Graphical representation of data
A table of numbers hides its story until you draw it. Graphical representation of data is the Class 9 toolkit for turning a frequency distribution into a picture: the bar graph for categories, the histogram for continuous class intervals, and the frequency polygon for the shape of a distribution. The three look similar on the page but obey different rules — bars touch or do not touch depending on whether the data is continuous, and histogram bars need a height correction whenever the class widths are unequal. Getting these rules right, especially the frequency-density correction, is the part that separates a correct diagram from a misleading one, and it is exactly the part examiners probe.
Bar graphs use gapped equal-width bars for categorical data, histograms use touching bars over continuous class intervals with heights corrected to frequency density when widths are unequal, and frequency polygons plot frequency against class mark to trace the shape of the distribution.
The core idea
Bar graph. Rectangular bars of equal width, drawn with uniform gaps between them, one bar per category. The height (or length) of a bar is proportional to the frequency or value it stands for. Use a bar graph when the underlying data is categorical or discrete — separate labels like favourite fruit or marks scored by named students — never for continuous class intervals.
Histogram. Bars for grouped, continuous data, drawn touching each other since the class boundaries run into one another with no gap between them. When every class has the same width, the bar height is simply the class frequency. When class widths differ, area (not height) must stay proportional to frequency, so each height is corrected to a frequency density:
where: frequency is the count in that class, width of the class is its interval length, and minimum class width is the smallest width among all classes, chosen as the common reference unit so every bar's area stays honestly comparable.
Frequency polygon. A line graph obtained by plotting each class's frequency against its class mark (the interval's midpoint),
and joining the points with straight lines. It can be drawn directly from these plotted points, or by joining the midpoints of the tops of an existing histogram's bars. To close the shape properly, an extra class of zero frequency is imagined on each side of the real data, and the polygon is brought down to touch the horizontal axis at the midpoints of those two invisible classes.
The goal is to show why plotting raw frequency directly as bar height becomes misleading the moment class widths differ, and to derive the corrected-height formula that keeps every bar's area — not its height — proportional to its frequency, which is the property a histogram must preserve for the picture to be honest.
Step 1. State what a histogram promises. In a histogram, the area of a bar is meant to represent the frequency of that class, not the height alone. When every class has the same width, area and height are proportional to each other automatically, so plotting frequency as height works without any correction.
Step 2. See where equal-width plotting breaks. If class widths differ, area equals height times width, so . Plotting raw frequency as height then makes area proportional to frequency times width — a wider class gets an inflated area purely because it is wider, even if it is no more "dense" with data than a narrow one.
Step 3. Fix it by rescaling to a common width. Choose the minimum class width among all the classes as the reference unit. For any class of width and frequency , define the corrected height so that area stays proportional to frequency alone:
A class exactly as wide as the minimum keeps its raw frequency as its height (the multiplier is 1); a class twice as wide gets its frequency halved before being plotted, correctly showing it is less densely packed per unit width.
Step 4. Confirm the fix restores fairness. With this correction, area of a bar — a constant multiple of the frequency for every class, regardless of . Area is once again proportional to frequency alone, which is exactly what a correct histogram requires.
Worked example
The number of children (in hundreds) in different age groups of a town is given below, with unequal class widths. Draw a histogram to represent this data correctly.
| Age group (years) | Number of children (in hundreds) |
|---|---|
| 0–5 | 15 |
| 5–10 | 15 |
| 10–15 | 12 |
| 15–20 | 6 |
| 20–30 | 8 |
Step 1. Find each class width: the first four classes are 5 years wide, and the last class (20–30) is 10 years wide — unequal, so raw frequencies cannot be plotted as heights directly.
Step 2. Take the minimum class width as the reference: here it is 5.
Step 3. Apply the corrected-height formula, , to every class: 0–5 gives ; 5–10 gives ; 10–15 gives ; 15–20 gives ; and 20–30 gives .
Step 4. Draw the histogram with these adjusted heights against the age-group boundaries, bars touching since the intervals are continuous. The corrected heights are 15, 15, 12, 6 and 4 — notably, the last bar drops to 4 rather than the raw 8, correctly showing that children are less densely packed in that wider 20–30 interval than the raw count would suggest.
Competitive stretch
If a sixth class 30–50 (width 20) had frequency 10, its corrected height would be — a useful check that the correction scales smoothly however wide a class becomes, always keeping area, not height, tied to frequency.
Common traps
- Leaving gaps in a histogram or closing them in a bar graph. Histogram bars must touch (continuous intervals); bar graph bars must have equal gaps (separate categories) — swapping these is one of the most common exam slips.
- Plotting raw frequency as height when class widths are unequal. This silently inflates wider classes; always correct to frequency density first using the minimum-width formula.
- Using class limits instead of class marks for a frequency polygon. Points must sit at each class's midpoint, not at its boundary.
- Forgetting the zero-frequency classes at the ends of a frequency polygon. Without them the polygon never meets the horizontal axis and looks like an unfinished, dangling line.
Ask the tutor: unsure whether your data needs a bar graph or a histogram, or want your unequal-width height calculation double-checked — ask the AI tutor to walk through it with you.
Why do histogram bars touch but bar graph bars have gaps?
A histogram represents continuous class intervals — the upper limit of one class is the lower limit of the next, so there is genuinely no gap in the data, and the bars are drawn touching to reflect that. A bar graph represents separate, unrelated categories, so the gaps are a deliberate visual cue that the categories are distinct and not part of a continuous scale. Drawing one like the other misrepresents the type of data.
Do I always need to correct histogram heights, even when the widths look almost equal?
Yes, correct whenever widths differ at all, however slightly — the formula still applies and the correction may just be small. Only when every class has exactly the same width can you skip the correction and plot raw frequency directly as height, because then the multiplier minimum width divided by class width equals 1 for every class, so it changes nothing.
How is a frequency polygon different from just joining the tops of histogram bars?
It is not fundamentally different — joining the midpoints of the tops of consecutive histogram bars with straight lines produces exactly a frequency polygon. The polygon can also be drawn independently, by plotting frequency against class mark directly without drawing the histogram first. Either method needs the extra zero-frequency classes at both ends so the shape closes down to the axis instead of ending abruptly mid-air.