BIOSTATISTICS | ZOOL-600-601 LECTURE 5
Instructor: Dr. Waheed Ali Panhwar | University of Sindh BS Zoology — Final Year
Scales of Measurement
What this lecture is about: Scales of measurement answer a precise question: exactly
how much mathematical information is carried by a piece of data? The answer controls
which statistical operations are legally valid. There are four scales arranged in a hierarchy:
Nominal → Ordinal → Interval → Ratio. Each level adds one new mathematical capability
that the level below did not have.
1. The Four Scales — Staircase Model
Key Point: Think of the four scales as a staircase. Each step upward adds exactly one
new mathematical property. A higher scale inherits EVERYTHING from all the scales below
it. Nominal has only names. Ordinal adds order. Interval adds equal gaps. Ratio adds a
true zero.
Scale Identity Order Equal Intervals True Zero (Ratios
(Same/Different?) (Greater/Lesser?) (How much?) valid?)
NOMINAL ✓ YES ✗ NO ✗ NO ✗ NO
ORDINAL ✓ YES ✓ YES ✗ NO ✗ NO
INTERVAL ✓ YES ✓ YES ✓ YES ✗ NO
RATIO ✓ YES ✓ YES ✓ YES ✓ YES
Exam Tip: This table is the most important thing to memorise in Lecture 5. Your exam
may ask you to compare scales, identify which operations are valid, or classify a variable
into its correct scale. If you know this table, you can answer all of those questions.
2. The Four Scales in Detail
SCALE 1 — NOMINAL SCALE "Names Only — the most basic level"
Definition: The nominal scale simply places individuals into named categories. The only
mathematical relationship between two nominal values is identity — they are either the
same category or a different category. No order, no ranking, no arithmetic of any kind.
New Capability Added: Identity — you can tell whether two measurements belong to the
same category or different ones.
Key Limitation: You cannot order the categories, measure gaps between them, or
perform any arithmetic. Even if numbers are used as labels, those numbers carry no
mathematical value.
✓ ✗ NOT PERMITTED
PER • Calculate mean or any average
MITT • Rank the categories
ED • Perform addition, subtraction, multiplication, or division
Oper
ation
s
•
Coun
, t
frequ
encie
s
(how
many
in
each
categ
ory)
•
Calcu
late
perce
ntage
s
• Find
the
mode
(most
com
mon
categ
ory)
General Examples: Species names, blood groups, sex (male/female), habitat type, animal ID
numbers, country of origin.
Zoology Example: Recording the species of 300 fish caught from Keenjhar Lake (Catla catla,
Labeo rohita, Cirrhinus mrigala). You can count how many of each species — you cannot
average the species names.
SCALE 2 — ORDINAL SCALE "Names Plus Ranking — order without equal gaps"
Definition: The ordinal scale has everything nominal has, plus a meaningful order or
ranking between categories. You now know which value is greater or lesser. However, the
gaps between ranks are NOT equal or measurable — you know the direction of the
difference, but not its size.
New Capability Added: Order — you can now rank categories from lowest to highest (or
highest to lowest).
Key Limitation: The gaps between ranks are unequal and cannot be measured. You
cannot perform meaningful arithmetic because a one-rank difference at the top may
represent a very different real-world gap than a one-rank difference at the bottom.
✓ ✗ NOT PERMITTED
PER • Calculate the mean — averaging rank numbers produces a meaningless result
MITT • Measure the exact distance between ranks
ED • Say 'Severe is twice as bad as Mild'
Oper
ation
s
•
Rank
indivi
duals
from
lowes
t to
highe
st
Instructor: Dr. Waheed Ali Panhwar | University of Sindh BS Zoology — Final Year
Scales of Measurement
What this lecture is about: Scales of measurement answer a precise question: exactly
how much mathematical information is carried by a piece of data? The answer controls
which statistical operations are legally valid. There are four scales arranged in a hierarchy:
Nominal → Ordinal → Interval → Ratio. Each level adds one new mathematical capability
that the level below did not have.
1. The Four Scales — Staircase Model
Key Point: Think of the four scales as a staircase. Each step upward adds exactly one
new mathematical property. A higher scale inherits EVERYTHING from all the scales below
it. Nominal has only names. Ordinal adds order. Interval adds equal gaps. Ratio adds a
true zero.
Scale Identity Order Equal Intervals True Zero (Ratios
(Same/Different?) (Greater/Lesser?) (How much?) valid?)
NOMINAL ✓ YES ✗ NO ✗ NO ✗ NO
ORDINAL ✓ YES ✓ YES ✗ NO ✗ NO
INTERVAL ✓ YES ✓ YES ✓ YES ✗ NO
RATIO ✓ YES ✓ YES ✓ YES ✓ YES
Exam Tip: This table is the most important thing to memorise in Lecture 5. Your exam
may ask you to compare scales, identify which operations are valid, or classify a variable
into its correct scale. If you know this table, you can answer all of those questions.
2. The Four Scales in Detail
SCALE 1 — NOMINAL SCALE "Names Only — the most basic level"
Definition: The nominal scale simply places individuals into named categories. The only
mathematical relationship between two nominal values is identity — they are either the
same category or a different category. No order, no ranking, no arithmetic of any kind.
New Capability Added: Identity — you can tell whether two measurements belong to the
same category or different ones.
Key Limitation: You cannot order the categories, measure gaps between them, or
perform any arithmetic. Even if numbers are used as labels, those numbers carry no
mathematical value.
✓ ✗ NOT PERMITTED
PER • Calculate mean or any average
MITT • Rank the categories
ED • Perform addition, subtraction, multiplication, or division
Oper
ation
s
•
Coun
, t
frequ
encie
s
(how
many
in
each
categ
ory)
•
Calcu
late
perce
ntage
s
• Find
the
mode
(most
com
mon
categ
ory)
General Examples: Species names, blood groups, sex (male/female), habitat type, animal ID
numbers, country of origin.
Zoology Example: Recording the species of 300 fish caught from Keenjhar Lake (Catla catla,
Labeo rohita, Cirrhinus mrigala). You can count how many of each species — you cannot
average the species names.
SCALE 2 — ORDINAL SCALE "Names Plus Ranking — order without equal gaps"
Definition: The ordinal scale has everything nominal has, plus a meaningful order or
ranking between categories. You now know which value is greater or lesser. However, the
gaps between ranks are NOT equal or measurable — you know the direction of the
difference, but not its size.
New Capability Added: Order — you can now rank categories from lowest to highest (or
highest to lowest).
Key Limitation: The gaps between ranks are unequal and cannot be measured. You
cannot perform meaningful arithmetic because a one-rank difference at the top may
represent a very different real-world gap than a one-rank difference at the bottom.
✓ ✗ NOT PERMITTED
PER • Calculate the mean — averaging rank numbers produces a meaningless result
MITT • Measure the exact distance between ranks
ED • Say 'Severe is twice as bad as Mild'
Oper
ation
s
•
Rank
indivi
duals
from
lowes
t to
highe
st
