FOXP2 is found in many vertebrates, where it plays an important role in mimicry in birds (such as birdsong) and echolocation in bats. FOXP2 is also required for the proper development of speech and language in humans. In humans, mutations in FOXP2 cause the severe speech and language disorder developmental verbal dyspraxia. Studies of the gene in mice and songbirds indicate that it is necessary for vocal imitation and the related motor learning. Outside the brain, FOXP2 has also been implicated in development of other tissues such as the lung and digestive system.
Initially identified in 1998 as the genetic cause of a speech disorder in a British family designated the KE family, FOXP2 was the first gene discovered to be associated with speech and language and was subsequently dubbed "the language gene". However, other genes are necessary for human language development, and a
2018 analysis confirmed that there was no evidence of recent positive evolutionary selection of FOXP2 in humans.
Structure and function
FOXP2 is expressed in the developing cerebellum and the hindbrain of the embryonic day-13.5 mouse. Allen Brain Atlases.
As a FOX protein, FOXP2 contains a forkhead-box domain. In addition, it contains a polyglutamine tract, a zinc finger and a leucine zipper.
The protein binds directly to DNA to control expression of other
proteins and controls their activity through the forkhead-box domain.
Only a few targeted genes have been identified, however researchers
believe that there could be up to hundreds of other genes targeted by
the FOXP2 gene. The forkhead box P2 protein is active in the brain and
other tissues before and after birth, and many studies show that it is
paramount for the growth of nerve cells and transmission between them.
The FOXP2 gene is also involved in synaptic plasticity, making it
imperative for learning and memory.
FOXP2 is required for proper brain and lung development. Knockout mice with only one functional copy of the FOXP2 gene have significantly reduced vocalizations as pups. Knockout mice with no functional copies of FOXP2 are runted, display abnormalities in brain regions such as the Purkinje layer, and die an average of 21 days after birth from inadequate lung development.
FOXP2 is expressed in many areas of the brain, including the basal ganglia and inferior frontal cortex, where it is essential for brain maturation and speech and language development. In mice, the gene was found to be twice as highly expressed in male
pups than female pups, which correlated with an almost double increase
in the number of vocalisations the male pups made when separated from
mothers. Conversely, in human children aged 4–5, the gene was found to
be 30% more expressed in the Broca's areas of female children. The researchers suggested that the gene is more active in "the more communicative sex".
Three amino acid substitutions distinguish the human FOXP2
protein from that found in mice, while two amino acid substitutions
distinguish the human FOXP2 protein from that found in chimpanzees, but only one of these changes is unique to humans. Evidence from genetically manipulated mice and human neuronal cell models suggests that these changes affect the neural functions of FOXP2.
Clinical significance
The FOXP2 gene has been implicated in several cognitive functions
including; general brain development, language, and synaptic plasticity.
The FOXP2 gene region acts as a transcription factor for the forkhead
box P2 protein. Transcription factors affect other regions, and the
forkhead box P2 protein has been suggested to also act as a
transcription factor for hundreds of genes. This prolific involvement
opens the possibility that the FOXP2 gene is much more extensive than
originally thought. Other targets of transcription have been researched without correlation
to FOXP2. Specifically, FOXP2 has been investigated in correlation with
autism and dyslexia, however with no mutation was discovered as the
cause. One well identified target is language. Although some research disagrees with this correlation, the majority of research shows that a mutated FOXP2 causes the observed production deficiency.
There is some evidence that the linguistic impairments associated
with a mutation of the FOXP2 gene are not simply the result of a
fundamental deficit in motor control. Brain imaging of affected
individuals indicates functional abnormalities in language-related
cortical and basal ganglia regions, demonstrating that the problems
extend beyond the motor system.
Mutations in FOXP2 are among several (26 genes plus 2 intergenic) loci which correlate to ADHD
diagnosis in adults – clinical ADHD is an umbrella label for a
heterogeneous group of genetic and neurological phenomena which may
result from FOXP2 mutations or other causes.
It is theorized that the translocation of the 7q31.2 region of the FOXP2 gene causes a severe language impairment called developmental verbal dyspraxia (DVD) or childhood apraxia of speech (CAS) So far this type of mutation has only been discovered in three families across the world including the original KE family. A missense mutation causing an arginine-to-histidine substitution
(R553H) in the DNA-binding domain is thought to be the abnormality in
KE. This would cause a normally basic residue to be fairly acidic and
highly reactive at the body's pH. A heterozygous nonsense mutation,
R328X variant, produces a truncated protein involved in speech and
language difficulties in one KE individual and two of their close family
members. R553H and R328X mutations also affected nuclear localization,
DNA-binding, and the transactivation (increased gene expression)
properties of FOXP2.
These individuals present with deletions, translocations, and
missense mutations. When tasked with repetition and verb generation,
these individuals with DVD/CAS had decreased activation in the putamen
and Broca's area in fMRI studies. These areas are commonly known as
areas of language function. This is one of the primary reasons that FOXP2 is known as a language
gene. They have delayed onset of speech, difficulty with articulation
including slurred speech, stuttering, and poor pronunciation, as well as
dyspraxia. It is believed that a major part of this speech deficit comes from an
inability to coordinate the movements necessary to produce normal speech
including mouth and tongue shaping. Additionally, there are more general impairments with the processing of the grammatical and linguistic aspects of speech. These findings suggest that the effects of FOXP2 are not limited to
motor control, as they include comprehension among other cognitive
language functions. General mild motor and cognitive deficits are noted
across the board. Clinically these patients can also have difficulty coughing, sneezing, or clearing their throats.
