MATH1024: Introduction toProbability
and Statistics
Prof Sujit Sahu
Edition: Autumn 2020
2
Contents
,Chapter 1
Introduction to Statistics
• Lecture 1: What is statistics?
• Early and modern definitions
• The word statistics has its roots in the Latin word status which means the state, and in
the middle of the 18th century was intended to mean:
collection, processing and use of data by the state.
• With the rapid industrialization of Europe in the first half of the 19th century, statistics
became established as a discipline. This led to the formation of the Royal Statistical
Society, the premier professional association of statisticians in the UK and also world-
wide, in 1834.
• During this 19th century growth period, statistics acquired a new meaning as the
interpre- tation of data or methods of extracting information from data for decision
making. Thus statistics has its modern meaning as the methods for:
collection, analysis and interpretation of data.
• Indeed, the Oxford English Dictionary defines statistics as: “The practice or science of col-
lecting and analysing numerical data in large quantities, especially for the purpose of
inferringproportions in a whole from those in a representative sample.”
• Note that the word ‘state’ has gone from its definition. Instead, statistical methods are
now essential for everyone wanting to answer questions using data.
, For example, will it rain tomorrow? Does eating red meat make us live longer? Is smoking
harmful during pregnancy? Is the new shampoo better than the old? Will the UK economy get
better after Brexit? At a more personal level: What degree classification will I get at graduation?
How long will I live for? What prospects do I have in the career I have chosen? How do I invest
my money to maximise the return? Will the stock market crash tomorrow?
9
• Uncertainty: the main obstacle to decision making
Uncertainty is the only certainty!
The main obstacle to answering the types of questions above is uncertainty, which means lack
of one-to-one correspondence between cause and effect. For example, having a diet of (well-
cooked) red meat for a period of time is not going to kill me immediately. The effect of smoking
during pregnancy is difficult to judge because of the presence of other factors, e.g. diet and
lifestyle; such effects will not be known for a long time, e.g. at least until the birth. Thus it
seems:
• Statistics tames uncertainty
• It is clear that we may never be able to get to the bottom of every case to learn the full
truth and so will have to make a decision under uncertainty; thus mistakes cannot be
avoided!
• If mistakes cannot be avoided, it is better to know how often we make mistakes (which
provides knowledge of the amount of uncertainty) by following a particular rule of
decision making.
• Such knowledge could be put to use in finding a rule of decision making which does
not betray us too often, or which minimises the frequency of wrong decisions, or
which minimises theloss due to wrong decisions.
Thus we have the equation:
Usable knowledge
Knowledge of the extent ofuncertainty in it
Uncertain knowledge
+ =
Researchers often make guesses about scientific quantities. For example, try to guess my
age: 65 or 45? These predictions are meaningless without the associated uncertainties. Instead,
appropriate data collection and correct application of statistical methods may enable us to
make statements like: I am 97% certain that the correct age is between 47 and 54 years.
Remember, “to guess is cheap, to guess incorrectly is expensive” – old Chinese proverb.
• Why should I study statistics as part of my degree?
, • Studying statistics will equip you with the basic skills in data analysis and doing science
with data.
• A decent level of statistical knowledge is required no matter what branch of
mathematics, engineering, science and social science you will be studying.
• Learning statistical theories gives you the opportunity to practice your deductive
mathemat- ical skills on real life problems. In this way, you will improve at mathematical
methods while studying statistical methods.
“All knowledge is, in final analysis, history.
All sciences are, in the abstract, mathematics.
All judgements are, in their rationale, statistics.”
Prof. C. R. Rao
• Lie, Damn Lie and Statistics?
Sometimes people say, “you can prove anything in statistics!” and many such jokes. Such
remarks bear testimony to the fact that often statistics and statistical methods are miss-quoted
without proper verification and robust justification. This is even more important in this year of
the global pandemic as everyday we are showered with a deluge of numbers. The front and
the back cover of this booklet plot two pandemic related diagrams that we plan to discuss as
we learn different topics in this module.
Returning to the criticisms of statistics, admittedly and regretfully, statistics can be very
much miss-used and miss-interpreted. However, we statisticians argue:
• Every number is guilty unless proved innocent.
• Figures won’t lie, but liars can figure!
Hence, although people may miss-use the tools of statistics, it is our duty to learn and sharpen
the those to develop scientifically robust and strong arguments.
As discussed before statistical methods are only viable tool whenever there is uncertainty
in decision making. In scientific investigations, statistics is an inevitable instrument in search of
truth when uncertainty cannot be totally removed from decision making. Off-course, a
statistical method may not yield the best predictions in a very particular situation, but a
systematic and robust application of statistical methods will eventually win over pure guesses.
For example, statistical methods prove that cigarette smoking is bad for human health.
• What’s in this module?
• Chapter 1: We will start with the basic statistics used in everyday life, e.g. mean, median,
mode, standard deviation, etc. Statistical analysis and report writing will be discussed.
We will also learn how to explore data using graphical methods.
• For this we will use the R statistical package. R is freely available to download.
