MATH10282: Introduction to Statistics
Supplementary Lecture Notes
1
,1 Introduction: What is Statistics?
Statistics is:
‘the science of learning from data, and of measuring, controlling, and communicating uncertainty;
and it thereby provides the navigation essential for controlling the course of scientific and societal
advances.’
Davidian, M. and Louis, T.A. (2012), Science.
http://dx.doi.org/10.1126/science.1218685
There are two basic forms: descriptive statistics and inferential statistics. In this course we will discuss both,
with inferential statistics being the major emphasis.
• Descriptive Statistics is primarily about summarizing a given data set through numerical summaries and
graphs, and can be used for exploratory analysis to visualize the information contained in the data and
suggest hypotheses etc.
It is useful and important. It has become more exciting nowadays with people regularly using fancy
interactive computer graphics to display numerical information (e.g. Hans Rosling’s visualisation of the
change in countries’ health and wealth over time – see Youtube).
• Inferential Statistics is concerned with methods for making conclusions about a population using infor-
mation from a sample, and assessing the reliability of, and uncertainty in, these conclusions.
This allows us to make judgements in the presence of uncertainty and variability, which is extremely
important in underpinning evidence-based decision making in science, government, business etc.
Many statistical analyses and calculations are easiest to perform using a computer. We will learn how to
use the statistical software R, which is freely available to download from http://r-project.org for use
on your own computer. A good introductory guide is ‘Introduction to R’ by Venables et al. (2006), which
can be downloaded as a PDF from the R project website, or accessed from the R software itself via the menu
(Help→Manuals).
To interact with R, we type commands into the console, or write script files which contain several commands
for longer analyses. These commands are written in the R computer programming language, whose syntax
is fairly easy to learn. In this way, we can perform mathematical and statistical calculations. R has many
existing built-in functions, and users are also able to create their own functions. The R software also has very
good graphical facilities, which can produce high quality statistical plots. Datasets for use in the R sessions
are available from the course website https://minerva.it.manchester.ac.uk/~saralees/intro.html You
can download these and store them for use in the lab sessions.
2 Populations and samples
A population is the collection of all individuals or items under consideration in the study. For a given
population there will typically be one or more variables in which we are interested. For example, consider the
following populations together with corresponding variables of interest:
(i) All adults in the UK who are eligible to vote; the variable of interest is the political party supported.
(ii) Car batteries of a particular type manufactured by a particular company; the variable of interest is the
lifetime of the battery before failure.
2
,(iii) All adult males working full-time in Manchester; the variable of interest is the person’s gross income.
(iv) All potential possible outcomes of a planned laboratory experiment; the variable of interest is the value
of a particular measurement.
In general, the variables of interest may be either qualitative or quantitative. Qualitative variables are
either nominal, e.g. gender or political party supported, or ordinal, e.g. a measurement of size grouped into
three categories: small, medium or large. Quantitative variables are either discrete, for example a count, or
continuous, such as the variables income and lifetime above.
We wish to make conclusions, or inferences, about the population characteristics of variables of interest.
One way to do so is to conduct a census, i.e. to collect data for each individual in the population. However
often this is not feasible, due to one or more of the following:
• It may be too expensive or time consuming to do so, e.g. (i)
• Testing may be destructive, e.g. (ii), and we need to have some products left to sell!
• The population may be purely conceptual, e.g. (iv)
Instead, we collect data only for a sample, i.e. a subset of the population. We then use the characteristics
of the sample to estimate the characteristics of the population. In order for this procedure to give a good
estimate, the sample must be representative of the population. Otherwise, if an unrepresentative or ‘biased’
sample is used the conclusions will be systematically incorrect.
Some examples of samples from populations are given below:
(i) In an opinion poll in May 2015, a sample of 1000 adults was obtained and asked which political party
they intended to vote for in the upcoming UK General Election on 7 May 2015. A summary of these
responses is:
Party Number of supporters
Conservative 369
Labour 314
Lib Dem 75
UKIP 118
Other 124
(ii) A random sample of 40 manufactured car batteries was taken from the production line, and their lifetimes
(in years) determined. The data are as follows, arranged in ascending order for convenience:
1.6, 1.9, 2.2, 2.5, 2.6, 2.6, 2.9, 3.0, 3.0, 3.1,
3.1, 3.1, 3.1, 3.2, 3.2, 3.2, 3.3, 3.3, 3.3, 3.4,
3.4, 3.4, 3.5, 3.5, 3.6, 3.7, 3.7, 3.7, 3.8, 3.8,
3.9, 3.9, 4.1, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.7
(iii) We could obtain a sample of 500 adult males working full-time in Manchester. The following table
summarizes a hypothetical data set of the annual incomes in thousands of pounds for such a sample.
