CDS Mathematics · Statistics

Data, Scales and Presentation

Data is primary or secondary, discrete or continuous, and measured on a nominal, ordinal, interval or ratio scale; grouped data is shown in classes, histograms and polygons.

All Statistics notes CDS Mathematics strategy

Why this matters

Eleven PYQs, nearly all EASY. They are vocabulary questions — which kind of data, which scale, which diagram — and each has one right term. Learn the table on this page and the inclusive/exclusive class distinction, and these marks are certain.

Concept 1 of 2: Kinds of data and scales of measurement

Two questions classify any data set: who collected it (you, or someone before you), and what kind of number it is (a count, a measurement, a rank, or just a label).

Definition

Primary data is collected first-hand for the enquiry; secondary data was collected by someone else (published reports, registers, a thesis).
Discrete data takes separate values (a count of cards); continuous data can take any value in a range (a height).
Qualitative (categorical) data is a label; quantitative data is a number.
Scales: nominal (labels only), ordinal (ranked, gaps not equal), interval (equal gaps, no true zero), ratio (equal gaps and a true zero).

Scale	What it allows	Example
Nominal	Naming only	Blood group, roll number
Ordinal	Ranking	Hotel star rating, class rank
Interval	Equal differences	Temperature in °C
Ratio	Equal differences and ratios	Height, income, marks

Each scale allows everything the one above it allows, and one thing more.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q98Easy

Example 1 · Statistics · Data, Scales and Presentation

The total number of live births in a specific locality during different months of a specific year was obtained from the office of the Birth Registrar. This set of data may be called

Ranked is not measured

Star ratings and ranks can be ordered, but a 4-star hotel is not 'twice' a 2-star one. That makes them ordinal, not interval or ratio.

Drill 3 more on kinds of data and scales of measurement

Concept 2 of 2: Classes, histograms and polygons

Grouping data into classes loses the individual values but shows the shape. In a histogram the AREA of each bar stands for the frequency, so when classes have different widths the height must be frequency per unit width.

Definition

Inclusive classes $15$ – $19$ , $20$ – $24$ : both limits belong to the class (there is a gap). Exclusive classes $15$ – $20$ , $20$ – $25$ : the upper limit belongs to the next class.
To use formulas on inclusive classes, convert to boundaries: $14.5$ – $19.5$ , $19.5$ – $24.5$ .
Frequency density = class frequency ÷ class width; it is the height of a histogram bar.
With equal widths, the bar height is simply the frequency.
A frequency polygon joins the mid-points of the bar tops; it smooths out as classes get narrower and data grows.
Bar diagrams, pie diagrams and pictograms are all diagrammatic presentation; a bar diagram suits discrete values.

Frequency density

\text{frequency density} = \dfrac{\text{class frequency}}{\text{class width}}

Visualization · change the bin width, watch the shape change

Bin width:

Same 30 data points each time — only the bin width changes. At width 50 the shape looks almost uniform; at width 5 it looks jagged. Choosing bin width is part of the analysis, not the data.

Worked example

The class

20

–

35

has frequency

45

. Find its frequency density, and its lower boundary if the classes were written

20

–

34

35

–

49

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — Elementary Mathematics · Q95Moderate

Example 2 · Statistics · Data, Scales and Presentation

Suppose the class interval 10–15 has frequency 30, then what is the frequency density of this class interval ?

Inclusive and exclusive are easy to swap

'15–19, 20–24' is INCLUSIVE (the upper limit is inside the class). '15–20, 20–25' is EXCLUSIVE. A statement question that names them the other way round is false.

Drill 6 more on classes, histograms and polygons

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 (1)

Classes, histograms and polygons
Frequency density
$\text{frequency density} = \dfrac{\text{class frequency}}{\text{class width}}$

Reference tables (1)

Kinds of data and scales of measurement4 rows

Scale	What it allows	Example
Nominal	Naming only	Blood group, roll number
Ordinal	Ranking	Hotel star rating, class rank
Interval	Equal differences	Temperature in °C
Ratio	Equal differences and ratios	Height, income, marks

Each scale allows everything the one above it allows, and one thing more.

Watch out for (2)

Ranked is not measured
→ Kinds of data and scales of measurement
Inclusive and exclusive are easy to swap
→ Classes, histograms and polygons

Test yourself on Statistics

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Statistics chapter test Drill all 11 questions