While FOXP2 has been proposed to play a critical role in the
development of speech and language, this view has been challenged by the
fact that the gene is also expressed in other mammals as well as birds
and fish that do not speak. It has also been proposed that the FOXP2 transcription-factor is not so
much a hypothetical 'language gene' but rather part of a regulatory
machinery related to externalization of speech.
Evolution
Human FOXP2 gene and evolutionary conservation is shown in a multiple alignment (at bottom of figure) in this image from the UCSC Genome Browser. Note that conservation tends to cluster around coding regions (exons).
The FOXP2 gene is highly conserved in mammals. The human gene differs from that in non-human primates by the substitution of two amino acids, a threonine to asparagine substitution at position 303 (T303N) and an asparagine to serine substitution at position 325 (N325S). In mice it differs from that of humans by three substitutions, and in zebra finch by seven amino acids. One of the two amino acid differences between human and chimps also arose independently in carnivores and bats. Similar FOXP2 proteins can be found in songbirds, fish, and reptiles such as alligators.
DNA sampling from Homo neanderthalensis bones indicates that their FOXP2 gene is a little different though largely similar to those of Homo sapiens (i.e. humans). Previous genetic analysis had suggested that the H. sapiens FOXP2 gene became fixed in the population around 125,000 years ago. Some researchers consider the Neanderthal findings to indicate that the
gene instead swept through the population over 260,000 years ago,
before our most recent common ancestor with the Neanderthals. Other researchers offer alternative explanations for how the H. sapiens version would have appeared in Neanderthals living 43,000 years ago.
According to a 2002 study, the FOXP2 gene showed indications of recent positive selection.Some researchers have speculated that positive selection is crucial for the evolution of language in humans. Others, however, were unable to find a clear association between
species with learned vocalizations and similar mutations in FOXP2. A 2018 analysis of a large sample of globally distributed genomes
confirmed there was no evidence of positive selection, suggesting that
the original signal of positive selection may be driven by sample
composition. Insertion of both human mutations into mice, whose version of FOXP2 otherwise differs from the human and chimpanzee
versions in only one additional base pair, causes changes in
vocalizations as well as other behavioral changes, such as a reduction
in exploratory tendencies, and a decrease in maze learning time. A
reduction in dopamine levels and changes in the morphology of certain
nerve cells are also observed.
FOXP2 downregulates CNTNAP2, a member of the neurexin family found in neurons. CNTNAP2 is associated with common forms of language impairment.
FOXP2 also downregulates SRPX2, the 'Sushi Repeat-containing Protein X-linked 2'. It directly reduces its expression, by binding to its gene's promoter. SRPX2 is involved in glutamatergicsynapse formation in the cerebral cortex
and is more highly expressed in childhood. SRPX2 appears to
specifically increase the number of glutamatergic synapses in the brain,
while leaving inhibitory GABAergic synapses unchanged and not affecting dendritic spine
length or shape. On the other hand, FOXP2's activity does reduce
dendritic spine length and shape, in addition to number, indicating it
has other regulatory roles in dendritic morphology.
In other animals
Chimpanzees
In chimpanzees, FOXP2 differs from the human version by two amino acids. A study in Germany sequenced FOXP2's complementary DNA in chimps and
other species to compare it with human complementary DNA in order to
find the specific changes in the sequence. FOXP2 was found to be functionally different in humans compared to
chimps. Since FOXP2 was also found to have an effect on other genes, its
effects on other genes is also being studied. Researchers deduced that there could also be further clinical
applications in the direction of these studies in regards to illnesses
that show effects on human language ability.
Mice
In a mouse FOXP2 gene knockouts, loss of both copies of the gene causes severe motor impairment related to cerebellar abnormalities and lack of ultrasonicvocalisations normally elicited when pups are removed from their mothers. These vocalizations have important communicative roles in
mother–offspring interactions. Loss of one copy was associated with
impairment of ultrasonic vocalisations and a modest developmental delay.
Male mice on encountering female mice produce complex ultrasonic
vocalisations that have characteristics of song. Mice that have the R552H point mutation carried by the KE family show cerebellar reduction and abnormal synaptic plasticity in striatal and cerebellar circuits.
Humanized FOXP2 mice display altered cortico-basal ganglia circuits. The human allele of the FOXP2 gene was transferred into the mouse embryos through homologous recombination
to create humanized FOXP2 mice. The human variant of FOXP2 also had an
effect on the exploratory behavior of the mice. In comparison to
knockout mice with one non-functional copy of FOXP2, the humanized mouse
model showed opposite effects when testing its effect on the levels of
dopamine, plasticity of synapses, patterns of expression in the striatum
and behavior that was exploratory in nature.
