Histogram and Ogive

Chapter in a nutshell: Grouped data is shown graphically by a histogram (adjoining bars) and an ogive (cumulative frequency curve). The mode is read from a histogram, and the median and quartiles are read from an ogive.

1. Key Ideas

• Histogram: bars (no gaps) whose widths = class intervals and heights = frequencies; used on continuous (exclusive) classes.
• Mode from a histogram: join the tops of the tallest bar to its neighbours crosswise; the x-value under the intersection is the mode.
• Ogive (cumulative frequency curve): plot cumulative frequency against class boundaries, joined by a smooth curve.

2. Reading Values from the Ogive (positions)

• Median: at cumulative frequency $\dfrac{n}{2}$ — read across to the curve, then down to the x-axis.
• Lower quartile $Q_1$: at $\dfrac{n}{4}$.
• Upper quartile $Q_3$: at $\dfrac{3n}{4}$.
• Inter-quartile range $=Q_3-Q_1$.
• The median is also where the two ogives intersect (their meeting point's x-value).

3. Prerequisite Formulas / Ideas (in full)

• Class boundaries (exclusive form): adjust inclusive limits by ±0.5; class mark $=\frac{l+u}{2}$; class size $h=u-l$.
• Cumulative frequency (cf): running total of frequencies; total cf $=\sum f=n$.
• Median (by formula, for cross-check): $l+\dfrac{n/2-cf}{f}\times h$.
• Choosing a suitable scale on both axes for an accurate graph.

4. Worked Example (one, for the method)

5. Common Mistakes to Avoid

• Leaving gaps between histogram bars (bars must touch — continuous data).
• Plotting an ogive against class marks instead of class boundaries.
• Confusing less-than (upper boundary, rising) and more-than (lower boundary, falling) ogives.
• Using $n/2$ for the quartiles — $Q_1$ is $n/4$, $Q_3$ is $3n/4$.
• Forgetting to convert inclusive classes to boundaries first.

6. Likely Exam Questions (with crisp answers)

1. What is a histogram? → A bar graph of grouped continuous data with no gaps.
2. How is the mode found from a histogram? → By the crosswise lines from the tallest bar's top.
3. What is an ogive? → A cumulative-frequency curve.
4. Where is the median read on the ogive? → At cumulative frequency $n/2$.
5. Cumulative frequency for $Q_1$ and $Q_3$? → $n/4$ and $3n/4$.
6. Define inter-quartile range. → $Q_3-Q_1$.
7. Less-than ogive uses which boundary? → The upper class boundary.
8. More-than ogive uses which boundary? → The lower class boundary.
9. What do the two ogives' intersection give? → The median.
10. Why must histogram bars touch? → The data is continuous (no gaps between classes).

Extended & Prerequisite Formulas (full)

• Median read at cf = n/2; Q₁ at n/4; Q₃ at 3n/4; IQR = Q₃−Q₁; median = x-coordinate where the two ogives cross.
• Less-than ogive at upper boundaries (rising); more-than ogive at lower boundaries (falling).
• Prereq: class boundaries (±0.5), class mark (l+u)/2, cumulative frequency; median formula $l+\tfrac{n/2-cf}{f}h$.