Big picture
Raw data, such as the marks of 50 students written in the order collected, are hard to understand. Organisation arranges them into groups (classification and frequency distributions); presentation shows them clearly in text, tables or diagrams so patterns stand out at a glance.
This chapter is practical: you must be able to build a frequency distribution, convert class intervals, calculate cumulative frequencies, draw a table with all its parts, and choose and draw the right diagram.
Learning goals
Explain the types of classification.
Construct a frequency distribution, using class limits, class marks and tally marks.
Distinguish the exclusive and inclusive methods and convert between them.
Prepare less-than and more-than cumulative frequency distributions.
Prepare a table with all its parts.
Draw and interpret bar diagrams, pie diagrams, histograms, frequency polygons, ogives and line graphs.
Core ideas
1. Classification
Classification arranges data into groups or classes according to their common characteristics.
Type | Basis | Example |
|---|---|---|
Chronological | Time | India's population in 1991, 2001, 2011 |
Spatial (geographical) | Place | Wheat production in Punjab, Haryana, Uttar Pradesh |
Qualitative | Attributes that can't be measured | Population by gender, literacy, religion |
Quantitative | Measurable characteristics | Students grouped by marks, families by income |
2. Frequency distribution: key terms
A frequency distribution shows how the values of a variable are distributed across classes.
Term | Meaning | For the class 20–30 |
|---|---|---|
Class interval | The group into which values are placed | 20–30 |
Class limits | The lowest (lower limit) and highest (upper limit) values of a class | Lower 20, upper 30 |
Class width (magnitude) | Upper limit − lower limit | 30 − 20 = 10 |
Class mark (mid-point) | (Lower limit + upper limit) ÷ 2 | (20 + 30) ÷ 2 = 25 |
Frequency | Number of observations in the class | Counted using tally marks |
Tally marks count observations in groups of five: four vertical strokes and a fifth across them (written as |||| crossed), which makes counting quick.
Worked example. Marks of 20 students: 12, 25, 37, 8, 45, 32, 18, 27, 41, 22, 35, 15, 29, 48, 33, 21, 39, 26, 31, 44. Make a frequency distribution with classes 0–10, 10–20 and so on.
Marks | Values in the class | Frequency |
|---|---|---|
0–10 | 8 | 1 |
10–20 | 12, 18, 15 | 3 |
20–30 | 25, 27, 22, 29, 21, 26 | 6 |
30–40 | 37, 32, 35, 33, 39, 31 | 6 |
40–50 | 45, 41, 48, 44 | 4 |
Total | 20 |
3. Exclusive and inclusive methods
Method | Classes | How a boundary value is treated |
|---|---|---|
Exclusive | 0–10, 10–20, 20–30 | The upper limit is excluded: a value of 20 goes in 20–30, not 10–20. Used for continuous variables |
Inclusive | 0–9, 10–19, 20–29 | Both limits are included: 19 goes in 10–19, 20 in 20–29. Used for discrete variables |
Converting inclusive to exclusive. Find the correction factor = (lower limit of a class − upper limit of the previous class) ÷ 2. Subtract it from every lower limit and add it to every upper limit.
Example. Classes 10–19, 20–29, 30–39. Correction factor = (20 − 19) ÷ 2 = 0.5. The exclusive classes are 9.5–19.5, 19.5–29.5, 29.5–39.5. This is necessary before drawing a histogram, which needs continuous classes.
Univariate and bivariate distributions: a univariate distribution covers one variable (marks); a bivariate distribution shows two variables together (for example, sales and advertising expenditure of firms) in a two-way table.
4. Cumulative frequency
Cumulative frequency adds frequencies successively.
Marks | Frequency | "Less than" cumulative frequency | "More than" cumulative frequency |
|---|---|---|---|
0–10 | 1 | Less than 10: 1 | More than 0: 20 |
10–20 | 3 | Less than 20: 4 | More than 10: 19 |
20–30 | 6 | Less than 30: 10 | More than 20: 16 |
30–40 | 6 | Less than 40: 16 | More than 30: 10 |
40–50 | 4 | Less than 50: 20 | More than 40: 4 |
"Less than" uses upper limits and adds downward; "more than" uses lower limits and subtracts downward from the total.
