 35 |
Class Intervals: Inclusive vs Exclusive
| Feature | Inclusive (Discrete) | Exclusive (Continuous) |
|---|---|---|
| Example | 0-9, 10-19, 20-29 | 0-10, 10-20, 20-30 |
| Upper limit | Included in the same class | Excluded from the class, goes to next class |
| Gap between classes | There is a gap (9 to 10) | No gap — upper limit of one class = lower limit of next |
| Suitability | Discrete data (whole numbers) | Continuous data (can take decimals) |
| Conversion to exclusive | Not needed if discrete | Already in exclusive form |
| Graphical use | Must be converted to exclusive for histograms | Can be used directly for histograms and ogives |
Converting inclusive to exclusive intervals:
When data is continuous but given in inclusive form, convert it by adjusting the limits:
- Adjustment factor = (lower limit of next class - upper limit of current class) / 2
- New lower limit = old lower limit - adjustment factor
- New upper limit = old upper limit + adjustment factor
Worked example: Inclusive classes 10-19, 20-29
- Gap = 20 - 19 = 1
- Adjustment factor = 1 / 2 = 0.5
- First class becomes: 10 - 0.5 = 9.5 to 19 + 0.5 = 19.5
- Second class becomes: 20 - 0.5 = 19.5 to 29 + 0.5 = 29.5
- So: 9.5-19.5, 19.5-29.5 (exclusive)
Key Terms in Continuous Frequency Distribution
| Term | Formula | Example (class 20-30, frequency 12) |
|---|---|---|
| Lower limit (L) | Smallest value in the class | 20 |
| Upper limit (U) | Largest value in the class | 30 |
| Class mark (x) | (L + U) / 2 | (20 + 30) / 2 = 25 |
| Class width (h) | U - L (for exclusive intervals) | 30 - 20 = 10 |
| Class frequency (f) | Number of observations in the class | 12 |
| Lower boundary | L - adjustment (for inclusive intervals) | 19.5 |
| Upper boundary | U + adjustment (for inclusive intervals) | 29.5 |
| Magnitude | Same as class width | 10 |
| Class density | Frequency / Class width | 12 / 10 = 1.2 |
Cumulative Frequency
Cumulative frequency is the running total of frequencies. It is of two types:
Less Than Cumulative Frequency
Start from the lowest class and keep adding frequencies going upward.
| Marks (less than) | Frequency | Cumulative frequency |
|---|---|---|
| Less than 10 | 4 | 4 |
| Less than 20 | 8 | 4 + 8 = 12 |
| Less than 30 | 12 | 12 + 12 = 24 |
| Less than 40 | 6 | 24 + 6 = 30 |
| Less than 50 | 5 | 30 + 5 = 35 |
More Than Cumulative Frequency
Start from the highest class and keep adding frequencies going downward.
| Marks (more than) | Frequency | Cumulative frequency |
|---|---|---|
| More than 0 | 35 | 35 |
| More than 10 | 4 | 35 - 4 = 31 |
| More than 20 | 8 | 31 - 8 = 23 |
| More than 30 | 12 | 23 - 12 = 11 |
| More than 40 | 6 | 11 - 6 = 5 |
Bivariate Frequency Distribution
A bivariate frequency distribution shows the joint distribution of two variables simultaneously. It is presented as a two-way table.
| Income / Expenditure | 0-5 | 5-10 | 10-15 | 15-20 | Total |
|---|---|---|---|---|---|
| 0-10 | 2 | 3 | 0 | 0 | 5 |
| 10-20 | 1 | 4 | 2 | 0 | 7 |
| 20-30 | 0 | 2 | 5 | 3 | 10 |
| 30-40 | 0 | 0 | 3 | 5 | 8 |
| Total | 3 | 9 | 10 | 8 | 30 |
This table shows how many households fall in each combination of income and expenditure class. It helps study the relationship between the two variables. The row totals and column totals are called marginal frequencies.
Types of Frequency Distribution Series
| Type | Description | When to use | Example |
|---|---|---|---|
| Individual series | Raw values listed without grouping | Small dataset (n < 20) | 5, 12, 8, 3, 15 |
| Discrete series | Values with their frequencies | Data takes specific repeated values | 5(3), 10(7), 15(5) |
| Continuous series | Class intervals with frequencies | Large dataset with continuous data | 0-10(4), 10-20(8) |
Rules for Forming Class Intervals
1. Number of classes: Usually 5 to 15. Too few classes lose detail; too many classes make the table unwieldy. 2. Class width: Should be uniform (equal width) for simplicity. If not, adjustments are needed for histograms. 3. No overlapping: Classes must be mutually exclusive — each value belongs to exactly one class. 4. No gaps: For continuous data, classes should cover the entire range without gaps. 5. Open-ended classes: Avoid if possible. Classes like "below 10" or "above 50" cause problems in mean/median calculations.
Worked Example: Converting Inclusive to Exclusive Intervals
Given the following inclusive class intervals: 0-9, 10-19, 20-29, 30-39
Step 1: Identify the gap between consecutive classes.
- Gap between first and second class = 10 - 9 = 1
Step 2: Calculate the adjustment factor.
- Adjustment factor = Gap / 2 = 1 / 2 = 0.5
Step 3: Subtract 0.5 from lower limits and add 0.5 to upper limits.
