Statistics | Types & Summary | A Level Maths Revision Notes

Statistics is the branch of mathematics used to collect, analyze, interpret, and present data. The purpose of studying statistics is to be able to develop critical and analytic thinking skills.

Statistics is further divided into two branches:

Descriptive statistics: It deals with describing a set of data graphically
Inference statistics: This obtains information about a large set of data or future outcomes from a smaller sample

Statistics is closely linked to probability theory. We can use statistics to work out probabilities, Probability basically tells us about a chance, that a certain event will occur.

Moving on, as far as advanced level statistics is concerned, we will go through both these branches mentioned above. The topics we will cover in the statistics revision articles are:

Topics and Brief description

Averages

Average can be described as the number that gives us a sense of a central tendency, or a number that is more representative of a set.

Skewness

Skewness measures the departure from symmetry.

If the mean > median it indicates that the distribution is positively skewed.
If the mean is < median it indicates that the distribution is negatively skewed.

Box & Whisker diagrams

A box and whiskers diagram is also known as box plot, it displays a summary of a set of data. Minimum, maximum, median, first quartile and third quartile, interquartile, upper limit and lower limit.

Permutations & Combinations

Both permutations and combinations are groups or arrangements of objects. With combinations, the order of the objects is insignificant, whereas in permutations the order of the objects makes a difference.

Probability

$Probability\quad =\quad \frac { favourable\quad outcomes }{ possible\quad outcomes }$

It is a measure of the likelihood that an event will happen.

Bayes Theorem

$P(\frac { B }{ A } )\quad =\quad \frac { P(B)\quad .\quad P(\frac { A }{ B } ) }{ P(\frac { B }{ A } )P(A)\quad +\quad P(B)\quad .\quad P(\frac { B }{ A } ) }$

Linear regression

Regression line helps us connect two or more variables together, its equation is y = ax + b.

Product moment correlation coefficient

Formula for correlation: $\frac { { S }_{ xy } }{ \sqrt { { S }_{ x }{ S }_{ y } } }$

Measures of dispersion

Measures the fluctuation/variation that is present in the data.

Expectation & variance

Expected value of a random variable:

$\mu \quad =\quad E(X)\quad =\quad \sum { x\quad .\quad P(X\quad =\quad x) }$

Var(X), is defined by:

${ \sigma }^{ 2 }\quad =\quad Var(X)\quad =\quad E({ X }^{ 2 })\quad -\quad E{ (X) }^{ 2 }$

Discrete random variables

Discrete random variable are random variable that can take on distinct and separate variable.

Normal distribution

The normal distribution is a theoretical distribution of values.

$z\quad =\quad \frac { x\quad -\quad \mu }{ \sigma }$ .

Central limit theorem

The Central Limit Theorem (CLT) basically tells us that the sampling distribution of the sample mean is, at least approximately, normally distributed.

Poisson distribution

The probability function of the poisson distribution is:

$P(X\quad =\quad x)\quad =\quad \frac { { e }^{ -\lambda }{ \lambda }^{ x } }{ x! }$ .

Uniform distribution

In uniform distribution we should know that random variable is a continuous random variable. Probability density function f(x) to give $\frac { x\quad -\quad a }{ b\quad -\quad a }$ .

Binomial distribution

The formula for binomial distribution is as follows:

$P(X\quad =\quad x)\quad =\quad { C }_{ x }^{ n }{ p }^{ x }{ q }^{ n-x }$

Geometric distribution

The geometric distribution are the trails needed to get the first success in repeated and independent binomial trial.

$P(X\quad =\quad x)\quad =\quad { q }^{ x-1 }p$

Histograms & Cumulative frequency

A histogram show the distribution of numerical data. Cumulative frequency is accumulation of the frequencies

Discrete uniform distribution

The values of a discrete random variable are obtained by counting, thus making it known as countable. Uniform distribution simply means that when all of the random variable occur with equal probability.

Continuous random variables

If [a, b] are the domain of the continuous random variable function f and f(x) > 0 then:

$\int _{ a }^{ b }{ f\left( x \right) } dx\quad =\quad 1$

Normal approximations

A normal approximation can be defined as a process where the shape of the binomial distribution is estimated by using the normal curve.

One & two tailed tests

A one-tailed test is a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both.

Sampling

Sampling is the selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population.

Estimators

It is a rule, method, or criterion for arriving at an estimate of the value of a parameter.

Confidence intervals

A confidence interval is a range of values we are fairly sure our true value lies in.

Hypothesis testing

Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter.

Random samples

Random samples is known as the collection of independent and identically distributed random variables such as ${ X }_{ 1 },\quad { X }_{ 2 },\quad { X }_{ 3 },\quad { X }_{ 4 }\quad ........\quad { X }_{ n }$ .