When FOXP2 expression was altered in mice, it affected many
different processes including the learning motor skills and the
plasticity of synapses. Additionally, FOXP2 is found more in the sixth layer of the cortex than in the fifth, and this is consistent with it having greater roles in sensory integration. FOXP2 was also found in the medial geniculate nucleus
of the mouse brain, which is the processing area that auditory inputs
must go through in the thalamus. It was found that its mutations play a
role in delaying the development of language learning. It was also found
to be highly expressed in the Purkinje cells and cerebellar nuclei of
the cortico-cerebellar circuits. High FOXP2 expression has also been
shown in the spiny neurons that express type 1 dopamine receptors in the striatum, substantia nigra, subthalamic nucleus and ventral tegmental area.
The negative effects of the mutations of FOXP2 in these brain regions
on motor abilities were shown in mice through tasks in lab studies. When
analyzing the brain circuitry in these cases, scientists found greater
levels of dopamine and decreased lengths of dendrites, which caused
defects in long-term depression, which is implicated in motor function learning and maintenance. Through EEG
studies, it was also found that these mice had increased levels of
activity in their striatum, which contributed to these results. There is
further evidence for mutations of targets of the FOXP2 gene shown to
have roles in schizophrenia, epilepsy, autism, bipolar disorder and intellectual disabilities.
Bats
FOXP2 has implications in the development of batecholocation. Contrary to apes and mice, FOXP2 is extremely diverse in echolocating bats. Twenty-two sequences of non-bat eutherian
mammals revealed a total number of 20 nonsynonymous mutations in
contrast to half that number of bat sequences, which showed 44
nonsynonymous mutations. All cetaceans share three amino acid substitutions, but no differences were found between echolocating toothed whales and non-echolocating baleen cetaceans. Within bats, however, amino acid variation correlated with different echolocating types. There is also evidence of ongoing intraspecific selection in at least one species of bat, in response to environmental changes.
Birds
In songbirds, FOXP2 most likely regulates genes involved in neuroplasticity. Gene knockdown of FOXP2 in area X of the basal ganglia in songbirds results in incomplete and inaccurate song imitation. Overexpression of FOXP2 was accomplished through injection of adeno-associated virus
serotype 1 (AAV1) into area X of the brain. This overexpression
produced similar effects to that of knockdown; juvenile zebra finch
birds were unable to accurately imitate their tutors. Similarly, in adult canaries, higher FOXP2 levels also correlate with song changes.
Levels of FOXP2 in adult zebra finches are significantly higher
when males direct their song to females than when they sing song in
other contexts. "Directed" singing refers to when a male is singing to a female usually
for a courtship display. "Undirected" singing occurs when for example, a
male sings when other males are present or is alone. Studies have found that FOXP2 levels vary depending on the social
context. When the birds were singing undirected song, there was a
decrease of FOXP2 expression in Area X. This downregulation was not
observed and FOXP2 levels remained stable in birds singing directed
song.
Differences between song-learning and non-song-learning birds have been shown to be caused by differences in FOXP2 gene expression, rather than differences in the amino acid sequence of the FOXP2 protein.
Zebrafish
In zebrafish, FOXP2 is expressed in the ventral and dorsal thalamus, telencephalon, diencephalon
where it likely plays a role in nervous system development. The
zebrafish FOXP2 gene has an 85% similarity to the human FOX2P ortholog.
History
FOXP2 and its gene were discovered as a result of investigations on an English family known as the KE family, half of whom (15 individuals across three generations) had a speech and language disorder called developmental verbal dyspraxia. Their case was studied at the Institute of Child Health of University College London. In 1990, Myrna Gopnik, Professor of Linguistics at McGill University,
reported that the disorder-affected KE family had severe speech
impediment with incomprehensible talk, largely characterized by
grammatical deficits. She hypothesized that the basis was not of learning or cognitive
disability, but due to genetic factors affecting mainly grammatical
ability. (Her hypothesis led to a popularised existence of "grammar gene" and a controversial notion of grammar-specific disorder.) In 1995, the University of Oxford and the Institute of Child Health researchers found that the disorder was purely genetic. Remarkably, the inheritance of the disorder from one generation to the next was consistent with autosomal dominant inheritance, i.e., mutation of only a single gene on an autosome (non-sex chromosome) acting in a dominant fashion. This is one of the few known examples of Mendelian
(monogenic) inheritance for a disorder affecting speech and language
skills, which typically have a complex basis involving multiple genetic
risk factors.
The FOXP2 gene is located on the long (q) arm of chromosome 7, at position 31.
In 1998, Oxford University geneticists Simon Fisher, Anthony Monaco, Cecilia S. L. Lai, Jane A. Hurst, and Faraneh Vargha-Khadem identified an autosomal dominant monogenic inheritance that is localized on a small region of chromosome 7 from DNA samples taken from the affected and unaffected members. The chromosomal region (locus) contained 70 genes. The locus was given the official name "SPCH1" (for
speech-and-language-disorder-1) by the Human Genome Nomenclature
committee. Mapping and sequencing of the chromosomal region was
performed with the aid of bacterial artificial chromosome clones. Around this time, the researchers identified an individual who was
unrelated to the KE family but had a similar type of speech and language
disorder. In this case, the child, known as CS, carried a chromosomal
rearrangement (a translocation)
in which part of chromosome 7 had become exchanged with part of
chromosome 5. The site of breakage of chromosome 7 was located within
the SPCH1 region.
