Data Handling — Bar Graphs, Histograms, Pie Charts, Ogives

What Data Handling Is About

Data handling means collecting information, organizing it, and representing it visually so patterns become obvious. Instead of staring at a list of 50 numbers, we draw a bar graph or pie chart — and the story in the data jumps out.

For Classes 6-8, this chapter builds year by year. Class 6 introduces pictographs and bar graphs. Class 7 adds mean, median, mode and introduces probability. Class 8 brings pie charts, histograms, and grouped data.

graph TD
    A{What type of data?} --> B[Categorical]
    A --> C[Numerical]
    B --> D{How to display?}
    D -->|Compare categories| E[Bar Graph]
    D -->|Parts of whole| F[Pie Chart]
    D -->|Simple count| G[Pictograph]
    C --> H{Continuous or discrete?}
    H -->|Continuous intervals| I[Histogram]
    H -->|Cumulative| J[Ogive]
    H -->|Individual values| K[Frequency Table]

Key Terms & Definitions

Data — Raw facts and figures collected for analysis. Example: marks of 30 students in a test.

Frequency — How many times a value (or range) appears in the data.

Frequency Distribution Table — A table showing each value or class interval with its frequency.

Class Interval — A range grouping values together (e.g., 10-20, 20-30). The difference between upper and lower limits is the class width.

Mean — The average: $\bar{x} = \frac{\sum x_i}{n}$ .

Median — The middle value in sorted data. For odd $n$ : the $\left(\frac{n+1}{2}\right)^\text{th}$ value. For even $n$ : the average of the two central values.

Mode — The most frequent value. A dataset can have no mode, one mode, or multiple modes.

Types of Graphs

Bar Graph

Each category gets a bar whose height represents frequency. Bars have equal width and equal gaps.

When to use: Comparing quantities across categories (favourite sports, production by year).

Bar graphs have gaps between bars (categories are separate). Histograms have no gaps (data is continuous). This is a favourite exam question.

Histogram

For continuous grouped data. The x-axis has class intervals, y-axis has frequencies. Adjacent bars with no gaps. Bar width equals class width.

When to use: Frequency distribution of continuous data (marks, heights, time intervals).

Pie Chart

The full circle (360°) represents the total. Each slice’s angle is proportional to its share.

Formula: Angle $= \frac{\text{Category value}}{\text{Total}} \times 360°$

Ogive

Plot cumulative frequency against upper class boundaries and connect smoothly. Two types: less-than ogive (cumulative from below) and more-than ogive (from above). Their intersection gives the median.

Worked Examples

Example 1 (Easy — Class 6)

Marks of 10 students: 45, 60, 55, 70, 45, 80, 55, 60, 55, 75. Find mean, median, mode.

Mean $= \frac{600}{10} = \mathbf{60}$

Sorted: 45, 45, 55, 55, 55, 60, 60, 70, 75, 80

Median $= \frac{55 + 60}{2} = \mathbf{57.5}$

Mode $= \mathbf{55}$ (appears 3 times)

Example 2 (Medium — Class 7)

Pie chart for: Cricket 40, Football 25, Tennis 15, Badminton 20 students.

Total = 100.

Sport	Students	Angle
Cricket	40	$144°$
Football	25	$90°$
Tennis	15	$54°$
Badminton	20	$72°$

Check: $144 + 90 + 54 + 72 = 360°$ .

Example 3 (Medium — Class 8)

Marks distribution — find the modal class.

Marks	10-20	20-30	30-40	40-50	50-60
Students	5	12	18	10	5

Modal class = 30-40 (highest frequency = 18).

Exam-Specific Tips

CBSE Class 6-7: Focus on reading data from graphs and computing mean/median/mode from small datasets. Double-check arithmetic — the most common mark loss is addition errors.

CBSE Class 8: Pie charts and histograms dominate. For pie charts, verify angles sum to $360°$ . For histograms, label both axes and use correct class intervals. Converting discontinuous to continuous intervals is frequently tested.

Common Mistakes to Avoid

Mistake 1 — Angles not summing to 360° in pie charts. Always verify the total. If it’s off, recheck your division.

Mistake 2 — Gaps in histograms. Histograms are for continuous data — bars must touch. Convert gaps (10-19, 20-29) to continuous form (9.5-19.5, 19.5-29.5).

Mistake 3 — Confusing mean and median. Mean uses all values (sum/count). Median is the middle value in sorted order. They can differ significantly for skewed data.

Mistake 4 — Wrong median position for even n. For even $n$ , median is the average of two middle values. Students often pick just one.

Mistake 5 — Wrong scale on axes. If frequencies go to 80 but the y-axis only reaches 50, bars won’t fit. Choose a sensible scale before drawing.

Practice Questions

Q1. Runs in 6 innings: 36, 35, 50, 46, 60, 55. Find the mean.

Mean $= \frac{282}{6} = 47$ .

Q2. Find the mode of: 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.

14 appears 4 times, 18 appears 3 times. Mode $= 14$ .

Q3. Draw a pie chart: Bus 20, Car 10, Bicycle 15, Walking 5 students.

Total 50. Bus $= 144°$ , Car $= 72°$ , Bicycle $= 108°$ , Walking $= 36°$ . Sum $= 360°$ .

Q4. If the median of 11 observations is 35 and each increases by 4, what’s the new median?

New median $= 35 + 4 = 39$ .

Q5. Histogram: 0-10 (f=5), 10-20 (f=8), 20-30 (f=12), 30-40 (f=7). Modal class?

20-30 (highest frequency = 12).

