Maths-9 Bar Graphs: Questions and Step-by-Step Solutions for all Competitive Exams

Maths-9 Bar Graphs : Questions and Step-by-Step Solutions for all Competitive Exams

Bar graphs & other graphs

Let’s make bar graphs, histograms, line graphs, pie charts and reading data from tables super clear. I’ll start from basics, show rules, then give worked examples (step-by-step) you can follow and practise.

1) What is a graph?

A graph is a picture that shows numbers or information so we can see patterns quickly. Different graphs are useful for different kinds of data.

2) Common types you must know (and when to use them)

• Bar graphs / Column graph — for comparing separate categories (e.g., marks of students, number of books). Bars are separated by small gaps.
• Histogram — for grouped continuous data (e.g., ages in ranges). Bars touch each other (no gaps) because ranges are continuous.
• Line graph / Time-series graph — for showing change with time (e.g., temperature over days).
• Pie chart — to show parts of a whole (percentages of total).
• Frequency polygon / Ogive (cumulative frequency curve) — for showing the shape of the distribution or cumulative totals.

3) Useful vocabulary / rules

• Category (or class) — the group name (e.g., “Math”, “Science”).
• Frequency (f) — how many times a value/category occurs.
• Class interval — e.g., 10–19, 20–29 (used in grouped data).
• Class width (h) — upper limit − lower limit (e.g., 10 in 10–19).
• Scale — choose units on axes (must cover the data, evenly spaced).
• Label axes and title the graph.
• Bars in a bar graph: same width, equal spacing, height = frequency or value.
• Histogram bars: width = class width; height = frequency density (if class widths differ) or frequency (if equal widths).

4) Quick formulas

For a frequency distribution with classes and frequencies:

• Mean (from frequency table)

(If you have class mid-points , use them.)

• Median (grouped data, formula)
  If = total frequency, median class lower boundary , cumulative frequency before median class , frequency of median class , class width :

• Mode (grouped data, formula)
  If modal class has frequency , previous class frequency , next class frequency , lower boundary , class width :

• Percentage for pie chart
  If category frequency = and total , then angle for pie sector:

5) Worked examples — step by step

Example A — Draw a bar graph (simple)

Data: Number of books read in a month by 5 students

Steps to draw (by hand):

1. Choose axes: horizontal (x) = students A–E; vertical (y) = number of books (0–8).
2. Mark equal spaces on x for each student and label A, B, C, D, E.
3. Choose scale on y (e.g., 1 small square = 1 book). Mark 0,1,2,…,8.
4. For each student draw a vertical bar of height = number of books (A → 4, B → 7…). Leave small gaps between bars.
5. Title: “Books read in a month”. Label axes.

Reading the graph: Who read most? B (7). Difference between B and C = 7 − 3 = 4 books.

Example B — Draw a pie chart (percentages)

Data: Favourite fruit among 60 students

Steps & solution:

1. Total .
2. Compute angles: angle = (f / N) × 360°.
   • Apple: .
   • Banana: .
   • Mango: .
   • Orange: .
     (Check: 108+72+120+60 = 360°.)
3. Draw a circle, use a protractor from the centre: mark sectors of those angles, label and colour.

Interpretation: Mango is most popular (120° sector).

Example C — Histogram and mean from grouped data

Data (ages of 20 children):

a) Draw histogram (by hand):

1. Horizontal axis = age groups (5–7, 8–10, …). Vertical = frequency (0–6).
2. Each bar width corresponds to class width (here, each group has a width 3 years). Bars touch (no gaps). Height = frequency.
3. Title & axes labels.

b) Find approximate mean using class mid-points:

1. Compute class midpoints :
   • 5–7 → midpoint = (5+7)/2 = 6
   • 8–10 → 9
   • 11–13 → 12
   • 14–16 → 15
   • 17–19 → 18
2. Multiply :
   • 4×6 = 24
   • 6×9 = 54
   • 5×12 = 60
   • 3×15 = 45
   • 2×18 = 36
3. Sum frequencies . Sum .
4. Mean ≈ years.

Answer: Approx mean age ≈ 10.95 years.

Example D — Median from grouped data (step by step)

Use the same table above.

Steps:

1. Total . Median position = th observation (for even N use ).
2. Build cumulative frequency (cf):
   • 5–7: cf = 4
   • 8–10: cf = 4+6 = 10 ← median class (because cf ≥ 10 here)
3. Median class = 8–10. Its lower boundary (if class written as 8–10, lower boundary is 7.5 in continuous data; for school problems you may use 8). Frequency of median class . Cumulative frequency before median class . Class width .
4. Apply grouped median formula:

Answer: Median ≈ 10.5 years.

(Note: Using inclusive integer ages, you might get median ≈ 10; the grouped formula is for continuous classes.)

Example E — Mode from grouped data

From the same table, find the mode (most frequent class is 8–10 with f_m = 6).

We need = frequency of previous class (5–7) = 4, and = frequency of next class (11–13) = 5. , , .

Mode formula:

Answer: Mode ≈ 9.5 years (modal value lies inside 8–10 group).

Example F — Line graph (time series)

Data: Temperature over 5 days

Steps to draw:

1. Horizontal axis = days; vertical axis = temperature scale (20–30°C).
2. Plot points (Mon,24), (Tue,26) etc.
3. Join points with straight lines. Title: “Temperature over a week”.
4. Read: highest on Thu (27°C), lowest on Wed (23°C).

6) Practical tips for drawing and exam speed

• Choose a neat scale: tick marks evenly spaced, include 0 unless instructed otherwise.
• Label everything: axes, units, title, legend (if multiple series).
• Keep bar widths equal and use gaps only for bar charts (not histograms).
• Check totals when drawing pie charts (angles must sum to 360°).
• For histograms with unequal class widths, use frequency density = frequency/class width as bar height.
• When asked “read value from graph”: use interpolation if needed and read carefully.

7) Short practice (you can do in 10 mins)

1. Draw a bar graph for: Sales (in ₹1000) — Mon: 5, Tue: 7, Wed: 6, Thu: 8, Fri: 4.
2. Convert to pie angles for students: A: 12, B: 8, C: 10, D: 10 (total 40 students). Find angles.
3. Given grouped data classes 0–9 (5), 10–19 (8), 20–29 (7), find mean using midpoints.
4. Build cf and find median for the grouped example in section C.
5. From the temperatures above, what is the average temperature for the 5 days?

8) Quick answers to practice

1. Bar graph — follow steps in Example A (scale 0–8 works).
2. Angles: total = 40. A angle = 12/40×360 = 108°, B = 8/40×360 = 72°, C = 90°, D = 90°.
3. Midpoints: 4.5, 14.5, 24.5. Mean = (5×4.5 + 8×14.5 + 7×24.5) / (5+8+7) = (22.5 + 116 + 171.5)/20 = 310/20 = 15.5.
4. Median from section C was 10.5 years.
5. Average temp = (24+26+23+27+25)/5 = 125/5 = 25°C.