What is data, types of measurement

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DESCRIPTION OF THE TOPIC
Items Description of the Topic
Course Data Analysis for Social Science Teachers
Topic What is data? Types of Measurement
Module Id 1.2
1. What is Measurement?
Measurement refers to attaching numbers or labels to objects or events.
We know how to measure heights and weights of individuals. We measure height in, say,
centimeters and weight in kilograms.
The following table contains Statistical data of weights for a class of 60 students:
Individual Height (cms)
1 170
2 168
3 172
-
-
60 176
This table presents the data relating to the Heights of a Finance class of 30 students:
Individual Weight (kg)
1 75
2 71
3 82
-
-
30 78
Through measurement, we can generate data. Data is a collection of measurements. Data is
analyzed to yield some insights.
We may describe the central tendency of the heights and weight data using mean, median and
mode. We can also describe the variability in the data using Range, Mean Deviation, Quartile
Deviation and Standard Deviation. We may also use such tools as skewness and kurtosis to

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describe the shape of the distribution. However, the choice of a particular measure of central
tendency depends on the way we measure the data.
Similarly, when we are trying to describe the relationship between two variables, the choice of the
statistical technique depends on the way the two variables have been measured.
Can we measure anything easily?
When we try to measure something, we realize that it is far easier to measure certain things
compared to the other.
Taking physical measurements like Height, Weight, Blood Pressure, and Blood Sugar is much
easier than measuring the constructs such as Attitude, Motivation, Honesty, and Punctuality.
Just as we use a measurement tape to measure height and a weighing machine to measure the
weight, we need some instrument, popularly known as scale, to measure the constructs such as
attitude, motivation, honesty, etc.
Before we understand how a scale is developed to measure constructs like attitude, motivation,
honesty, etc., let us devote some time to understand different levels at which something can be
measured.
Different Ways (Levels) of Measuring a Construct
We can measure a particular thing in different ways (or at different levels).
Let us say, we are trying to measure the age of our respondents in a survey. we may measure the
age in the following ways (or levels):
a) We may just categorize the respondents into three groups: Young, Middle Age, and Old
Age;
b) We may categorize the respondents into certain age intervals: Below 30, 31 to 60, 61 to 90,
91 and above;and
c) We might also record everyone’s age in the completed years/completed months/completed
days, etc.
2. Levels of Measurement
Broadly there are four levels or scales of measurement: Nominal, Ordinal, Interval and Ratio.
Before we proceed to understand different levels or scales of measurement, let us reemphasize
the choice of statistical tools to analyze the data is primarily dependent on the way the
underlying construct/variables were measured.

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2.1 Nominal Scale
When we measure something using a nominal scale, we use numbers as just labels or identifiers.
For example, each player in a cricket team is labeled with a number (printed on their jersey) to
easily identify them when they are playing. These numbers are just labels and do not make any
difference even when we interchange the numbers given to two players. Further, we may not assign
consecutive numbers to players. Even when they are assigned consecutive numbers, say, 1 to 11,
the order will not have any meaning.
I used to watch Indian Cricket from 2001 to 2010. I remember Rahul Dravid wearing a jersey
numbered 5 and Sachin Tendulkar wearing a jercy numbered 10. Did these numbers have any
significance? Can it be said that Tendulkar was twice efficient as Dravid? For that matter Ganguly
used to wear a jersey numbered 1. Can it be said that Tendulkar was ten times better than Ganguly?
Further, we can also assign alphabets like A, B, C, D instead of 1,2,3,4. It too does the same
purpose. Now we realize that when we are measuring something using a nominal scale, we are just
assigning labels, or we are categorizing the subjects into predefined bins.
Let us take another example. A market research firm wanted to do a survey of brand loyalty. Let
us say this survey in three cities, namely, Hyderabad, Delhi, and Mumbai and we are interested in
the city of origin of our respondents. When we try to measure the city of or origin of respondents,
we simply pick a respondent, note him/her and put him/her in one of the bins of Hyderabad, Delhi,
and Mumbai.
Gender, color preference, nationality, personality type, native state are some of the examples of
variables that are generally measured on a nominal scale
Now let us turn to the Ordinal Scale.
2.2 Ordinal scale
When we measure something using an ordinal scale, again we attach labels to the
objects/respondents. However here the labels can be arranged in an order. Consider the example
of the designations of teachers in a university: Assistant Professors, Associate Professors, and
Professors. These are just labels. While we pick a teacher, we will be able to categorize him/her
into one of the three bins. However, these labels have some order. When organized from the lowest
to the highest, we will get the order starting with Assistant Professor, Associate Professor, and
then Professor. Similarly the designations in police or military organizations are on an ordinal
scale. These designations have some order.
Sometimes we also assign numbers as labels. Think of ranks of students in the NEET examination.
The label 1st
rank is assigned to the student who has scored the highest marks; the next highest
scorer gets the label 2nd
rank and the next, the third rank and so on.