Search download R or go to: https://cran.r-project.org/. We will use it as a
calculator and also as a graphics package to explore data, perform statistical
analysis, illustrate theorems and calculate probabilities. You do not need to learn
and Statistics
Prof Sujit Sahu
Edition: Autumn 2020
2
Contents
,Chapter 1
Introduction to Statistics
• Lecture 1: What is statistics?
• Early and modern definitions
• The word statistics has its roots in the Latin word status which means the state, and in
the middle of the 18th century was intended to mean:
collection, processing and use of data by the state.
• With the rapid industrialization of Europe in the first half of the 19th century, statistics
became established as a discipline. This led to the formation of the Royal Statistical
Society, the premier professional association of statisticians in the UK and also world-
wide, in 1834.
• During this 19th century growth period, statistics acquired a new meaning as the
interpre- tation of data or methods of extracting information from data for decision
making. Thus statistics has its modern meaning as the methods for:
collection, analysis and interpretation of data.
• Indeed, the Oxford English Dictionary defines statistics as: “The practice or science of col-
lecting and analysing numerical data in large quantities, especially for the purpose of
inferringproportions in a whole from those in a representative sample.”
• Note that the word ‘state’ has gone from its definition. Instead, statistical methods are
now essential for everyone wanting to answer questions using data.
, For example, will it rain tomorrow? Does eating red meat make us live longer? Is smoking
harmful during pregnancy? Is the new shampoo better than the old? Will the UK economy get
better after Brexit? At a more personal level: What degree classification will I get at graduation?
How long will I live for? What prospects do I have in the career I have chosen? How do I invest
my money to maximise the return? Will the stock market crash tomorrow?
9
• Uncertainty: the main obstacle to decision making
Uncertainty is the only certainty!
The main obstacle to answering the types of questions above is uncertainty, which means lack
of one-to-one correspondence between cause and effect. For example, having a diet of (well-
cooked) red meat for a period of time is not going to kill me immediately. The effect of smoking
during pregnancy is difficult to judge because of the presence of other factors, e.g. diet and
lifestyle; such effects will not be known for a long time, e.g. at least until the birth. Thus it
seems:
• Statistics tames uncertainty
• It is clear that we may never be able to get to the bottom of every case to learn the full
truth and so will have to make a decision under uncertainty; thus mistakes cannot be
avoided!
• If mistakes cannot be avoided, it is better to know how often we make mistakes (which
provides knowledge of the amount of uncertainty) by following a particular rule of
decision making.
• Such knowledge could be put to use in finding a rule of decision making which does
not betray us too often, or which minimises the frequency of wrong decisions, or
which minimises theloss due to wrong decisions.
Thus we have the equation:
Usable knowledge
Knowledge of the extent ofuncertainty in it
Uncertain knowledge
+ =
Researchers often make guesses about scientific quantities. For example, try to guess my
age: 65 or 45? These predictions are meaningless without the associated uncertainties. Instead,
appropriate data collection and correct application of statistical methods may enable us to
make statements like: I am 97% certain that the correct age is between 47 and 54 years.
Remember, “to guess is cheap, to guess incorrectly is expensive” – old Chinese proverb.
• Why should I study statistics as part of my degree?
, • Studying statistics will equip you with the basic skills in data analysis and doing science
with data.
• A decent level of statistical knowledge is required no matter what branch of
mathematics, engineering, science and social science you will be studying.
• Learning statistical theories gives you the opportunity to practice your deductive
mathemat- ical skills on real life problems. In this way, you will improve at mathematical
methods while studying statistical methods.
“All knowledge is, in final analysis, history.
All sciences are, in the abstract, mathematics.
All judgements are, in their rationale, statistics.”
Prof. C. R. Rao
• Lie, Damn Lie and Statistics?
Sometimes people say, “you can prove anything in statistics!” and many such jokes. Such
remarks bear testimony to the fact that often statistics and statistical methods are miss-quoted
without proper verification and robust justification. This is even more important in this year of
the global pandemic as everyday we are showered with a deluge of numbers. The front and
the back cover of this booklet plot two pandemic related diagrams that we plan to discuss as
we learn different topics in this module.
Returning to the criticisms of statistics, admittedly and regretfully, statistics can be very
much miss-used and miss-interpreted. However, we statisticians argue:
• Every number is guilty unless proved innocent.
• Figures won’t lie, but liars can figure!
Hence, although people may miss-use the tools of statistics, it is our duty to learn and sharpen
the those to develop scientifically robust and strong arguments.
As discussed before statistical methods are only viable tool whenever there is uncertainty
in decision making. In scientific investigations, statistics is an inevitable instrument in search of
truth when uncertainty cannot be totally removed from decision making. Off-course, a
statistical method may not yield the best predictions in a very particular situation, but a
systematic and robust application of statistical methods will eventually win over pure guesses.
For example, statistical methods prove that cigarette smoking is bad for human health.
• What’s in this module?
• Chapter 1: We will start with the basic statistics used in everyday life, e.g. mean, median,
mode, standard deviation, etc. Statistical analysis and report writing will be discussed.
We will also learn how to explore data using graphical methods.
• For this we will use the R statistical package. R is freely available to download.
Search download R or go to: https://cran.r-project.org/. We will use it as a
calculator and also as a graphics package to explore data, perform statistical
analysis, illustrate theorems and calculate probabilities. You do not need to learn
