Three ways to present data
- Textual (descriptive): data written in sentences. Good only when there are few numbers.
- Tabular: data arranged in rows and columns.
- Diagrammatic: data shown as pictures: bars, circles and graphs. Easy to understand at a glance.
Tables and their parts
- Table number: to find it easily.
- Title: short, clear, tells what, where and when.
- Captions (column headings): names of the columns.
- Stubs (row headings): names of the rows.
- Body: the main numbers.
- Unit of measurement: like ₹ crore or lakh tonnes.
- Source: where the data came from.
- Note (footnote): any special point about the data.
Tables can classify data qualitatively, quantitatively, by time (temporal) or by place (spatial).
Bar diagrams
Bars are rectangles of equal width with equal gaps. Only the height (length) shows the value.
- Simple bar diagram: one bar per item or year.
- Multiple bar diagram: two or more bars side by side for each item (wheat and rice for each year).
- Component (sub-divided) bar diagram: one bar split into parts to show the make-up of a total. A percentage bar has every bar equal to 100.
Pie diagram
A circle is cut into slices. Each slice shows a part of the total.
Angle of a slice = (value ÷ total) × 360°. If you already have percentages, angle = percentage × 3.6°.
Draw the biggest slice first, starting from the 12 o'clock line, and go clockwise. Label each slice.
Histogram, frequency polygon and frequency curve
Histogram
Rectangles with no gaps, drawn on class intervals. With equal classes, height = frequency. With unequal classes, adjust height = frequency ÷ width (frequency density). Classes must be exclusive, so adjust inclusive ones first. The mode can be found from a histogram.
Frequency polygon
Join the midpoints of the tops of the rectangles with straight lines. Close the shape by joining to the midpoints of an empty class at each end on the x-axis. It can also be drawn without a histogram by plotting (class mark, frequency).
Frequency curve
Join the same points with a smooth freehand curve instead of straight lines.
Ogives (cumulative frequency curves)
- Less than ogive: plot less-than cumulative frequencies against upper limits. It rises.
- More than ogive: plot more-than cumulative frequencies against lower limits. It falls.
The two ogives cross at height N/2. Drop a line to the x-axis: that value is the median.
Arithmetic line graph (time series graph)
Used for data that changes over time. Put time (years, months) on the x-axis and the value on the y-axis. Plot the points and join them with straight lines. Rising lines show growth; falling lines show decline. Two or more lines can be drawn on the same graph to compare, such as exports and imports.
If the y-axis does not start from 0, show a false base line (a small zig-zag) near the origin.
Try it: your family budget pie
Ask at home how the monthly money is spent, in rough parts: food, rent, travel, school, others. Find each part's angle with (part ÷ total) × 360°. Draw the circle with a bangle, and slices with a protractor. Then track the price of one vegetable for 5 weeks and put the values into the sliders on the last 3D step to see your own line graph.
Board exam pattern
Expect: parts of a table (3 marks), angle calculation and drawing a pie diagram (4 marks), histogram with frequency polygon (4 to 6 marks), ogives and median (4 marks). Label axes, give a title and choose a clear scale.
Key formulas and definitions
- Angle of a slice = (component value ÷ total) × 360°
- Angle from percentage = percentage × 3.6°
- Histogram (unequal classes): adjusted height = frequency ÷ class width
- Class mark = (lower + upper limit) ÷ 2 (for polygon points)
- Less than ogive: plot (upper limit, less-than cf)
- More than ogive: plot (lower limit, more-than cf); ogives meet at N/2 → median
Worked examples
1. A family spends ₹20,000: food ₹8,000, rent ₹6,000, transport ₹2,000, education ₹4,000. Find the pie angles.
Food = 8,000 ÷ 20,000 × 360 = 144°. Rent = 108°. Transport = 36°. Education = 72°. Check: 144 + 108 + 36 + 72 = 360°.
2. Education gets 15% of a budget. What angle does it get in a pie diagram?
15 × 3.6 = 54°.
3. Classes 0–10, 10–20, 20–30 have frequencies 5, 9, 6. Write the points for the frequency polygon.
Class marks 5, 15, 25 with f: (5, 5), (15, 9), (25, 6). Add end points (−5, 0) and (35, 0) on the axis to close it.
4. For classes 0–10, 10–20, 20–30, 30–40 with f = 3, 7, 6, 4, write the less-than and more-than cumulative frequencies.
Less than 10: 3, less than 20: 10, less than 30: 16, less than 40: 20. More than 0: 20, more than 10: 17, more than 20: 10, more than 30: 4.
5. Classes 0–10 (f = 5) and 10–30 (f = 12). What heights should the histogram rectangles have?
Use frequency per unit width. 0–10: 5 ÷ 10 = 0.5. 10–30: 12 ÷ 20 = 0.6. Scaled to width 10: 5 and 6.
6. Which diagram fits each case: (a) share of states in rice output in one year, (b) exports over 10 years, (c) marks of 100 students in classes?
(a) Pie diagram (parts of a whole). (b) Arithmetic line graph (time series). (c) Histogram (frequency distribution).
Common mistakes
- Leaving gaps between histogram bars. Histogram bars touch; bar diagram bars have gaps.
- Forgetting to adjust heights when class widths are unequal.
- Plotting less-than ogive at lower limits. It uses upper limits.
- Pie angles that do not add up to 360°. Always check the total.