- 0 - 0.5 = -0.5 to 9 + 0.5 = 9.5 → -0.5 to 9.5 (but we use 0-9.5 if data starts at 0)
- 10 - 0.5 = 9.5 to 19 + 0.5 = 19.5 → 9.5 to 19.5
- 20 - 0.5 = 19.5 to 29 + 0.5 = 29.5 → 19.5 to 29.5
- 30 - 0.5 = 29.5 to 39 + 0.5 = 39.5 → 29.5 to 39.5
The exclusive intervals are: 0-9.5, 9.5-19.5, 19.5-29.5, 29.5-39.5 (adjusted for 0 start)
Important Points for Exams
1. Classification organises raw data into meaningful groups based on common characteristics. 2. Four bases of classification: chronological, geographical, qualitative, quantitative. 3. Frequency distribution shows how many times each value or class occurs. 4. Exclusive class intervals have no gap; inclusive class intervals have a gap between upper and lower limits. 5. Class mark = (lower limit + upper limit) / 2; it represents the midpoint of a class. 6. Cumulative frequency can be "less than" type or "more than" type. 7. Bivariate frequency distribution shows two variables together in a two-way table. 8. To convert inclusive to exclusive intervals, use the adjustment factor = gap / 2. 9. Usually 5 to 15 class intervals are formed; class width should preferably be uniform. 10. Marginal frequencies in a bivariate distribution are the row totals and column totals.
Common Mistakes
| Mistake | Why it is wrong | Correct approach |
|---|---|---|
| Confusing class limits with class boundaries | Limits are stated values; boundaries are adjusted to remove gaps | Boundaries are used for continuous data; limits are used for discrete data |
| Including the upper limit in exclusive intervals | In exclusive intervals, a value equal to the upper limit belongs to the next class | Place boundary values in the next higher class |
| Calculating class mark using boundaries instead of limits | Class mark uses stated limits, not adjusted boundaries, for standard calculation | Class mark = (lower limit + upper limit) / 2 |
| Forgetting to convert inclusive to exclusive before drawing graphs | Histograms and ogives require continuous (exclusive) intervals | Always convert inclusive intervals to exclusive before graphical presentation |
| Confusing "less than" and "more than" cumulative frequency | They start from opposite ends and serve different purposes | "Less than" starts from lowest; "more than" starts from highest |
| Using too many or too few classes | Too few classes lose information; too many make the table complex | Use 5-15 classes depending on data size |
NCERT Exercise Focus
Focus on these NCERT exercise questions while preparing:
1. "Distinguish between inclusive and exclusive class intervals" — Most predictable question. Prepare a comparison table. 2. "Construct a frequency distribution from the given raw data" — Practice grouping data into class intervals. 3. "Calculate cumulative frequency (less than and more than)" — Both types are tested. Practice thoroughly. 4. "What is class mark? Calculate class marks for the given data." — Simple but commonly asked 1-2 mark question. 5. "Prepare a bivariate frequency distribution" — Understand two-way table construction.
Exam Tips
1. Know the four bases of classification — Chronological, geographical, qualitative, quantitative. These appear in 1-mark and 3-mark questions. 2. Inclusive vs Exclusive intervals — This is the most tested distinction. Prepare a comparison table and know the conversion method with a worked example. 3. Cumulative frequency calculation — Practice both "less than" and "more than" types. They are used for drawing ogives in Chapter 4. 4. Class mark, class width — Simple formula-based 1-mark questions. Class mark = (L+U)/2, class width = U-L. 5. Bivariate frequency distribution — Understand how to read a two-way table. You may be asked to find marginal totals. 6. Numerical questions — Practice converting inclusive to exclusive intervals and calculating cumulative frequencies. These are common 3-mark numericals. 7. Rules for forming class intervals — Know the guidelines: 5-15 classes, uniform width, no overlapping, no gaps.
Practice Questions for Self-Test
1. Distinguish between inclusive and exclusive class intervals with examples. (3 marks) 2. What are the four bases of classification of data? Explain with examples. (3 marks) 3. Convert the following inclusive class intervals to exclusive: 0-9, 10-19, 20-29. (2 marks) 4. Calculate class marks for the classes: 10-20, 20-30, 30-40, 40-50. (2 marks) 5. What is cumulative frequency? Explain with an example. (3 marks) 6. What is a bivariate frequency distribution? When is it used? (2 marks) 7. Differentiate between individual, discrete, and continuous series. (3 marks) 8. What are the rules for forming class intervals? (3 marks) 9. What is the difference between class limits and class boundaries? (2 marks) 10. Calculate less than and more than cumulative frequencies for the data: 0-10(4), 10-20(8), 20-30(12), 30-40(6), 40-50(5). (3 marks)
Key Statistics and Facts for Quick Reference
| Fact | Detail |
|---|---|
| Four bases of classification | Chronological, Geographical, Qualitative, Quantitative |
| Inclusive example | 0-9, 10-19 (gap exists, upper limit included) |
| Exclusive example | 0-10, 10-20 (no gap, upper limit excluded) |
| Class mark formula | (L + U) / 2 |
| Class width formula | U - L |
| Adjustment factor | Gap / 2 |
| Recommended classes | 5 to 15 |
| Cumulative frequency types | Less than (ascending), More than (descending) |
Quick Revision Notes
- Classification types: chronological (time), geographical (place), qualitative (attribute), quantitative (value).
- Frequency distribution: shows frequency of each value or class — discrete or continuous.
- Exclusive intervals: no gap, upper limit excluded (0-10, 10-20). Inclusive intervals: gap, upper limit included (0-9, 10-19).
- Conversion: inclusive to exclusive — subtract and add adjustment factor = gap/2 to limits.
- Class mark = (L + U) / 2. Class width = U - L.
- Cumulative frequency: "less than" (ascending), "more than" (descending).
- Bivariate distribution: two-way table showing joint frequencies of two variables.
- Individual series: raw values; Discrete series: values with frequencies; Continuous series: class intervals with frequencies.
- Rules: 5-15 classes, uniform width, no overlap, no gaps, avoid open-ended classes.