In 2001, the team identified in CS that the mutation is in the middle of a protein-coding gene. Using a combination of bioinformatics and RNA analyses, they discovered that the gene codes for a novel protein belonging to the forkhead-box (FOX) group of transcription factors.
As such, it was assigned with the official name of FOXP2. When the
researchers sequenced the FOXP2 gene in the KE family, they found a heterozygouspoint mutation shared by all the affected individuals, but not in unaffected members of the family and other people. This mutation is due to an amino-acid substitution that inhibits the DNA-binding domain of the FOXP2 protein. Further screening of the gene identified multiple additional cases of FOXP2 disruption, including different point mutations and chromosomal rearrangements, providing evidence that damage to one copy of this gene is sufficient to derail speech and language development.
Statistics (from German: Statistik, orig. "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model
to be studied. Populations can be diverse groups of people or objects
such as "all people living in a country" or "every atom composing a
crystal". Statistics deals with every aspect of data, including the
planning of data collection in terms of the design of surveys and experiments.
When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey samples.
Representative sampling assures that inferences and conclusions can
reasonably extend from the sample to the population as a whole. An experimental study
involves taking measurements of the system under study, manipulating
the system, and then taking additional measurements using the same
procedure to determine if the manipulation has modified the values of
the measurements. In contrast, an observational study does not involve experimental manipulation.
Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences made using mathematical statistics employ the framework of probability theory, which deals with the analysis of random phenomena.
A standard statistical procedure involves the collection of data leading to a test of the relationship
between two statistical data sets, or a data set and synthetic data
drawn from an idealized model. A hypothesis is proposed for the
statistical relationship between the two data sets, an alternative to an idealized null hypothesis
of no relationship between two data sets. Rejecting or disproving the
null hypothesis is done using statistical tests that quantify the sense
in which the null can be proven false, given the data that are used in
the test. Working from a null hypothesis, two basic forms of error are
recognized: Type I errors (null hypothesis is rejected when it is in fact true, giving a "false positive") and Type II errors
(null hypothesis fails to be rejected when it is in fact false, giving a
"false negative"). Multiple problems have come to be associated with
this framework, ranging from obtaining a sufficient sample size to
specifying an adequate null hypothesis.
Statistical measurement processes are also prone to error with
regard to the data they generate. Many of these errors are classified as
random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence of missing data or censoring may result in biased estimates, and specific techniques have been developed to address these problems.
"Statistics is both the science of
uncertainty and the technology of extracting information from data." -
featured in the International Encyclopedia of Statistical Science.
Statistics is the discipline that deals with data,
facts and figures with which meaningful information is inferred. Data
may represent a numerical value, in form of quantitative data, or a
label, as with qualitative data. Data may be collected, presented and
summarised, in one of two methods called descriptive statistics. Two
elementary summaries of data, singularly called a statistic, are the
mean and dispersion. Whereas inferential statistics interprets data from
a population sample to induce statements and predictions about a
population.
Statistics is regarded as a body of science or a branch of mathematics. It is based on probability, a branch of mathematics that studies random
events. Statistics is considered the science of uncertainty. This
arises from the ways to cope with measurement and sampling error as well
as dealing with uncertanties in modelling. Although probability and
statistics were once paired together as a single subject, they are
conceptually distinct from one another. The former is based on deducing
answers to specific situations from a general theory of probability,
meanwhile statistics induces statements about a population based on a
data set. Statistics serves to bridge the gap between probability and
applied mathematical fields.
Some consider statistics to be a distinct mathematical science
rather than a branch of mathematics. While many scientific
investigations make use of data, statistics is generally concerned with
the use of data in the context of uncertainty and decision-making in the
face of uncertainty. Statistics is indexed at 62, a subclass of probability theory and
stochastic processes, in the Mathematics Subject Classification. Mathematical statistics is covered in the range 276-280 of subclass QA
(science > mathematics) in the Library of Congress Classification.
The word statistics ultimately comes from the Latin word Status,
meaning "situation" or "condition" in society, which in late Latin
adopted the meaning "state". Derived from this, political scientist
Gottfried Achenwall, coined the German word statistik (a summary of how
things stand). In 1770, the term entered the English language through
German and referred to the study of political arrangements. The term
gained its modern meaning in the 1790s in John Sinclair's works. In modern German, the term statistik is synonymous with mathematical
statistics. The term statistic, in singular form, is used to describe a
function that returns its value of the same name.
Statistical data
Data collection
Sampling
When full census data cannot be collected, statisticians collect sample data by developing specific experiment designs and survey samples. Statistics itself also provides tools for prediction and forecasting through statistical models.
To use a sample as a guide to an entire population, it is
important that it truly represents the overall population.
Representative sampling
ensures that inferences and conclusions can safely be extended from the
sample to the population as a whole. A major problem lies in
determining the extent to which the chosen sample is actually
representative. Statistics offers methods to estimate and correct for
any bias in the sample and data collection procedures. There are also
methods of experimental design that can lessen these issues at the
outset of a study, strengthening its ability to discern truths about the
population.
Sampling theory is part of the mathematical discipline of probability theory. Probability is used in mathematical statistics to study the sampling distributions of sample statistics and, more generally, the properties of statistical procedures.
The use of any statistical method is valid only when the system or
population under consideration satisfies the assumptions of the method.
