Data Presentation: Bar Charts, Pie Charts and Histograms

Turning Numbers into Pictures

A list of 200 numbers tells you very little at a glance. The same data drawn as a bar chart, a pie chart or a histogram instantly shows which category is largest, how values are spread, and whether there are unusual results. Governments, businesses, newspapers and researchers all communicate through charts, and WAEC regularly asks students to draw and interpret them. This tutorial covers data presentation as tested in WAEC, NECO, BECE and JAMB mathematics: types of data, frequency tables and tally marks, bar charts, pictograms, pie charts (drawing and reading), line graphs, histograms with equal and unequal class widths, frequency polygons and stem-and-leaf diagrams, plus how to spot misleading graphs.

Types of Data

The right choice of data presentation depends on the type of data:

Qualitative (categorical) data describes categories: favourite subject, blood group, state of origin. It is shown in bar charts, pictograms and pie charts.
Quantitative (numerical) data is made of numbers, and it comes in two kinds, discrete and continuous.
Discrete data can take only particular values, usually whole numbers: the number of children in a family, or goals scored.
Continuous data can take any value in a range: heights, masses, times and temperatures. It is grouped into classes and shown in histograms and frequency polygons.

Frequency Tables and Tally Marks

Raw data is organised into a frequency table. Each value is recorded with a tally mark, and tallies are grouped in fives (four strokes crossed by a fifth) to make counting easy. The frequency is the number of times each value or class occurs.

Worked example 1

The numbers of children in 20 families are: 2, 3, 1, 4, 2, 2, 0, 3, 5, 2, 1, 3, 2, 4, 2, 3, 1, 2, 3, 2. Make a frequency table.

0 children: frequency 1
1 child: frequency 3
2 children: frequency 8
3 children: frequency 5
4 children: frequency 2
5 children: frequency 1

The frequencies add up to 20, the number of families, which is a useful check. The mode, the most common value, is 2 children.

Bar Charts and Pictograms

A bar chart shows each category as a bar whose height (or length) represents its frequency.

All the bars have the same width, and there are gaps between them, because the categories are separate.
Label both axes, start the frequency axis at zero, and give the chart a title.
A compound (component) bar chart stacks parts within each bar. A multiple bar chart places bars for different groups side by side, for example boys and girls in each class.

A pictogram uses repeated symbols, each representing a fixed number of items, with a key. For example, one bag symbol might stand for 10 bags of rice, and half a symbol for 5.

Pie Charts

A pie chart shows how a whole is divided into parts, as sectors of a circle. Each sector's angle is proportional to its frequency:

Sector angle = (frequency ÷ total frequency) × 360°

Worked example 2: drawing a pie chart

A student's monthly budget of ₦60,000 is spent as follows: food ₦24,000, rent ₦15,000, transport ₦9,000, savings ₦6,000 and other items ₦6,000. Calculate the sector angles.

Each naira is worth 360 ÷ 60,000 = 0.006°, so:

Food: 24,000 × 0.006 = 144°
Rent: 15,000 × 0.006 = 90°
Transport: 9,000 × 0.006 = 54°
Savings: 6,000 × 0.006 = 36°
Others: 6,000 × 0.006 = 36°

Check that the angles add up to 360°: 144 + 90 + 54 + 36 + 36 = 360. Draw a circle, draw a radius, and measure each angle in turn with a protractor, labelling each sector.

Worked example 3: reading a pie chart

In a pie chart showing the 1,200 students of a school by class, the sector for SS 3 has an angle of 72°. How many SS 3 students are there?

Number = (72 ÷ 360) × 1,200 = ⅕ × 1,200 = 240 students.

Worked example 4: finding the total

In another pie chart, the sector for the 45 students who walk to school has an angle of 54°. How many students were surveyed?

54° represents 45 students, so 1° represents 45 ÷ 54 students, and 360° represents 360 × 45 ÷ 54 = 300 students.

Pie charts show proportions clearly, but they are poor for comparing similar-sized sectors or showing changes over time. For those, use a bar chart or a line graph.

Line Graphs

A line graph shows how a quantity changes over time, such as monthly rainfall, a patient's temperature or a company's yearly sales. Plot the points, with time on the horizontal axis, and join them with straight lines. The intermediate points on the lines are only estimates, so the lines show the trend.

Histograms

A histogram displays grouped continuous data. It looks like a bar chart, but with important differences:

There are no gaps between the bars, because the data is continuous and the classes run into one another.
The horizontal axis is a continuous number scale, and the bars are drawn between the class boundaries.
It is the area of each bar, not its height, that represents the frequency.