5. Presentation of data
Textual presentation: data are described in sentences. Suitable for small amounts of data. Example: "Of the 50 students, 30 passed and 20 failed."
Tabular presentation: data are arranged in rows and columns. Parts of a table:
Part | Purpose |
|---|---|
Table number | Identifies the table for reference |
Title | Describes the contents briefly and clearly |
Captions (column headings) | Describe each column |
Stubs (row headings) | Describe each row |
Body | The actual data |
Unit of measurement | States the unit (₹ crore, tonnes) |
Source | Where the data were taken from |
Note (footnote) | Explains any special point about the data |
Diagrammatic presentation makes data attractive and easy to compare.
6. Diagrams
Bar diagrams (bars of equal width, with heights proportional to values; gaps between bars):
Type | Use | Example |
|---|---|---|
Simple bar diagram | One variable across categories or years | Wheat output of four states |
Multiple bar diagram | Two or more variables compared side by side | Exports and imports of India over five years |
Component (sub-divided) bar diagram | Parts of a total within each bar | Expenditure of families divided into food, rent and education |
Pie diagram: a circle divided into sectors showing parts of a whole. Each sector's angle = (value ÷ total) × 360°.
Example. A family's monthly expenditure: food ₹6,000, rent ₹3,000, education ₹1,500, others ₹1,500. Total ₹12,000.
Item | Amount ₹ | Angle |
|---|---|---|
Food | 6,000 | 6,000 ÷ 12,000 × 360 = 180° |
Rent | 3,000 | 90° |
Education | 1,500 | 45° |
Others | 1,500 | 45° |
Total | 12,000 | 360° |
Frequency diagrams (for frequency distributions):
Diagram | How it is drawn |
|---|---|
Histogram | Rectangles drawn on continuous (exclusive) classes, with no gaps; area is proportional to frequency. If class widths are unequal, frequencies must be adjusted (frequency density) |
Frequency polygon | Join the mid-points of the tops of histogram bars with straight lines (or plot frequencies against class marks), closing it at the base on both sides |
Frequency curve | A smooth freehand curve through the polygon's points |
Ogive (cumulative frequency curve) | Plot cumulative frequencies against class limits: a less-than ogive (rising, using upper limits) and a more-than ogive (falling, using lower limits). The two ogives intersect at the median |
Arithmetic line graph (time series graph): time on the X-axis and the variable on the Y-axis, points joined by lines. Useful to show trends, such as India's GDP or prices over years.
A reliable approach to questions
For a frequency distribution, choose classes, use tally marks, and check that frequencies add up to the total.
Convert inclusive classes to exclusive before drawing a histogram, polygon or ogive.
For a pie diagram, convert each value to degrees and check that they total 360°.
Choose the diagram: comparison (bar), parts of a whole (pie or component bar), frequency distribution (histogram, polygon), cumulative (ogive), trend (line graph).
Connections that matter
Collection of data produces the raw data organised here.
Measures of central tendency use frequency distributions and class marks; the median can be found from ogives.
Correlation uses bivariate data, shown in a scatter diagram.
Common mistakes
Leaving gaps between histogram bars. A histogram has no gaps.
Drawing a histogram with inclusive classes without converting them.
Calculating class mark wrongly as the class width.
Forgetting the source and unit in a table.
Placing a boundary value in the wrong class in the exclusive method.
Revision checklist
I can classify data by time, place, quality and quantity.
I can build a frequency distribution with class marks and tally marks.
I can convert inclusive classes to exclusive.
I can prepare less-than and more-than cumulative frequencies.
I can prepare a table with all its parts.
I can draw and choose the right diagram, including a pie diagram with correct angles.