The difference in point of view between classic probability theory and
sampling theory is, roughly, that probability theory starts with the
given parameters of a total population to deduce probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction—inductively inferring from samples to the parameters of a larger or total population.
Experimental and observational studies
A common goal for a statistical research project is to investigate causality, and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables. There are two major types of causal statistical studies: experimental studies and observational studies.
In both types of studies, the effect of differences of an independent
variable (or variables) on the behavior of the dependent variable are
observed. The difference between the two types lies in how the study is
actually conducted. Each can be very effective. An experimental study
involves taking measurements of the system under study, manipulating the
system, and then taking additional measurements with different levels
using the same procedure to determine if the manipulation has modified
the values of the measurements. In contrast, an observational study does
not involve experimental manipulation.
Instead, data are gathered and correlations between predictors and
response are investigated. While the tools of data analysis work best on
data from randomized studies, they are also applied to other kinds of data—like natural experiments and observational studies—for which a statistician would use a modified, more structured estimation method (e.g., difference in differences estimation and instrumental variables, among many others) that produce consistent estimators.
Experiments
The basic steps of a statistical experiment are:
Planning the research, including finding the number of
replicates of the study, using the following information: preliminary
estimates regarding the size of treatment effects, alternative hypotheses, and the estimated experimental variability.
Consideration of the selection of experimental subjects and the ethics
of research is necessary. Statisticians recommend that experiments
compare (at least) one new treatment with a standard treatment or
control, to allow an unbiased estimate of the difference in treatment
effects.
Further examining the data set in secondary analyses, to suggest new hypotheses for future study.
Documenting and presenting the results of the study.
Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line
workers. The researchers first measured the productivity in the plant,
then modified the illumination in an area of the plant and checked if
the changes in illumination affected productivity. It turned out that
productivity indeed improved (under the experimental conditions).
However, the study is heavily criticized today for errors in
experimental procedures, specifically for the lack of a control group and blindness. The Hawthorne effect
refers to finding that an outcome (in this case, worker productivity)
changed due to observation itself. Those in the Hawthorne study became
more productive not because the lighting was changed but because they
were being observed.[20]
Observational study
An example of an observational study is one that explores the
association between smoking and lung cancer. This type of study
typically uses a survey to collect observations about the area of
interest and then performs statistical analysis. In this case, the
researchers would collect observations of both smokers and non-smokers,
perhaps through a cohort study, and then look for the number of cases of lung cancer in each group. A case-control study
is another type of observational study in which people with and without
the outcome of interest (e.g. lung cancer) are invited to participate
and their exposure histories are collected.
Various attempts have been made to produce a taxonomy of levels of measurement. The psychophysicist Stanley Smith Stevens
defined nominal, ordinal, interval, and ratio scales. Nominal
measurements do not have meaningful rank order among values, and permit
any one-to-one (injective) transformation. Ordinal measurements have
imprecise differences between consecutive values, but have a meaningful
order to those values, and permit any order-preserving transformation.
Interval measurements have meaningful distances between measurements
defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in Celsius or Fahrenheit),
and permit any linear transformation. Ratio measurements have both a
meaningful zero value and the distances between different measurements
defined, and permit any rescaling transformation.
Because variables conforming only to nominal or ordinal
measurements cannot be reasonably measured numerically, sometimes they
are grouped together as categorical variables, whereas ratio and interval measurements are grouped together as quantitative variables, which can be either discrete or continuous, due to their numerical nature. Such distinctions can often be loosely correlated with data type in computer science, in that dichotomous categorical variables may be represented with the Boolean data type, polytomous categorical variables with arbitrarily assigned integers in the integral data type, and continuous variables with the real data type involving floating-point arithmetic.
But the mapping of computer science data types to statistical data
types depends on which categorization of the latter is being
implemented.
Other categorizations have been proposed. For example, Mosteller and Tukey (1977) distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998), van den Berg (1991).)
The issue of whether it is appropriate to apply different kinds
of statistical methods to data obtained from different kinds of
measurement procedures is complicated by issues concerning the
transformation of variables and the precise interpretation of research
questions. "The relationship between the data and what they describe
merely reflects the fact that certain kinds of statistical statements
may have truth values that are not invariant under some transformations.
Whether a transformation is sensible to contemplate depends on the
question one is trying to answer."
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features of a collection of information, while descriptive statistics in the mass noun sense is the process of using and analyzing those statistics. Descriptive statistics is distinguished from inferential statistics (or inductive statistics), in that descriptive statistics aims to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent.
Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics.
Descriptive statistics is solely concerned with properties of the
observed data, and it does not rest on the assumption that the data come
from a larger population.
A statistic is a random variable that is a function of the random sample, but not a function of unknown parameters.
The probability distribution of the statistic, though, may have unknown
parameters. Consider now a function of the unknown parameter: an estimator is a statistic used to estimate such function. Commonly used estimators include sample mean, unbiased sample variance and sample covariance.
A random variable that is a function of the random sample and of the unknown parameter, but whose probability distribution does not depend on the unknown parameter is called a pivotal quantity or pivot. Widely used pivots include the z-score, the chi square statistic and Student's t-value.
Between two estimators of a given parameter, the one with lower mean squared error is said to be more efficient. Furthermore, an estimator is said to be unbiased if its expected value is equal to the true value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at the limit to the true value of such parameter.
