Statistics is the study of collecting, analyzing, and interpreting data. It involves determining measures of central tendency like the mean, median, and mode to describe patterns in data. Forensic statistics applies these statistical techniques to scientific evidence used in legal cases. It aims to evaluate likelihoods in an unbiased way using likelihood ratios. Professor Aitken developed Bayesian methods for interpreting forensic evidence and determining optimal sample sizes, helping forensic scientists evaluate uncertainties and provide statistically sound evidence for trials.

2.  Statistics is a branch of mathematics dealing with the collection,
analysis, interpretation, presentation, and organization of data. In
applying statistics to, e.g., a scientific, industrial, or social
problem, it is conventional to begin with a statistical population or
a statistical model process to be studied. Populations can be
diverse topics such as "all people living in a country" or "every
atom composing a crystal". Statistics deals with all aspects of data
including the planning of data collection in terms of the design
of surveys and experiments.
 Some popular definitions are:
 Merriam-Webster dictionary defines statistics as "a branch of
mathematics dealing with the collection, analyze]".sis,
interpretation, and presentation of masses of numerical data"
 Statistician Sir Arthur Lyon Bowler defines statistics as
"Numerical statements of facts in any department of inquiry
placed in relation to each other]".

3.  In statistics, a central tendency (or measure of central tendency)
is a central or typical value for a probability distribution. It may
also be called a center or location of the distribution. Colloquially,
measures of central tendency are often called averages. The
term central tendency dates from the late 1920s.
 The most common measures of central tendency are the arithmetic
mean, the median and the mode. A central tendency can be
calculated for either a finite set of values or for a theoretical
distribution, such as the normal distribution. Occasionally authors
use central tendency to denote "the tendency of
quantitative data to cluster around some central value."
 The central tendency of a distribution is typically contrasted with
its dispersion or variability; dispersion and central tendency are the
often characterized properties of distributions. Analysts may
judge whether data has a strong or a weak central tendency based
on its dispersion.

4. In mathematics, mean has several different definitions
depending on the context.
In probability and statistics, mean and expected
value are used synonymously to refer to one measure
of the central tendency either of a probability
distribution or of the random variable characterized by
that distribution. In the case of a discrete probability
distribution of a random variable X, the mean is equal
to the sum over every possible value weighted by the
probability of that value; that is, it is computed by
taking the product of each possible value x of X and its
probability P(x), and then adding all these products
together, giving An analogous formula applies to the
case of a continuous probability distribution. Not
every probability distribution has a defined mean; see
the Cauchy distribution for an example. Moreover, for
some distributions the mean is infinite: for example,
when the probability of the value for n = 1, 2, 3, ....

5.  The median is the value separating the higher half of a
data sample, a population, or a probability distribution,
from the lower half. In simple terms, it may be thought of as
the "middle" value of a data set. For example, in the data set
{1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth number in the
sample. The median is a commonly used measure of the
properties of a data set in statistics and probability theory.
 The basic advantage of the median in describing data
compared to the mean (often simply described as the
"average") is that it is not skewed so much by extremely
large or small values, and so it may give a better idea of a
'typical' value. For example, in understanding statistics like
household income or assets which vary greatly, a mean may
be skewed by a small number of extremely high or low
values. Median income, for example, may be a better way to
suggest what a 'typical' income is.

6.  The mode is the value that appears most often in a set
of data. The mode of a discrete probability
distribution is the value x at which its probability mass
function takes its maximum value. In other words, it is
the value that is most likely to be sampled. The mode
of a continuous probability distribution is the
value x at which its probability density function has its
maximum value, so the mode is at the peak.

Like the statistical mean and median, the mode is a
way of expressing, in a (usually) single number,
important information about a random variable or
a population. The numerical value of the mode is the
same as that of the mean and median in a normal
distribution, and it may be very different in highly
skewed distributions.

7. In statistics, dispersion (also called variability, scatter,
or spread) is the extent to which a distribution is
stretched or squeezed.[ Common examples of
measures of statistical dispersion are
the variance, standard deviation, and interquartile
range.

8. Forensic statistics is the application of probability models and statistical
techniques to scientific evidence, such as DNA evidence, and the law. In
contrast to "everyday" statistics, to not engender bias or unduly draw
conclusions, forensic statisticians report likelihoods as likelihood ratios (LR).
This ratio of probabilities is then used by juries or judges to draw inferences
or conclusions and decide legal matters.
Computer programs have been implemented with forensic DNA statistics for
assessing the biological relationships between two or more people. Forensic
science uses several approaches for DNA statistics with computer programs
such as; match probability, exclusion probability, likelihood ratios, Bayesian
approaches, and paternity and kinship testing.
Although the precise origin of this term remains unclear, it is apparent that
the term was used in the 1980s and 1990s. Among the first forensic statistics
conferences were two held in 1991 and 1993.

10.  An understanding of the value of forensic evidence relies heavily on an
assessment of uncertainty.
 Imagine, for example, that fragments of glass found at a crime scene are
believed to come from a broken bottle found in possession of a suspect.
The chemical composition of the glass in the fragments and in the bottle is
analyzed. What is the value of similarities between the composition of the
two samples?
 In order to improve the evaluation of such evidence, Professor Aitken
and other researchers from the Maxwell Institute for Mathematical
Sciences developed new Bayesian statistical methods. These methods
enable forensic scientists worldwide to interpret their data reliably.
 The glass fragments can come from the bottle found in possession of the
suspect, or they may come from another bottle. To help the court estimate
the relative likelihood of these two possibilities, Professor Aitken and
collaborators calculate a so-called likelihood ratio (LR) that takes into
account variations within glass bottles and between multiple bottles, first
assuming the fragments came from the suspect’s bottle, and second
assuming they came from another bottle

11.  Transformative impact
 In another scenario, when large consignments of potentially incriminating
material are seized (such as pills suspected of containing illegal drugs, or
computer files suspected of containing illegal material), the police want to
estimate the proportion that is illicit.
 Examination of every item is time-consuming, costly and stressful.
Professor Aitken developed procedures for determining the optimal size
of samples that should be examined. Through a careful assessment of the
uncertainties associated with an examination of only a fraction of a
consignment, his research ensures that investigators can sample fewer
items and still provide evidence that is fit for purpose in a criminal trial.
 Professor Aitken’s sampling protocols have been widely adopted. They
have been recommended to forensic laboratories by the Crown Office in
Scotland and in guidelines by the United Nations Office on Drugs and
Crime.
 Sampling software based on Professor Aitken’s statistical methods is
available through the European Network of Forensic Science Institutes.
This software allows forensic scientists without a strong background in
statistics to benefit from cutting-edge Bayesian statistical methods.