LECTURE 1 AND 2 Introduction to Statistics
Statistics
Statistics" (preferably with a capital letter) to refer to the academic discipline concerned
with the collection, description, analysis and interpretation of numerical data. As such,
the subject of Statistics may be divided into two main categories:
(a) Descriptive Statistics
This is mainly concerned with collecting and summarising data, and presenting the
results in appropriate tables and charts. For example, companies collect and
summarise their financial data in tables (and occasionally charts) in their annual
reports, but there is no attempt to go "beyond the data".
(b) Statistical Inference
This is concerned with analysing data and then interpreting the results (attempting
to go "beyond the data"). The main way in which this is done is by collecting data
from a sample and then using the sample results to infer conclusions about the
population.
For example, prior to general elections in the Kenya and many other countries,
statisticians conduct opinion polls in which samples of potential voters are asked
which political party they intend to vote for. The sample proportions are then used
to predict the voting intentions of the entire population.
Of course, before any descriptive statistics can be calculated or any statistical inferences
made, appropriate data has to be collected. We will start the course, therefore, by seeing
how we collect data. This study unit looks at the various types of data, the main sources
of data and some of the numerous methods available to collect data.
B. MEASUREMENT SCALES AND TYPES OF DATA
Measurement Scales
Quantitative methods use quantitative data which consists of measurements of various
kinds. Quantitative data may be measured in one of four measurement scales, and it is
important to be aware of the measurement scale that applies to your data before
commencing any data description or analysis. The four measurement scales are:
(a) Nominal Scale
The nominal scale uses numbers simply to identify members of a group or category.
For example, in a questionnaire, respondents may be asked whether they are male
or female and the responses may be given number codes (say 0 for males and 1 for
females). Similarly, companies may be asked to indicate their ownership form and
again the responses may be given number codes (say 1 for public limited
companies, 2 for private limited companies, 3 for mutual organizations, etc.). In
these cases, the numbers simply indicate the group to which the respondents
belong and have no further arithmetic meaning.
(b) Ordinal Scale
The ordinal scale uses numbers to rank responses according to some criterion, but
has no unit of measurement. In this scale, numbers are used to represent "more
than" or "less than" measurements, such as preferences or rankings. For example, it
is common in questionnaires to ask respondents to indicate how much they agree
with a given statement and their responses can be given number codes (say 1 for
"Disagree Strongly", 2 for "Disagree", 3 for "Neutral", 4 for "Agree" and 5 for "Agree
Strongly"). This time, in addition to indicating to which category a respondent
1| Jeff Arodi
, LECTURE 1 AND 2 Introduction to Statistics
belongs, the numbers measure the degree of agreement with the statement and
tell us whether one respondent agrees more or less than another respondent.
However, since the ordinal scale has no units of measurement, we cannot say that
the difference between 1 and 2 (i.e. between disagreeing strongly and just
disagreeing) is the same as the difference between 4 and 5 (i.e. between agreeing
and agreeing strongly).
(b) Interval Scale
The interval scale has a constant unit of measurement, but an arbitrary zero point.
Good examples of interval scales are the Fahrenheit and Celsius temperature
scales. As these scales have different zero points (i.e. 0 degrees F is not the same
as 0 degrees C), it is not possible to form meaningful ratios. For example, although
we can say that 30 degrees C (86 degrees F) is hotter than 15 degrees C (59 degrees
F), we cannot say that it is twice as hot (as it clearly isn't in the Fahrenheit scale).
(d) Ratio Scale
The ratio scale has a constant unit of measurement and an absolute zero point. So
this is the scale used to measure values, lengths, weights and other characteristics
where there are well-defined units of measurement and where there is an absolute
zero where none of the characteristic is present. For example, in values measured
in pounds, we know (all too well) that a zero balance means no money. We can
also say that Sh 30 is twice as much as Sh15, and this would be true whatever
currency were used as the unit of measurement. Other examples of ratio scale
measurements include the average petrol consumption of a car, the number of
votes cast at an election, the percentage return on an investment, the profitability
of a company, and many others. The measurement scale used gives us one way of
distinguishing between different types of data. For example, a set of data may be
described as being "nominal scale", "ordinal scale", "interval scale" or "ratio scale"
data. More often, a simpler distinction is made between categorical data (which
includes all data measured using nominal or ordinal scales) and quantifiable data
(which includes all data measured using interval or ratio scales).
Variables and Data
Any characteristic on which observations can be made is called a variable or variate. For
example, height is a variable because observations taken are of the heights of a number of
people. Variables, and therefore the data which observations of them produce, can be
categorised in various ways:
(a) Quantitative and Qualitative Variables
Variables may be either quantitative or qualitative. Quantitative variables, to
which we shall restrict discussion here, are those for which observations are
numerical in nature. Qualitative variables have non-numeric observations, such as
colour of hair, although of course each possible non-numeric value may be
associated with a numeric frequency.
(b) Continuous and Discrete Variables
Variables may be either continuous or discrete. A continuous variable may take any
value between two stated limits (which may possibly be minus and plus infinity).
Height, for example, is a continuous variable, because a person's height may (with
appropriately accurate equipment) be measured to any minute fraction of a
millimetre.
