Diagrams and graphs: histogram, ogives, pie and bar diagrams

1. Diagrams and graphs: general rules

Diagrams make data attractive, easy to grasp and easy to compare; they need no training to read. Their limits: they show only approximate values, cannot show many characteristics at once, and can mislead if the scale is chosen badly.

Rules: a title and a number; a suitable scale stated on the paper (“1 cm = 10 students”); bars of equal width with equal gaps; an index (key) where shades or colours are used; neatness; and the source.

2. Bar diagrams (one-dimensional)

Only the length (height) of the bar matters; width is uniform. The syllabus limits you to one-dimensional diagrams.

• Simple bar: one bar per item (sales for five years).
• Multiple bar: bars for two or more related variables side by side for each period. August 2024: results of 2nd B.Com LL.B. for December 2016, 2017 and 2018, with four bars (1st class, 2nd class, 3rd class, failed) in each year’s group, a gap between groups and a key for the shading. Take a scale of 1 cm = 100 students; the tallest bar is 700 (3rd class, 2017).
• Sub-divided (component) bar: one bar for the total, divided into its parts.
• Percentage bar: every bar is of the same height (100) and is divided by percentages, so that families or firms of different size can be compared.

Worked: percentage bar (August 2024). Family A spends Rs. 4,100 and Family B Rs. 3,100 a month. Convert each item to a percentage of the family’s total and cumulate.

Draw two bars of 10 cm each (1 cm = 10%) and mark the divisions at the cumulative percentages. Family B spends a larger share on food (38.7% against 31.7%), as Engel’s law predicts for the poorer family.

3. Pie diagram (June 2025)

A circle of 360° is divided into sectors in proportion to the values: angle = (value ÷ total) × 360°.

Start from the 12 o’clock line, draw the sectors clockwise in descending order of size with a protractor, and shade each with a key.

4. Histogram

A histogram is a set of adjoining rectangles drawn on the class intervals (x-axis) with heights equal to the frequencies (y-axis). Classes must be exclusive and continuous; if widths are unequal, heights are adjusted (frequency density = frequency ÷ width).

For the January 2025 distribution (previous note): rectangles on 0–10, 10–20 … 50–60 with heights 17, 7, 5, 6, 4, 1; scale 1 cm = 10 marks on x and 1 cm = 2 students on y.

Locating the mode on a histogram (100-mark January 2026): classes 0–10 to 40–50 with frequencies 2, 10, 40, 20, 15. In the tallest rectangle (20–30) join its top left corner to the top left corner of the next rectangle and its top right corner to the top right corner of the previous one. From the crossing point drop a perpendicular to the x-axis; it meets it at the mode. Check by formula: Mode = 20 + (40 − 10) / (80 − 10 − 20) × 10 = 20 + 30/50 × 10 = 26.

A frequency polygon joins the mid-points of the tops of the rectangles; a frequency curve is the same drawn freehand and smoothly.

5. Ogives and the median (June 2025)

Draw less-than and more-than ogives for: 10–20 (2), 20–30 (5), 30–40 (7), 40–50 (12), 50–60 (9), 60–70 (4), and determine the median graphically.

Plot the less-than points against the upper limits, starting from (10, 0), and join them by a smooth rising curve. Plot the more-than points against the lower limits, ending at (70, 0), giving a falling curve. The two curves cross; the perpendicular from the crossing point meets the x-axis at the median. (Alternatively, mark N/2 = 19.5 on the y-axis on the less-than ogive alone.)

Verification by formula. N/2 = 19.5 lies in 40–50 (c.f. before it 14, f = 12): Median = 40 + (19.5 − 14)/12 × 10 = 40 + 4.58 = 44.58. The graph should read about 44.6.

January 2026 data (10–50: 10, 50–100: 30, 100–150: 50, 150–200: 40, 200–250: 20; the paper prints the last class as “200–2500”, an obvious misprint for 200–250 — say so in a line). Less-than c.f.: 10, 40, 90, 130, 150 at 50, 100, 150, 200, 250. More-than c.f.: 150, 140, 110, 60, 20 at 10, 50, 100, 150, 200. N/2 = 75: Median = 100 + (75 − 40)/50 × 50 = 135. Note the first class is 40 wide, not 50; ogives plot cumulative frequencies, so no adjustment is needed.

How this is examined

Ogives with the median: January 2026 and June 2025 (ten marks each). Histogram from raw data: January 2025; histogram with the mode located: the 100-mark January 2026 paper. Pie chart: June 2025. Percentage bar and multiple bar: August 2024. Use graph paper, state the scale, title the graph and always verify a graphical median or mode by the formula.