Equal class widths

When all the classes are the same width, the bar heights can simply be the frequencies.

Unequal class widths: frequency density

When classes have different widths, the height of each bar must be the frequency density:

Frequency density = frequency ÷ class width

Otherwise, a wide class would look misleadingly large.

Worked example 5

The ages of 75 people at a clinic are grouped as follows: 0–10 years (frequency 10), 10–20 years (20), 20–40 years (30) and 40–70 years (15). Find the heights of the histogram bars.

0–10: width 10, so density = 10 ÷ 10 = 1.0
10–20: width 10, so density = 20 ÷ 10 = 2.0
20–40: width 20, so density = 30 ÷ 20 = 1.5
40–70: width 30, so density = 15 ÷ 30 = 0.5

Although the 20–40 class has the highest frequency, the 10–20 bar is the tallest, because people are most concentrated in that age range. Label the vertical axis "frequency density".

Class boundaries

For data recorded to the nearest whole number in classes such as 10–19, 20–29, the true class boundaries are 9.5, 19.5, 29.5 and so on, and the class width is 10. Bars are drawn between the boundaries, so that there are no gaps.

Frequency Polygons

A frequency polygon is made by plotting the frequency of each class against its class midpoint and joining the points with straight lines. It can be drawn by joining the midpoints of the tops of the bars of a histogram. The polygon is usually closed by joining it to the horizontal axis at the midpoints of the empty classes at each end.

Frequency polygons are especially useful for comparing two distributions on the same graph, such as the test scores of two classes.

Stem-and-Leaf Diagrams

A stem-and-leaf diagram keeps the original data values while showing their shape. Each value is split into a stem (the leading digits) and a leaf (the last digit).

Worked example 6

Show these test scores in a stem-and-leaf diagram: 45, 52, 58, 61, 63, 63, 67, 70, 74, 78, 81, 95.

4 | 5
5 | 2 8
6 | 1 3 3 7
7 | 0 4 8
8 | 1
9 | 5

Key: 6 | 1 means 61. The leaves are written in order, so the median and the mode can be read directly: the mode is 63, and the median is the mean of the 6th and 7th values, (63 + 67) ÷ 2 = 65.

Spotting Misleading Graphs

A vertical axis that does not start at zero can make small differences look huge.
Pictograms that make a symbol bigger in both directions exaggerate the change, because area grows with the square of the size.
Uneven scales, missing labels, or a 3D effect that distorts pie sectors can all mislead.

Always read the axes, the scale and the key before drawing conclusions from a chart.

Common Mistakes with Data Presentation

Leaving gaps between histogram bars, or no gaps between the bars of a bar chart.
Using frequency instead of frequency density for unequal class widths.
Pie-chart angles that do not add up to 360°, usually because of rounding. Check the total.
Plotting frequency polygons at the class boundaries instead of the midpoints.
Missing titles, axis labels or keys.

Data Presentation: Practice Questions

Classify each type of data: (a) shoe size (b) a student's height (c) favourite colour.
In a survey of 90 people, 25 prefer football. What angle represents football on a pie chart?
In a pie chart of a farmer's 600 hectares, maize takes up 150°. How many hectares are planted with maize?
A class of width 5 has a frequency of 30. Find its frequency density.
What is the midpoint of the class 20–29, with data recorded to the nearest whole number?
State one difference between a bar chart and a histogram.

Answers: 1. (a) Discrete quantitative. (b) Continuous quantitative. (c) Qualitative. 2. 25 ÷ 90 × 360 = 100°. 3. 150 ÷ 360 × 600 = 250 hectares. 4. 30 ÷ 5 = 6. 5. (19.5 + 29.5) ÷ 2 = 24.5. 6. A bar chart has gaps between its bars and shows categories, with height representing frequency. A histogram has no gaps and shows continuous data, with area representing frequency.

Conclusion

Data presentation turns raw numbers into clear pictures. Organise data with tally marks in a frequency table, then choose the right chart. Bar charts and pictograms compare categories, pie charts show parts of a whole with sector angle = frequency/total × 360°, and line graphs show change over time. Histograms show continuous data with no gaps between the bars, and for unequal classes their heights are the frequency density (frequency ÷ class width). Frequency polygons join the class midpoints, and stem-and-leaf diagrams keep the original values. Always label charts fully, and read the scales critically.