Presentation of Data

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Based on NCERT Economics Class XI (Statistics for Economics), Chapter 4.

Which would you rather read?

A radio news bulletin reads out the literacy rate of twelve states in one breath. By the end you remember none of them. The same twelve numbers drawn as bars are understood in two seconds: which state is highest, which is lowest and how big the gap is.

Organising data (the previous chapter) puts them in order. Presentation makes them quick to grasp. This chapter is a toolbox, and the exam skill is picking the right tool: which diagram answers which question, what each one is made of, and which statistical measure each one can give back.

Three ways to present data

A diagram trades some accuracy for a lot of communication. That is why reports pair every chart with the table beneath it.

Tabulation: how a table is classified

A table can be built on four bases. Notice that the first chapter on organising data called the time-based type chronological, while tabulation calls it temporal. They are the same idea.

The eight parts of a statistical table

Here is a small table built to show every part. The data are illustrative.

Table 2. Daily sales at a campus tea stall in 2024 (in hundreds of cups)

Unit of measurement: hundreds of cups. Source: stall register. Note: figures are rounded to the nearest hundred.

Memory hook: the caption sits on the ceiling (top) and the stub is on the side. Source and Note are easy to swap, so fix them in your mind: Source names where the data came from, Note explains the table.

The diagram families

Bar diagrams: three types

A simple bar diagram has equally wide, equally spaced bars standing on a common base line, and only the height carries information. It is one-dimensional. Because the width is arbitrary, the eye compares heights alone. If the widths varied, the eye would compare areas and be misled. Bars must start from zero for the same reason.

Choosing between multiple and component comes down to the headline. If the parts matter and you want them side by side, use a multiple bar. If the whole matters and you also want its breakdown, use a component bar.

The pie diagram

A pie diagram is a circle divided into sectors whose angles are proportional to the shares. The full circle is 360°, so one per cent is $360° / 100 = 3.6°$. First convert each value to a percentage, then multiply by 3.6°.

$$\text{angle} = \dfrac{\text{value}}{\text{total}} \times 360° = \text{percentage} \times 3.6°$$

Worked example: a household budget as a pie. A family spends ₹20,000 a month: food ₹8,000, rent ₹6,000, education ₹3,000, transport ₹2,000 and other items ₹1,000. Find the angle of each sector.

Common wrong step. Using the percentage itself as the angle (40°, 30°, …). These add up to 100°, not 360°. Another slip is multiplying by 36 instead of 3.6. The sum-to-360° check catches both.

Pause and predict: what angle at the centre of a pie shows a share of 25 per cent? If the total value doubles but the shares stay the same, do the angles change?

Check your answer: $25 \times 3.6° = 90°$. The angles do not change, because they depend only on the shares. The size of the whole pie says nothing about the total value.

Histogram: area is frequency

A histogram is used for a continuous frequency distribution. It is a set of adjacent rectangles, with the class intervals as bases and the area of each rectangle proportional to the class frequency. The bars touch because the classes are continuous. It is two-dimensional: width and height both carry information.

When all classes have the same width, heights are proportional to frequencies and you can plot frequency directly. When widths differ, a wide class would look too big, so the height must be the frequency density, the frequency per unit of class width.

$$\text{frequency density} = \dfrac{\text{class frequency}}{\text{class width}}, \qquad \text{area of bar} = \text{width} \times \text{density} = \text{frequency}$$

Worked example: unequal classes. A survey of 110 workers gives the data below. Draw the histogram by finding the correct heights.

Each bar's area (width times height) gives back its frequency: $10 \times 2 = 20$, $10 \times 3 = 30$, $20 \times 2 = 40$, $40 \times 0.5 = 20$. The busiest part of the distribution is therefore 10–20.

Common wrong step. Plotting the raw frequencies 20, 30, 40, 20 as heights. The 20–40 bar then looks tallest (40) even though its density is lower than that of 10–20, and its double width inflates its area further. Frequency density shows the real peak at 10–20.

Pause and predict: class A has width 10 and frequency 30, class B has width 30 and frequency 30. Which bar is taller on the histogram?

Check your answer: class A. Its density is $30/10 = 3$, against $30/30 = 1$ for class B. The same count spread over a wider class is less dense.

Frequency polygon and frequency curve

To draw a frequency polygon, first draw the histogram. Then join the mid-points of the tops of the bars with straight lines. Add one class with zero frequency at each end so that the polygon returns to the base line. Equivalently, plot (class mark, frequency) points and join them. The polygon has the same area as the histogram, and that area stands for the total frequency.