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The ordinal scale measurement is one step better than the nominal scale measurement in that the
labels assigned can be arranged in an order. However, we notice that the distance between two
consecutive labels may not be the same.
Let us get back to the example of the NEET examination. The differences between 1st
rank holder
and the 2nd
rank holder may not be the same and the same is true between the 2nd
rank holder and
3rd
rank holder. Let us say the first three rankers have scored the 98, 96, and 92 respectively. The
distance between 1st
and 2nd
ranker is 2 marks, whereas the distance between 2nd
rank and 3rd
rank
holder is 4 marks.
Individual Rank Marks
A 1 98
B 2 96
C 3 92
2.3 Interval Scale
An ordinal scale wherein the interval between any two consecutive data points is the same can be
termed as an interval scale. That means in an interval scale, the data points are ordered and the
distance between any two consecutive data points is the same. The popular example of interval
scale is temperature. We know the difference between 10 degree Celsius and 11 degree Celsius is
same as 18 degree Celsius and 19 degree Celsius. Similarly, the difference between any two
consecutive points on a Fahrenheit scale is also the same. The intervals are same.
However, we cannot say 20 degree Celsius is twice as much as 10 degree Celsius. This is not
possible because we do not have any absolute zero on a Celsius scale. The Celsius scale was
developed noting the freezing point of water as 0-Degree celsius; and boiling point of water as 100
degree Celsius and dividing the distance into 100 equal parts. That’s why the Celsius scale was
originally called the centigrade scale; that is ‘cent’ means 100. On the contrary, on a Fahrenheit
scale, the distance between the freezing point of water and boiling point of water is divided into
180 equal parts. The freezing point on Fahrenheit scale is noted as 32 degrees Fahrenheit and the
boiling point is 212 degrees Fahrenheit. The distance is divided into 180 equal parts (212-32=180).
However, on both the scales, the zero is arbitrary. In fact, the original centigrade scale has 0 as the
boiling point and 100 as the freezing point, that means it is running in the opposite direction. 100
degree Celsius then would have meant very cool.
As there is no absolute zero, the ratios of numbers on an interval scale are not meaningful. That’s
the reason why we can say that 20-degree celsius is not twice as 10-degree celsius.
Now, let us examine in detail why the ratios are not meaningful on an interval scale.

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We know the equation to convert Celsius to Fahrenheit, i.e
F= 32+1.8C
Let us look at the following data on two scales.
Celsius
Scale
Fahrenheit
Scale
Freezing Point of Water 0 32
10 50
20 68
Boiling Point of Water 100 212
The ratio of 20/10 on the Celsius Scale is:
20
10
= 2,
whereas the ratio of equivalent data points on the Fahrenheit scale is:
68
50
=1.36.
Incompatibility between the Ratios
We can take another example of an interval scale as a time of the day. We have a 12-hour scale
and 24-hour scale
12-hour scale 2pm 3pm 4pm 5pm 6pm 7pm 8pm
24-hour scale 14 15 16 17 18 19 20
On the 12-hour scale, the difference between 4pm and 2pm is: 2 hours
Also
On the 24 hour scale, the difference between 16 and 14 is: 2 hours
However, the ratios are not equal
4
2
= 2 and
16
14
= 1.14
The reason why the ratios are not meaningful in an interval scale is because of the absence of a
natural or absolute zero.
The zero on an interval scale is just another point and is arbitrary. The freezing point of water is
fixed at zero degree centigrade. We have already noted in the original scale that the zero is boiling
point of water. Similarly, midnight 12 am is fixed at 0 hours. It is arbitrary too.

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2.4 Ratio Scale
The interval scale wherein the ratios are equal is called a ratio scale. Measurement of height,
weight, distance, etc., are examples of a ratio scale. The other examples include sales, profits,
costs, etc.
Lets us take the measurement of distance.
Meters Kilometers
0 0
1000 1
2000 2
3000 3
The intervals between any two consecutive points on both the scales are the same:
3km-2km=2km-1km=1km
2000meters-1999 meters=1999meters-1998meters= 1mtr
At the same time the ratios are also equal.
However, the ratios are not equal:
3000 𝑚𝑒𝑡𝑒𝑟𝑠
2000 𝑚𝑒𝑡𝑒𝑟𝑠
= 1.5 and
3 𝑘𝑚
2 𝑘𝑚
= 1.5
In the same way the ratios of sales, profits and costs are also meaningful. In a ratio scale the ratios
are meaningful for there exists an absolute zero. In comparison the interval scale will have an
arbitrary zero
3. Level of Measurement and Choice of Statistical Measures
When the data is on a ratio scale, we can compute all three measures of central tendency -- Mean,
Median and Mode -- and all measures of variance. However, in case of the data on nominal scale,
we can use only one measure of central tendency, i.e. mode.
The following table presents the summary of the choice of measures of central tendency and
variance for different levels of measurement:
Leave of Measurement Choice of Measures of Central
Tendency
Choice of Measures of
Spread/Variance
Nominal Mode only None
Ordinal Mode & Median Inter Quartile Range(IQR)
Interval Mode, Median & Mean IQR, Range, Variance and
Standard Deviation
Ratio Mode, Median & Mean IQR, Range, Variance and
Standard Deviation

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What Level of Measurement is desirable?
Where possible it is always desirable to measure by the ratio scale. Because the ratio scale data
allows the application of many statistical tools on it.
4. Different Types of Data
Univariate Data is data that concerns only one variable. The data concerning the heights of a
Finance class of 30 students presented in the following table is an example of Univariate Data
Individual Weight (kg)
1 75
2 71
3 82
-
-
30 78
Bivariate Data is data concerning only two variables. Continuing with our earlier example, if we
add the weight of each student along with his/her height, it will be a bivariate data.
Individual Weight (kg) Height(cms)
1 75 172
2 71 169
3 82 174
-
-
30 78 176
Multivariate Data is data concerning more than two variables.
Individual Weight (kgs) Height(cms) Marks in a
Exam(max 100)
Gender
1 75 172 80 Male
2 71 169 75 Male
3 82 174 82 Female
-
-
30 78 176 69 Male

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Categorical Data
The data measured on Numerical or Ordinal Scale is generally referred to as Categorical Data.
Categorical data is also referred to as Non-parametric data.
Numerical Data
The data measured on Interval or Ratio Scale is generally referred to as Numerical Data. The words
Numerical Data, Metric Data, Continuous Data, Parametric Data are generally used
interchangeably.