3
, Interval Frequency Percentage
5 to 15 83 16.6
15 to 25 142 28.4
25 to 35 90 18.0
35 to 45 79 15.8
45 to 55 46 9.2
55 to 65 28 5.6
65 to 75 13 2.6
75 to 85 6 1.2
85 to 95 4 0.8
95 to 105 3 0.6
105 to 115 0 0.0
115 to 125 2 0.4
125 to 135 0 0.0
135 to 145 0 0.0
145 to 155 1 0.2
155 to 165 0 0.0
165 to 175 1 0.2
175 to 185 1 0.2
185 to 195 1 0.2
Totals 500 100.0
The intervals in the table are open on the left and closed on the right, e.g. the first row gives the count
of incomes in the range (5, 15].
2.1 Finite population sampling
In modern Statistics, the most common way of guaranteeing representativeness is to use a random sample
of size n chosen according to a probabilistic sampling rule. This probabilistic sampling is objective and
eliminates investigator bias. For a population of finite size N , the most common method is to use simple
random sampling. This takes two main forms: sampling without replacement and sampling with
replacement.
• Sampling without replacement: each of the N
n possible samples of n distinct individuals from the population
N −1
has equal probability of selection, n . No individual appears more than once in the sample.
This can be implemented by choosing individuals sequentially, one at a time, as follows. For i = 1, . . . , n:
Step 1. Select an individual at random with equal probability from the remaining population of size N −i+1
Step 2. Include the selected individual as the ith member of the sample, and remove the selected individual
from the population, leaving N − i individuals remaining.
The above steps are repeated until a sample of size n is obtained.
• Sampling with replacement: each individual may appear any number of times in the sample, leading to N n
possible samples. The probability of selecting any particular sample is N −n . This can be implemented using
a similar sequential algorithm to before, where instead in Step 2 the selected individual is not removed from
the population.
Example. Let v1 , . . . , vN denote the values of the variable X for the 1st, . . ., N th individuals in the population.
4
Supplementary Lecture Notes
1
,1 Introduction: What is Statistics?
Statistics is:
‘the science of learning from data, and of measuring, controlling, and communicating uncertainty;
and it thereby provides the navigation essential for controlling the course of scientific and societal
advances.’
Davidian, M. and Louis, T.A. (2012), Science.
http://dx.doi.org/10.1126/science.1218685
There are two basic forms: descriptive statistics and inferential statistics. In this course we will discuss both,
with inferential statistics being the major emphasis.
• Descriptive Statistics is primarily about summarizing a given data set through numerical summaries and
graphs, and can be used for exploratory analysis to visualize the information contained in the data and
suggest hypotheses etc.
It is useful and important. It has become more exciting nowadays with people regularly using fancy
interactive computer graphics to display numerical information (e.g. Hans Rosling’s visualisation of the
change in countries’ health and wealth over time – see Youtube).
• Inferential Statistics is concerned with methods for making conclusions about a population using infor-
mation from a sample, and assessing the reliability of, and uncertainty in, these conclusions.
This allows us to make judgements in the presence of uncertainty and variability, which is extremely
important in underpinning evidence-based decision making in science, government, business etc.
Many statistical analyses and calculations are easiest to perform using a computer. We will learn how to
use the statistical software R, which is freely available to download from http://r-project.org for use
on your own computer. A good introductory guide is ‘Introduction to R’ by Venables et al. (2006), which
can be downloaded as a PDF from the R project website, or accessed from the R software itself via the menu
(Help→Manuals).
To interact with R, we type commands into the console, or write script files which contain several commands
for longer analyses. These commands are written in the R computer programming language, whose syntax
is fairly easy to learn. In this way, we can perform mathematical and statistical calculations. R has many
existing built-in functions, and users are also able to create their own functions. The R software also has very
good graphical facilities, which can produce high quality statistical plots. Datasets for use in the R sessions
are available from the course website https://minerva.it.manchester.ac.uk/~saralees/intro.html You
can download these and store them for use in the lab sessions.
2 Populations and samples
A population is the collection of all individuals or items under consideration in the study. For a given
population there will typically be one or more variables in which we are interested. For example, consider the
following populations together with corresponding variables of interest:
(i) All adults in the UK who are eligible to vote; the variable of interest is the political party supported.