2| Jeff Arodi
Statistics
Statistics" (preferably with a capital letter) to refer to the academic discipline concerned
with the collection, description, analysis and interpretation of numerical data. As such,
the subject of Statistics may be divided into two main categories:
(a) Descriptive Statistics
This is mainly concerned with collecting and summarising data, and presenting the
results in appropriate tables and charts. For example, companies collect and
summarise their financial data in tables (and occasionally charts) in their annual
reports, but there is no attempt to go "beyond the data".
(b) Statistical Inference
This is concerned with analysing data and then interpreting the results (attempting
to go "beyond the data"). The main way in which this is done is by collecting data
from a sample and then using the sample results to infer conclusions about the
population.
For example, prior to general elections in the Kenya and many other countries,
statisticians conduct opinion polls in which samples of potential voters are asked
which political party they intend to vote for. The sample proportions are then used
to predict the voting intentions of the entire population.
Of course, before any descriptive statistics can be calculated or any statistical inferences
made, appropriate data has to be collected. We will start the course, therefore, by seeing
how we collect data. This study unit looks at the various types of data, the main sources
of data and some of the numerous methods available to collect data.
B. MEASUREMENT SCALES AND TYPES OF DATA
Measurement Scales
Quantitative methods use quantitative data which consists of measurements of various
kinds. Quantitative data may be measured in one of four measurement scales, and it is
important to be aware of the measurement scale that applies to your data before
commencing any data description or analysis. The four measurement scales are:
(a) Nominal Scale
The nominal scale uses numbers simply to identify members of a group or category.
For example, in a questionnaire, respondents may be asked whether they are male
or female and the responses may be given number codes (say 0 for males and 1 for
females). Similarly, companies may be asked to indicate their ownership form and
again the responses may be given number codes (say 1 for public limited
companies, 2 for private limited companies, 3 for mutual organizations, etc.). In
these cases, the numbers simply indicate the group to which the respondents
belong and have no further arithmetic meaning.
(b) Ordinal Scale
The ordinal scale uses numbers to rank responses according to some criterion, but
has no unit of measurement. In this scale, numbers are used to represent "more
than" or "less than" measurements, such as preferences or rankings. For example, it
is common in questionnaires to ask respondents to indicate how much they agree
with a given statement and their responses can be given number codes (say 1 for
"Disagree Strongly", 2 for "Disagree", 3 for "Neutral", 4 for "Agree" and 5 for "Agree
Strongly"). This time, in addition to indicating to which category a respondent
1| Jeff Arodi
, LECTURE 1 AND 2 Introduction to Statistics
belongs, the numbers measure the degree of agreement with the statement and
tell us whether one respondent agrees more or less than another respondent.
However, since the ordinal scale has no units of measurement, we cannot say that
the difference between 1 and 2 (i.e. between disagreeing strongly and just
disagreeing) is the same as the difference between 4 and 5 (i.e. between agreeing
and agreeing strongly).
(b) Interval Scale
The interval scale has a constant unit of measurement, but an arbitrary zero point.
Good examples of interval scales are the Fahrenheit and Celsius temperature
scales. As these scales have different zero points (i.e. 0 degrees F is not the same
as 0 degrees C), it is not possible to form meaningful ratios. For example, although
we can say that 30 degrees C (86 degrees F) is hotter than 15 degrees C (59 degrees
F), we cannot say that it is twice as hot (as it clearly isn't in the Fahrenheit scale).
(d) Ratio Scale
The ratio scale has a constant unit of measurement and an absolute zero point. So
this is the scale used to measure values, lengths, weights and other characteristics
where there are well-defined units of measurement and where there is an absolute
zero where none of the characteristic is present. For example, in values measured
in pounds, we know (all too well) that a zero balance means no money. We can
also say that Sh 30 is twice as much as Sh15, and this would be true whatever
currency were used as the unit of measurement. Other examples of ratio scale
measurements include the average petrol consumption of a car, the number of
votes cast at an election, the percentage return on an investment, the profitability
of a company, and many others. The measurement scale used gives us one way of
distinguishing between different types of data. For example, a set of data may be
described as being "nominal scale", "ordinal scale", "interval scale" or "ratio scale"
data. More often, a simpler distinction is made between categorical data (which
includes all data measured using nominal or ordinal scales) and quantifiable data
(which includes all data measured using interval or ratio scales).
Variables and Data
Any characteristic on which observations can be made is called a variable or variate. For
example, height is a variable because observations taken are of the heights of a number of
people. Variables, and therefore the data which observations of them produce, can be
categorised in various ways:
(a) Quantitative and Qualitative Variables
Variables may be either quantitative or qualitative. Quantitative variables, to
which we shall restrict discussion here, are those for which observations are
numerical in nature. Qualitative variables have non-numeric observations, such as
colour of hair, although of course each possible non-numeric value may be
associated with a numeric frequency.
(b) Continuous and Discrete Variables
Variables may be either continuous or discrete. A continuous variable may take any
value between two stated limits (which may possibly be minus and plus infinity).
Height, for example, is a continuous variable, because a person's height may (with
appropriately accurate equipment) be measured to any minute fraction of a
millimetre.
2| Jeff Arodi
