AFCAT Numerical Ability · Statistics

Data, Class Intervals and Pie Charts

Tell primary data from secondary, turn class limits into boundaries and widths, and turn values into the angles of a pie chart or the sides of a square diagram.

All Statistics notes

Why this matters

Every question on this page is EASY, and the marks are lost on a definition rather than on arithmetic: a boundary half a unit below the printed limit, or an angle taken out of 100 instead of 360. Each pie chart question is one fraction of the whole, so write that fraction down first.

Concept 1 of 2: Data and class intervals

Statistical data are numbers collected for a purpose, so that they can be compared. When there are many of them they are grouped into classes. Each class has two printed limits, and two boundaries that close the gap between one class and the next so that the classes touch.

Definition

Primary data are collected first-hand by the investigator: a survey, door-to-door visits, telephone interviews. Secondary data are taken from records someone else collected, such as census reports or published tables.
A list of names is not statistical data. Measurements or counts collected for comparison are.
Class limits are the printed ends, such as 20-29. If one class ends at 29 and the next starts at 30, the gap is 1. Take half the gap off every lower limit and add it to every upper limit to get the class boundaries: 19.5 to 29.5.
Class width is the upper boundary minus the lower boundary. With $k$ classes of width $w$ starting at the lower boundary $L$ , the last class ends at $L + kw$ .
To make $k$ equal classes from raw data, the range (highest minus lowest) divided by $k$ gives the width. Start the first class at the lowest value.

Class width from the range

w = \dfrac{H - L}{k}, \qquad \text{last boundary} = L + kw

Worked example

The classes of a distribution are 5-9, 10-14, 15-19 and 20-24. Find the class boundaries of the third class and the class width.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Statistics · Data, Classes and Pie Charts

If the width of each of the ten classes in a frequency distribution is 2.5 and the lower class boundary of the lowest class is 5.1, then the upper class boundary of the highest class is

Every limit moves by the same half-gap

For the classes 15-24 and 25-34 the actual lower limits are 14.5 and 24.5. Moving only the first one, or moving the upper limits down, mixes limits with boundaries.

Census figures are secondary

Data copied from the census or a published report are secondary, even though someone once collected them first-hand. Primary means the investigator collected them.

Drill 4 more on data and class intervals

Concept 2 of 2: Pie charts and square diagrams

A pie chart shares the 360° of a circle among the items in proportion to their values, so each angle is a fraction of a full turn. A square diagram does the same with area: the square's area stands for the value, so its side grows only as the square root of the value.

Definition

The angle of an item is $\dfrac{v}{T} \times 360^\circ$ , where $v$ is its value and $T$ the total.
Its share as a percentage is $\dfrac{\theta}{360^\circ} \times 100\%$ .
Back from an angle to a value: $\dfrac{\theta}{360^\circ} \times T$ . Two items differ by $\dfrac{\theta_1 - \theta_2}{360^\circ} \times T$ .
If one item's value is known, the others scale by their angles: an item with twice the angle has twice the value. The total is not needed.
If the values are given as a ratio, the angle is that item's share of the ratio sum times 360°, whatever the total.
Square diagram: the area is the value divided by the scale (the value one square centimetre stands for), and the side is the square root of that area.

Angle of a sector

\theta = \dfrac{v}{T} \times 360^\circ, \qquad \text{side} = \sqrt{\dfrac{v}{s}}

Worked example

A family spends ₹12,000 a month: food ₹4,800, rent ₹3,600, travel ₹1,200 and savings ₹2,400. Find the angle for rent on a pie chart, and the share of travel as a percentage.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Statistics · Data, Classes and Pie Charts

The total number of cellphones sold for Motorola, Samsung and Sony was 45664. The number of cellphones sold for these companies were in the ratio 3:5:8 respectively. If these data were shown on a pie chart, calculate the angle represented by the number of cellphones sold by Motorola.

Out of 360, not out of 100

A sector of 90° is a quarter of the chart, which is 25%. Reading the angle as a percentage gives 90%.

The side is a square root

With 1 cm² for 10 units, 160 units is an area of 16 cm² and a side of 4 cm. Dividing 160 by 10 and calling 16 the side makes the side four times too long.

A ratio needs no total

When the items are given as a ratio, the angle depends on the ratio alone. A large total printed in the question can be ignored.

Drill 7 more on pie charts and square diagrams

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Data and class intervals
Class width from the range
$w = \dfrac{H - L}{k}, \qquad \text{last boundary} = L + kw$
Pie charts and square diagrams
Angle of a sector
$\theta = \dfrac{v}{T} \times 360^\circ, \qquad \text{side} = \sqrt{\dfrac{v}{s}}$

Watch out for (5)

Every limit moves by the same half-gap
→ Data and class intervals
Census figures are secondary
→ Data and class intervals
Out of 360, not out of 100
→ Pie charts and square diagrams
The side is a square root
→ Pie charts and square diagrams
A ratio needs no total
→ Pie charts and square diagrams

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