Other desirable properties for estimators include: UMVUE
estimators that have the lowest variance for all possible values of the
parameter to be estimated (this is usually an easier property to verify
than efficiency) and consistent estimators which converges in probability to the true value of such parameter.
This still leaves the question of how to obtain estimators in a
given situation and carry the computation, several methods have been
proposed: the method of moments, the maximum likelihood method, the least squares method and the more recent method of estimating equations.
Null hypothesis and alternative hypothesis
Interpretation of statistical information can often involve the development of a null hypothesis which is usually (but not necessarily) that no relationship exists among variables or that no change occurred over time. The alternative hypothesis is the name of the hypothesis that contradicts the null hypothesis.
The best illustration for a novice is the predicament encountered by a criminal trial. The null hypothesis, H0, asserts that the defendant is innocent, whereas the alternative hypothesis, H1, asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H0 (the status quo) stands in opposition to H1 and is maintained unless H1 is supported by evidence "beyond a reasonable doubt". However, "failure to reject H0"
in this case does not imply innocence, but merely that the evidence was
insufficient to convict. So the jury does not necessarily accept H0 but fails to reject H0. While one can not "prove" a null hypothesis, one can test how close it is to being true with a power test, which tests for type II errors.
Error
Working from a null hypothesis, two broad categories of error are recognized:
Type I errors where the null hypothesis is falsely rejected, giving a "false positive".
Type II errors
where the null hypothesis fails to be rejected and an actual difference
between populations is missed, giving a "false negative".
Standard deviation
refers to the extent to which individual observations in a sample
differ from a central value, such as the sample or population mean,
while Standard error refers to an estimate of difference between sample mean and population mean.
A statistical error is the amount by which an observation differs from its expected value. A residual
is the amount an observation differs from the value the estimator of
the expected value assumes on a given sample (also called prediction).
A least squares fit: in red the points to be fitted, in blue the fitted line.
Many statistical methods seek to minimize the residual sum of squares, and these are called "methods of least squares" in contrast to Least absolute deviations.
The latter gives equal weight to small and big errors, while the former
gives more weight to large errors. Residual sum of squares is also differentiable, which provides a handy property for doing regression. Least squares applied to linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called non-linear least squares.
Also in a linear regression model the non deterministic part of the
model is called error term, disturbance or more simply noise. Both
linear regression and non-linear regression are addressed in polynomial least squares,
which also describes the variance in a prediction of the dependent
variable (y axis) as a function of the independent variable (x axis) and
the deviations (errors, noise, disturbances) from the estimated
(fitted) curve.
Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias),
but other types of errors (e.g., blunder, such as when an analyst
reports incorrect units) can also be important. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems.
Confidence intervals: the red line is true value for the mean in this example, the blue lines are random confidence intervals for 100 realizations.
Most studies only sample part of a population, so results do not
fully represent the whole population. Any estimates obtained from the
sample only approximate the population value. Confidence intervals
allow statisticians to express how closely the sample estimate matches
the true value in the whole population. Often they are expressed as 95%
confidence intervals. Formally, a 95% confidence interval for a value is
a range where, if the sampling and analysis were repeated under the
same conditions (yielding a different dataset), the interval would
include the true (population) value in 95% of all possible cases. This
does not imply that the probability that the true value is in the confidence interval is 95%. From the frequentist perspective, such a claim does not even make sense, as the true value is not a random variable.
Either the true value is or is not within the given interval. However,
it is true that, before any data are sampled and given a plan for how
to construct the confidence interval, the probability is 95% that the
yet-to-be-calculated interval will cover the true value: at this point,
the limits of the interval are yet-to-be-observed random variables.
One approach that does yield an interval that can be interpreted as
having a given probability of containing the true value is to use a credible interval from Bayesian statistics: this approach depends on a different way of interpreting what is meant by "probability", that is as a Bayesian probability.
In principle, confidence intervals can be symmetrical or
asymmetrical. An interval can be asymmetrical because it works as a
lower or upper bound for a parameter (left-sided interval or right sided
interval), but it can also be asymmetrical if a two-sided interval is
built violating symmetry around the estimate. Sometimes the bounds of a
confidence interval are reached asymptotically, and these are used to
approximate the true bounds.
Statistics rarely give a simple Yes/No type answer to the question
under analysis. Interpretation often comes down to the level of
statistical significance applied to the numbers and often refers to the
probability of a value accurately rejecting the null hypothesis
(sometimes referred to as the p-value).
In this graph the black line is probability distribution for the test statistic, the critical region is the set of values to the right of the observed data point (observed value of the test statistic) and the p-value is represented by the green area.
The standard approach is to test a null hypothesis against an alternative hypothesis. A critical region
is the set of values of the estimator that leads to refuting the null
hypothesis. The probability of type I error is therefore the probability
that the estimator belongs to the critical region given that null
hypothesis is true (statistical significance)
and the probability of type II error is the probability that the
estimator does not belong to the critical region given that the
alternative hypothesis is true. The statistical power of a test is the probability that it correctly rejects the null hypothesis when the null hypothesis is false.
Referring to statistical significance does not necessarily mean
that the overall result is significant in real-world terms. For example,
in a large study of a drug it may be shown that the drug has a
statistically significant but very small beneficial effect, such that it
is unlikely to help the patient noticeably.