A frequency curve is the same figure smoothed by hand into a curve through the polygon's points, so that it again encloses about the same area.

Worked example: polygon points. A distribution of 50 workers has these frequencies: 10–20 (5), 20–30 (10), 30–40 (15), 40–50 (12), 50–60 (8). List the points of the frequency polygon.

Common wrong step. Leaving the polygon open at both ends. Without the two zero-frequency end classes, the polygon does not meet the base line and its area no longer matches the histogram.

Ogives: the cumulative curves

An ogive is a cumulative frequency curve. A less than ogive plots the number of observations below each upper limit and never decreases. A more than ogive plots the number at or above each lower limit and never increases. Both are drawn on the same axes, and they cross at the median. At the crossing point, the Y value is $N/2$ and the X value is the median.

Worked example: median from the two ogives. Use the 50 workers above. Find the median graphically.

Plot both curves. The Y value where they cross must be $N/2 = 25$. Less than ogive crosses 25 between 30 (value 15) and 40 (value 30), rising 1.5 per unit. So $15 + 1.5(x - 30) = 25$ gives $x - 30 = 6.67$, and the median is $36.67$. The more than ogive falls from 35 at 30 to 20 at 40, and meets 25 at the same point.

Common wrong step. Plotting the less than frequencies at the lower limits instead of the upper limits. The curve shifts one class to the left and the crossing point is wrong. A less than count always belongs to its upper limit.

Beyond the textbook. For the exam: the median is read from the crossing of the two ogives. In reality: you can read it from either ogive alone: find $N/2$ on the Y-axis, go across to the curve and drop to the X-axis. The second curve is a cross-check. Why it matters: the same idea gives any percentile or quartile, such as the value below which 25 per cent of the observations lie.

Reading the mode from a histogram

The mode is the value with the highest frequency, so it sits in the modal class, the tallest bar. To locate it within that bar, join the top right corner of the modal bar to the top right corner of the bar before it, and the top left corner of the modal bar to the top left corner of the bar after it. The two lines cross, and the X value of the crossing point is the mode.

Worked example: mode by construction. In the 50 worker distribution, the tallest bar is 30–40 (frequency 15), with 10 before it and 12 after it. Find the mode.

Common wrong step. Taking the class mark of the modal class (35) as the mode. The class mark ignores the neighbours. The mode is pulled toward the side with the taller neighbour: here, the 40–50 side, since 12 is bigger than 10.

Beyond the textbook. For the exam: the mode is read from a histogram. In reality: the construction is a drawing of the mode formula you will meet in the next chapter, $L + \dfrac{D_1}{D_1 + D_2} \times h$. The crossing point works out to exactly this value. Why it matters: the graph and the formula are the same calculation, so you can use one to check the other.

The arithmetic line graph

An arithmetic line graph (time-series graph) puts time on the X-axis and the variable on the Y-axis, then joins the points. Its job is to show the long-term trend. When two series are drawn together, the gap between them is the story. If imports run above exports year after year, the vertical gap between the lines is the trade deficit. A widening gap means the deficit is growing, and a narrowing gap means it is shrinking.

Pause and predict: on a line graph of a country's exports and imports, the imports line stays above the exports line and the two lines move further apart. What is happening to the trade deficit?

Check your answer: it is growing, because the gap between imports and exports is the deficit and the gap is widening.

Which diagram answers which question?

The mean cannot be read graphically from any of these. That is a classic trap.

Beyond the textbook. For the exam: a diagram is less accurate than a table but communicates better. In reality: diagrams can mislead as easily as inform. A bar chart whose axis starts at 80 instead of 0 makes a rise from 90 to 95 look enormous, which is why bars must stand on zero. People also judge lengths more accurately than angles, so a pie with many thin slices is hard to read and a bar chart is often better. Why it matters: examiners reward the diagram that suits the question, not the prettiest one.

Exam traps

Formula sheet

60-second revision

Where this chapter connects

• Previous: Organisation of Data builds the frequency tables that every frequency diagram here is drawn from.
• Next: Measures of Central Tendency calculates the median and the mode that this chapter only reads off graphs.
• Later use: Correlation uses scatter diagrams, which are plotted much like the line graphs here, and Index Numbers series are usually shown on an arithmetic line graph.