Q6. Find the median: 17, 2, 7, 27, 15, 5, 14, 8, 10, 24, 48, 10, 8, 7, 18.

Sorted: 2, 5, 7, 7, 8, 8, 10, 10, 14, 15, 17, 18, 24, 27, 48. $n = 15$ , median is 8th value $= 10$ .

Q7. A die thrown 100 times: 1 (18), 2 (15), 3 (22), 4 (12), 5 (16), 6 (17). Probability of getting more than 4?

Favourable = $16 + 17 = 33$ . Probability $= 33/100 = 0.33$ .

Q8. Mean of 5 numbers is 27. Excluding one number, mean becomes 25. Find the excluded number.

Total $= 135$ . Remaining total $= 100$ . Excluded $= 35$ .

FAQs

What’s the difference between a bar graph and a histogram?

Bar graphs are for categorical data with gaps between bars. Histograms are for continuous data with adjacent bars. Bar width in histograms represents the class interval.

When to use mean vs. median vs. mode?

Mean for symmetric data without outliers. Median when extreme values exist (income data). Mode when you want the most common value (shoe sizes).

How to find median from a frequency table?

Compute cumulative frequency. Find $n/2$ . The class where cumulative frequency first exceeds $n/2$ is the median class.

What is a double bar graph?

Two bars side by side per category for comparing two datasets (boys vs. girls in each sport).

Why convert discontinuous to continuous class intervals?

In histograms, bars must touch. Convert by subtracting 0.5 from lower limits and adding 0.5 to upper limits.

What’s the relationship between mean, median, and mode?

For moderately skewed data: Mode $\approx 3 \times$ Median $- 2 \times$ Mean. This is empirical, not exact, but useful for quick checks.

Advanced Concepts

Grouped data — mean by three methods

When data is in class intervals, we cannot see individual values. Three methods to find the mean:

Direct method: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ where $x_i$ is the midpoint of each class.

Assumed mean method: Choose $a$ (assumed mean, usually the midpoint of the central class). $d_i = x_i - a$ . Then $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$ .

Step deviation method: If all class widths are $h$ , let $u_i = \frac{x_i - a}{h}$ . Then $\bar{x} = a + h \cdot \frac{\sum f_i u_i}{\sum f_i}$ .

The step deviation method is the fastest for grouped data with equal class widths. It reduces all numbers to small integers, making calculation nearly error-free.

Median from a frequency table

\text{Median} = l + \left(\frac{N/2 - cf}{f}\right) \times h

where $l$ = lower limit of median class, $N$ = total frequency, $cf$ = cumulative frequency before median class, $f$ = frequency of median class, $h$ = class width.

Mode from a frequency table

\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h

where $l$ = lower limit of modal class, $f_1$ = frequency of modal class, $f_0$ = frequency of class before, $f_2$ = frequency of class after, $h$ = class width.

Worked Example — Grouped Data

Marks	0–10	10–20	20–30	30–40	40–50
Students	5	8	15	12	10

$N = 50$ , so $N/2 = 25$ .

Cumulative frequencies: 5, 13, 28, 40, 50.

$N/2 = 25$ falls in the 20–30 class (cf jumps from 13 to 28 here).

Median $= 20 + \frac{25 - 13}{15} \times 10 = 20 + \frac{12}{15} \times 10 = 20 + 8 = 28$ .

Probability basics (Class 7–8)

Experimental probability: $P(E) = \frac{\text{Number of times E occurs}}{\text{Total trials}}$

Theoretical probability: $P(E) = \frac{\text{Favourable outcomes}}{\text{Total outcomes}}$

Properties: $0 \leq P(E) \leq 1$ . $P(\text{sure event}) = 1$ . $P(\text{impossible event}) = 0$ . $P(E) + P(\bar{E}) = 1$ .

Additional Practice Questions

Q9. Find the median class for: 0–10 (f=3), 10–20 (f=7), 20–30 (f=15), 30–40 (f=10), 40–50 (f=5).

$N = 40$ , $N/2 = 20$ . Cumulative: 3, 10, 25, 35, 40. The 20th value falls in 20–30. Median class is 20–30.

Q10. The mean of 5 numbers is 20. If one number is excluded, the mean becomes 18. Find the excluded number.

Total = $5 \times 20 = 100$ . Remaining total = $4 \times 18 = 72$ . Excluded number = $100 - 72 = 28$ .

Range and Quartiles

Range = Maximum value $-$ Minimum value. It measures total spread but is sensitive to outliers.

Quartiles divide sorted data into four equal parts:

$Q_1$ (first quartile): median of the lower half
$Q_2$ (second quartile): the median itself
$Q_3$ (third quartile): median of the upper half

Interquartile range (IQR) = $Q_3 - Q_1$ . This measures spread while ignoring extreme values — more robust than range.

Effect of adding or multiplying on mean, median, mode

If each data value is increased by a constant $c$ : mean, median, and mode all increase by $c$ . Range and standard deviation stay the same.

If each value is multiplied by $k$ : mean, median, mode, and range all get multiplied by $k$ . Standard deviation gets multiplied by $|k|$ .

These transformation properties are frequently tested in competitive exams. The key rule: adding a constant shifts the distribution (changes location measures) but does not change spread measures. Multiplying scales everything proportionally.

When mean, median, and mode are equal

For a perfectly symmetric distribution (like the normal distribution), mean = median = mode. When data is skewed right (tail on the right), mean > median > mode. When skewed left, mean < median < mode. This relationship helps you quickly identify skewness from summary statistics.