Although in principle the acceptable level of statistical significance may be subject to debate, the significance level
is the largest p-value that allows the test to reject the null
hypothesis. This test is logically equivalent to saying that the p-value
is the probability, assuming the null hypothesis is true, of observing a
result at least as extreme as the test statistic. Therefore, the smaller the significance level, the lower the probability of committing type I error.
A difference that is highly statistically significant can still
be of no practical significance, but it is possible to properly
formulate tests to account for this. One response involves going beyond
reporting only the significance level to include the p-value when reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate the size
or importance of the observed effect and can also seem to exaggerate
the importance of minor differences in large studies. A better and
increasingly common approach is to report confidence intervals. Although these are produced from the same calculations as those of hypothesis tests or p-values, they describe both the size of the effect and the uncertainty surrounding it.
Fallacy of the transposed conditional, aka prosecutor's fallacy: criticisms arise because the hypothesis testing approach forces one hypothesis (the null hypothesis)
to be favored, since what is being evaluated is the probability of the
observed result given the null hypothesis and not probability of the
null hypothesis given the observed result. An alternative to this
approach is offered by Bayesian inference, although it requires establishing a prior probability.
Rejecting the null hypothesis does not automatically prove the alternative hypothesis.
As everything in inferential statistics it relies on sample size, and therefore under fat tails p-values may be seriously mis-computed.
Examples
Some well-known statistical tests and procedures are:
For statistically modelling purposes, Bayesian models tend to be hierarchical, for example, one could model each YouTube channel as having video views distributed as a normal distribution with channel dependent mean and variance ,
while modeling the channel means as themselves coming from a normal
distribution representing the distribution of average video view counts
per channel, and the variances as coming from another distribution.
Exploratory data analysis (EDA) is an approach to analyzingdata sets to summarize their main characteristics, often with visual methods. A statistical model
can be used or not, but primarily EDA is for seeing what the data can
tell us beyond the formal modeling or hypothesis testing task.
Formal discussions on inference date back to the mathematicians and cryptographers of the Islamic Golden Age between the 8th and 13th centuries. Al-Khalil (717–786) wrote the Book of Cryptographic Messages, which contains one of the first uses of permutations and combinations, to list all possible Arabic words with and without vowels. Al-Kindi's Manuscript on Deciphering Cryptographic Messages gave a detailed description of how to use frequency analysis to decipher encrypted messages, providing an early example of statistical inference for decoding. Ibn Adlan (1187–1268) later made an important contribution on the use of sample size in frequency analysis.
Although the term statistic was introduced by the Italian scholar Girolamo Ghilini in 1589 with reference to a collection of facts and information about a state, it was the German Gottfried Achenwall in 1749 who started using the term as a collection of quantitative information, in the modern use for this science. The earliest writing containing statistics in Europe dates back to 1663, with the publication of Natural and Political Observations upon the Bills of Mortality by John Graunt.
Early applications of statistical thinking revolved around the
needs of states to base policy on demographic and economic data, hence
its stat- etymology.
The scope of the discipline of statistics broadened in the early 19th
century to include the collection and analysis of data in general.
Today, statistics is widely employed in government, business, and
natural and social sciences.
Carl Friedrich Gauss made major contributions to probabilistic methods leading to statistics.
The mathematical foundations of statistics developed from discussions concerning games of chance among mathematicians such as Gerolamo Cardano, Blaise Pascal, Pierre de Fermat, and Christiaan Huygens. Although the idea of probability was already examined in ancient and medieval law and philosophy (such as the work of Juan Caramuel), probability theory as a mathematical discipline only took shape at the very end of the 17th century, particularly in Jacob Bernoulli's posthumous work Ars Conjectandi. This was the first book where the realm of games of chance and the
realm of the probable (which concerned opinion, evidence, and argument)
were combined and submitted to mathematical analysis. The method of least squares was first described by Adrien-Marie Legendre in 1805, though Carl Friedrich Gauss presumably made use of it a decade earlier in 1795.
Karl Pearson, a founder of mathematical statistics
In the 1830s-1850s, "statistical offices" and national "statistical
societies" were founded in Europe and America, and in the mid-19th
century, the idea arose of "organized contacts between the statisticians
of different countries although informal contacts occurred earlier". In those days, the name "statistics" referred mainly to "matters of
state", and British statisticians were often called "statists".
The modern field of statistics emerged in the late 19th and early 20th century in three stages. The first wave, at the turn of the century, was led by the work of Francis Galton and Karl Pearson,
who transformed statistics into a rigorous mathematical discipline used
for analysis, not just in science, but in industry and politics as
well. Galton's contributions included introducing the concepts of standard deviation, correlation, regression analysis
and the application of these methods to the study of the variety of
human characteristics—height, weight and eyelash length among others. Pearson developed the Pearson product-moment correlation coefficient, defined as a product-moment, the method of moments for the fitting of distributions to samples and the Pearson distribution, among many other things. Galton and Pearson founded Biometrika as the first journal of mathematical statistics and biostatistics (then called biometry), and the latter founded the world's first university statistics department at University College London.
The final wave, which mainly saw the refinement and expansion of
earlier developments, emerged from the collaborative work between Egon Pearson and Jerzy Neyman in the 1930s. They introduced the concepts of "Type II" error, power of a test and confidence intervals.
