Why the Bars Touch in One Graph and Not in the Other

Because its horizontal axis is a continuous scale, and there is nothing in between one class and the next for a gap to represent.

A bar graph shows separate categories — cricket, football, badminton. Nothing lies between cricket and football, so the bars are drawn apart to show they are unconnected.

A histogram shows classes of a measured quantity — marks from $10$ to $15$, then $15$ to $20$. A mark of $14.8$ exists and belongs to the first class; a mark of $15.2$ exists and belongs to the second. The scale runs without interruption, so the bars must touch.

That single difference tells you which graph a question wants. Categories mean gaps; measurements mean no gaps — and drawing a histogram with spaces between the bars loses marks for exactly this reason. This page covers the second part of the ICSE Class 8 Mathematics chapter on data handling.

Draw bars of equal width with equal gaps, one per category, heights proportional to the values.

Worked example 1 — a bar graph. Out of $720$ students, favourite sports are cricket $300$, football $180$, badminton $120$, hockey $90$ and others $30$.

To draw it: put the sports along the horizontal axis, and choose a vertical scale — here $1$ cm to $50$ students works, giving bar heights of $6$, $3.6$, $2.4$, $1.8$ and $0.6$ cm.

State the scale on the graph. Without *$1$ cm $=50$ students* written beside the axis, the bars are just rectangles and the graph cannot be read.

Start the vertical axis at zero. Beginning at $80$ would make the cricket bar look several times the football one instead of roughly $1.7$ times, and comparisons by bar height only work from a zero baseline.

Reading it back: the tallest bar names the most popular sport, and the total of all bars should return $300+180+120+90+30=720$.

Worked example 2 — a double bar graph. Monthly rainfall in two cities, in millimetres, over four months:

- City A: $120$, $250$, $300$, $180$
- City B: $90$, $200$, $350$, $140$

A double bar graph puts the two bars for each month side by side and touching each other, with a gap before the next month's pair. Shade the two cities differently and include a key saying which shading is which.

Reading it: City A's total is $120+250+300+180=850$ mm and City B's is $90+200+350+140=780$ mm, so A had more rain overall. But in the third month B exceeded A, $350$ mm against $300$ mm — the only month in which it did.

That is the whole point of a double bar graph. A single total of $850$ against $780$ hides the reversal in month three. Comparing two data series month by month is something a pair of separate bar graphs does badly and a double bar graph does at a glance.

The two bars in a pair touch; the pairs are separated. Within a month the two cities are being compared directly, so their bars are adjacent. Between months the comparison is a different one, so a gap is left — and a graph drawn with all eight bars equally spaced would lose that structure entirely.

Put the class boundaries on the horizontal axis, the frequency on the vertical, and draw adjacent bars with no gaps.

Worked example — the marks data. Twenty students' marks, grouped in fives:

- $10-15$: frequency $5$
- $15-20$: frequency $9$
- $20-25$: frequency $4$
- $25-30$: frequency $2$

To draw it, mark $10,15,20,25,30$ along the horizontal axis at equal spacing, choose a vertical scale such as $1$ cm to $2$ students, and draw four touching rectangles of heights $5$, $9$, $4$ and $2$.

Label the axes with what they measure, not just with numbers: marks horizontally and number of students vertically.

Reading a histogram back.

- Which class has the most students? The tallest bar, $15-20$, with $9$.
- How many scored less than $20$? The first two bars: $5+9=14$.
- How many scored $20$ or more? $4+2=6$. Check: $14+6=20$, the whole class.
- What fraction scored at least $25$? $\frac{2}{20}=\frac{1}{10}$, or $10\%$.

A histogram cannot tell you an individual value. It says nine students scored between $15$ and $20$, but not that six of them scored exactly $15$. That information was given away when the data was grouped, and a question asking *how many scored exactly $18$?* cannot be answered from the histogram at all — which is a fair question to be asked, and the correct answer is that the graph does not say.

Where the horizontal axis does not start at zero. If the classes began at $100$ rather than $10$, drawing the axis all the way from zero would waste most of the page. The convention is to draw a kink — a short zig-zag — in the axis just after the origin, showing that a stretch has been left out. Without the kink the graph implies classes that do not exist.

Equal class widths keep it simple. All four classes here are $5$ wide, so bar height can be read directly as frequency. If the widths differed, equal-frequency classes would need bars of different heights to keep the areas honest — and that is why examination questions at this level keep the class size constant.