(ii) Car batteries of a particular type manufactured by a particular company; the variable of interest is the
lifetime of the battery before failure.
2
,(iii) All adult males working full-time in Manchester; the variable of interest is the person’s gross income.
(iv) All potential possible outcomes of a planned laboratory experiment; the variable of interest is the value
of a particular measurement.
In general, the variables of interest may be either qualitative or quantitative. Qualitative variables are
either nominal, e.g. gender or political party supported, or ordinal, e.g. a measurement of size grouped into
three categories: small, medium or large. Quantitative variables are either discrete, for example a count, or
continuous, such as the variables income and lifetime above.
We wish to make conclusions, or inferences, about the population characteristics of variables of interest.
One way to do so is to conduct a census, i.e. to collect data for each individual in the population. However
often this is not feasible, due to one or more of the following:
• It may be too expensive or time consuming to do so, e.g. (i)
• Testing may be destructive, e.g. (ii), and we need to have some products left to sell!
• The population may be purely conceptual, e.g. (iv)
Instead, we collect data only for a sample, i.e. a subset of the population. We then use the characteristics
of the sample to estimate the characteristics of the population. In order for this procedure to give a good
estimate, the sample must be representative of the population. Otherwise, if an unrepresentative or ‘biased’
sample is used the conclusions will be systematically incorrect.
Some examples of samples from populations are given below:
(i) In an opinion poll in May 2015, a sample of 1000 adults was obtained and asked which political party
they intended to vote for in the upcoming UK General Election on 7 May 2015. A summary of these
responses is:
Party Number of supporters
Conservative 369
Labour 314
Lib Dem 75
UKIP 118
Other 124
(ii) A random sample of 40 manufactured car batteries was taken from the production line, and their lifetimes
(in years) determined. The data are as follows, arranged in ascending order for convenience:
1.6, 1.9, 2.2, 2.5, 2.6, 2.6, 2.9, 3.0, 3.0, 3.1,
3.1, 3.1, 3.1, 3.2, 3.2, 3.2, 3.3, 3.3, 3.3, 3.4,
3.4, 3.4, 3.5, 3.5, 3.6, 3.7, 3.7, 3.7, 3.8, 3.8,
3.9, 3.9, 4.1, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 4.7
(iii) We could obtain a sample of 500 adult males working full-time in Manchester. The following table
summarizes a hypothetical data set of the annual incomes in thousands of pounds for such a sample.
3
, Interval Frequency Percentage
5 to 15 83 16.6
15 to 25 142 28.4
25 to 35 90 18.0
35 to 45 79 15.8
45 to 55 46 9.2
55 to 65 28 5.6
65 to 75 13 2.6
75 to 85 6 1.2
85 to 95 4 0.8
95 to 105 3 0.6
105 to 115 0 0.0
115 to 125 2 0.4
125 to 135 0 0.0
135 to 145 0 0.0
145 to 155 1 0.2
155 to 165 0 0.0
165 to 175 1 0.2
175 to 185 1 0.2
185 to 195 1 0.2
Totals 500 100.0
The intervals in the table are open on the left and closed on the right, e.g. the first row gives the count
of incomes in the range (5, 15].
2.1 Finite population sampling
In modern Statistics, the most common way of guaranteeing representativeness is to use a random sample
of size n chosen according to a probabilistic sampling rule. This probabilistic sampling is objective and
eliminates investigator bias. For a population of finite size N , the most common method is to use simple
random sampling. This takes two main forms: sampling without replacement and sampling with
replacement.
• Sampling without replacement: each of the N
n possible samples of n distinct individuals from the population
N −1
has equal probability of selection, n . No individual appears more than once in the sample.
This can be implemented by choosing individuals sequentially, one at a time, as follows. For i = 1, . . . , n:
Step 1. Select an individual at random with equal probability from the remaining population of size N −i+1
Step 2. Include the selected individual as the ith member of the sample, and remove the selected individual
from the population, leaving N − i individuals remaining.
The above steps are repeated until a sample of size n is obtained.
• Sampling with replacement: each individual may appear any number of times in the sample, leading to N n
possible samples. The probability of selecting any particular sample is N −n . This can be implemented using
a similar sequential algorithm to before, where instead in Step 2 the selected individual is not removed from
the population.
Example. Let v1 , . . . , vN denote the values of the variable X for the 1st, . . ., N th individuals in the population.
4