Jerzy Neyman in 1934 showed that stratified random sampling was in
general a better method of estimation than purposive (quota) sampling.
Among the early attempts to measure national economic activity were those of William Petty in the 17th century. In the 20th century the uniform System of National Accounts was developed.
Today, statistical methods are applied in all fields that involve
decision making, for making accurate inferences from a collated body of
data and for making decisions in the face of uncertainty based on
statistical methodology. The use of modern computers
has expedited large-scale statistical computations and has also made
possible new methods that are impractical to perform manually.
Statistics continues to be an area of active research, for example on
the problem of how to analyze big data.
Applications
Applied statistics, theoretical statistics and mathematical statistics
Applied statistics, sometimes referred to as Statistical science, comprises descriptive statistics and the application of inferential statistics. Theoretical statistics concerns the logical arguments underlying justification of approaches to statistical inference, as well as encompassing mathematical statistics. Mathematical statistics includes not only the manipulation of probability distributions necessary for deriving results related to methods of estimation and inference, but also various aspects of computational statistics and the design of experiments.
Statistical consultants can help organizations and companies that do not have in-house expertise relevant to their particular questions.
Machine learning and data mining
Machine learning models are statistical and probabilistic models that capture patterns in the data through use of computational algorithms.
A typical statistics course covers descriptive statistics, probability, binomial and normal distributions, test of hypotheses and confidence intervals, linear regression, and correlation. Modern fundamental statistical courses for undergraduate students focus
on correct test selection, results interpretation, and use of free
statistics software.
The rapid and sustained increases in computing power starting from
the second half of the 20th century have had a substantial impact on the
practice of statistical science. Early statistical models were almost
always from the class of linear models, but powerful computers, coupled with suitable numerical algorithms, caused an increased interest in nonlinear models (such as neural networks) as well as the creation of new types, such as generalized linear models and multilevel models.
Increased computing power has also led to the growing popularity of computationally intensive methods based on resampling, such as permutation tests and the bootstrap, while techniques such as Gibbs sampling have made use of Bayesian models
more feasible. The computer revolution has implications for the future
of statistics with a new emphasis on "experimental" and "empirical"
statistics. A large number of both general and special purpose statistical software are now available. Examples of available software capable of complex statistical computation include programs such as Mathematica, SAS, SPSS, and R.
Statistics form a key basis tool in business and manufacturing as
well. It is used to understand measurement systems variability, control
processes (as in statistical process control or SPC), for summarizing data, and to make data-driven decisions.
Misuse of statistics
can produce subtle but serious errors in description and
interpretation—subtle in the sense that even experienced professionals
make such errors, and serious in the sense that they can lead to
devastating decision errors. For instance, social policy, medical
practice, and the reliability of structures like bridges all rely on the
proper use of statistics.
Even when statistical techniques are correctly applied, the
results can be difficult to interpret for those lacking expertise. The statistical significance
of a trend in the data—which measures the extent to which a trend could
be caused by random variation in the sample—may or may not agree with
an intuitive sense of its significance. The set of basic statistical
skills (and skepticism) that people need to deal with information in
their everyday lives properly is referred to as statistical literacy.
There is a general perception that statistical knowledge is all-too-frequently intentionally misused by finding ways to interpret only the data that are favorable to the presenter. A mistrust and misunderstanding of statistics is associated with the quotation, "There are three kinds of lies: lies, damned lies, and statistics". Misuse of statistics can be both inadvertent and intentional, and the book How to Lie with Statistics, by Darrell Huff,
outlines a range of considerations. In an attempt to shed light on the
use and misuse of statistics, reviews of statistical techniques used in
particular fields are conducted (e.g. Warne, Lazo, Ramos, and Ritter
(2012)).
Ways to avoid misuse of statistics include using proper diagrams and avoiding bias. Misuse can occur when conclusions are overgeneralized
and claimed to be representative of more than they really are, often by
either deliberately or unconsciously overlooking sampling bias. Bar graphs are arguably the easiest diagrams to use and understand, and
they can be made either by hand or with simple computer programs. Most people do not look for bias or errors, so they are not noticed.
Thus, people may often believe that something is true even if it is not
well represented. To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole. According to Huff, "The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism."
To assist in the understanding of statistics Huff proposed a series of questions to be asked in each case:
Who says so? (Do they have an axe to grind?)
How do they know? (Do they have the resources to know the facts?)
What's missing? (Do they give us a complete picture?)
Did someone change the subject? (Do they offer us the right answer to the wrong problem?)
Does it make sense? (Is their conclusion logical and consistent with what we already know?)
The confounding variable problem: X and Y may be correlated, not because there is causal relationship between them, but because both depend on a third variable Z. Z is called a confounding factor.
The concept of correlation is particularly noteworthy for the potential confusion it can cause. Statistical analysis of a data set
often reveals that two variables (properties) of the population under
consideration tend to vary together, as if they were connected. For
example, a study of annual income that also looks at age of death, might
find that poor people tend to have shorter lives than affluent people.
The two variables are said to be correlated; however, they may or may
not be the cause of one another. The correlation phenomena could be
caused by a third, previously unconsidered phenomenon, called a lurking
variable or confounding variable. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables.
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