**Each component gets a slice of the $360^{\circ }$ in proportion to its share:**

$$\text{central angle}=\frac{\text{component}}{\text{total}}\times 360^{\circ }$$

Worked example 1 — a monthly budget. A family's monthly income of ₹$36000$ is spent as food ₹$12000$, rent ₹$9000$, education ₹$6000$, travel ₹$4500$ and savings ₹$4500$.

$$\text{food}=\frac{12000}{36000}\times 360=\frac{1}{3}\times 360=120^{\circ }$$

$$\text{rent}=\frac{9000}{36000}\times 360=\frac{1}{4}\times 360=90^{\circ }$$

$$\text{education}=\frac{6000}{36000}\times 360=60^{\circ }$$

$$\text{travel}=\frac{4500}{36000}\times 360=45^{\circ },\text{savings}=45^{\circ }$$

Always total the angles.

$$120+90+60+45+45=360^{\circ }$$

**If the angles do not total $360^{\circ }$, something is wrong** — and that check costs one addition. A total of $355^{\circ }$ or $362^{\circ }$ means a rounding mistake or a component left out.

Worked example 2 — the sports data. Out of $720$ students:

$$\text{cricket}=\frac{300}{720}\times 360=150^{\circ },\text{football}=\frac{180}{720}\times 360=90^{\circ }$$

$$\text{badminton}=60^{\circ },\text{hockey}=45^{\circ },\text{others}=15^{\circ }$$

Totalling: $150+90+60+45+15=360^{\circ }$. Correct.

A shortcut worth knowing. Since the total is $720$ and $\frac{360}{720}=\frac{1}{2}$, every angle here is simply half the number of students. Looking for a simple fraction between the total and $360$ before computing five separate divisions saves real time.

To draw it: draw a circle, mark a radius, then measure each angle from the previous radius with a protractor, going round in one direction. Label every sector with its name and either its value or its percentage.

Percentages are a useful cross-check. Food is $\frac{12000}{36000}=33\frac{1}{3}\%$ and rent $25\%$, and since $1\%$ of the circle is $3.6^{\circ }$, food should be $33\frac{1}{3}\times 3.6=120^{\circ }$ — matching. Either route is acceptable, but mixing them halfway through a question is how rounding errors creep in, so pick one and stay with it.

Reverse the formula:

$$\text{component}=\frac{\text{central angle}}{360^{\circ }}\times \text{total}$$

Worked example 1. A pie chart represents $1080$ people, and one sector has a central angle of $100^{\circ }$. How many people does it stand for?

$$\frac{100}{360}\times 1080=\frac{100\times 3}{1}=300\text{ people}$$

Worked example 2. A pie chart shows a monthly expenditure of ₹$45000$, and the travel sector measures $72^{\circ }$.

$$\frac{72}{360}\times 45000=\frac{1}{5}\times 45000=₹9000$$

Worked example 3 — finding the total from one sector. In a pie chart, a sector of $90^{\circ }$ represents $150$ students. How many students altogether?

$$\frac{90}{360}\times \text{total}=150\Rightarrow \frac{1}{4}\times \text{total}=150\Rightarrow \text{total}=600$$

Worked example 4 — comparing two sectors. In the budget chart, food is $120^{\circ }$ and rent is $90^{\circ }$. How many times the rent is the food?

$$\frac{120}{90}=\frac{4}{3}\approx 1.33$$

The ratio of the angles is the ratio of the amounts, so food is $\frac{4}{3}$ times the rent — and checking against the figures, $\frac{12000}{9000}=\frac{4}{3}$. Correct. This means you can answer a ratio question straight from the angles, without knowing the total at all.

Worked example 5 — a missing angle. A pie chart has four sectors, three of them $100^{\circ }$, $85^{\circ }$ and $65^{\circ }$. Find the fourth.

$$360-(100+85+65)=360-250=110^{\circ }$$

A pie chart shows proportions, not amounts. Two schools could produce identical pie charts of favourite sports while one has $720$ students and the other $72$. So a question asking which school has more cricket players? cannot be answered from the charts alone — the totals are needed. That is the single most important limitation of the pie chart, and it is precisely why the total is always printed alongside one.

Which graph for which job. A pie chart shows how a whole divides into parts. A bar graph compares separate categories, and does it more accurately, because the eye judges lengths better than it judges angles. A histogram shows the shape of a continuous distribution. Choosing the wrong one is not a calculation error but it does make a correct calculation useless — so a question naming the graph is telling you what the data is like.