Mathematical Statistics

Mathematical statistics is the application of Mathematics to Statistics, which was originally conceived as the science of the state — the collection and analysis of facts about a country: its economy, and, military, population, and so forth.

Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equation and measure-theoretic probability theory.

Scope

Statistics is used in many sectors such as psychology, geology, sociology, weather forecasting, probability and much more. The goal of statistics is to gain understanding from data it focuses on applications and hence, it is distinctively considered as a Mathematical science.

Methods

The methods involve collecting, summarizing, analyzing, and interpreting variable numerical data. Here are some of the methods provided below.

Data collection
Data summarization
Statistical analysis

Data

Data is a collection of facts, such as numbers, words, measurements, observations etc.

Types of Data-

Qualitative data- it is descriptive data.
- Example- She can run fast, He is thin.
Quantitative data- it is numerical information.
- Example- An Octopus is an Eight legged creature.

Types of quantitative data

Discrete data- has a particular fixed value. It can be counted
Continuous data- is not fixed but has a range of data. It can be measured.

Representation of Data-

	Bar Graph A Bar Graph represents grouped data with rectangular bars with lengths proportional to the values that they represent. The bars can be plotted vertically or horizontally.
	Pie Chart A type of graph in which a circle is divided into Sectors that each represent a proportion of the whole.
	Line graph The line chart is represented by a series of data-points connected with a straight line The series of data points are called ‘markers.’
	Pictograph A pictorial symbol for a word or phrase, i.e. showing data with the help of pictures. Such as Apple, Banana & Cherry can have different numbers, it is just a representation of data.
	Histogram A diagram consisting of rectangles whose area is proportional to the frequency of a variable and whose width is equal to the class interval.
	Frequency Distribution The frequency of a data value is often represented by “f.” A frequency table is constructed by arranging collected data values in ascending order of magnitude with their corresponding frequencies.

Formulas used

Sample Mean (x¯)	∑xn
Population Mean (μ)	∑xN
Sample Standard Deviation (s)	∑(x−x¯)2n−1−−−−−−√
Population Standard Deviation (σ)	σ=(x−μ)2N−−−−−√
Sample Variance (s2)	s2=∑(xi−x¯)n−1
Population Variance (σ2)	σ2=∑(xi–x¯)N
Range (R)	Largest data value – smallest data value