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Given a dataset of When a sample is derived from a population, parameters of which observed values the sample can be calculated, and distribution (e.g., normal distribution, binary distribution, logistic distribution, etc.) they are known, its the estimates of parameters (e.g., mean and standard deviation, success probability and number of trials, location and scale, etc. ) can be obtainedthe population.
Given a dataset of When a sample is derived from a population, observations in the sample should follow the distribution God knows with the parameters God knows, which are impossible to know. And there are multiple possibility of sets of parameters which observed values data in the sample can follow, and different sets of parameters have different chances to exist. Those chances are ''likelihood''. A set of parameters can be followed by observed data in the sample with very low chance, another set of parameters can be followed by the sample with relatively high chance, and yet another set of parameters can be followed by the sample with the highest chance. As a natural sense, the highest chance of set of parameters, i.e., the most likely set of parameters (the parameters with the maximum ''likelihood'') is taken into account and is used to make the relevant equation. That is the maximum likelihood estimation method. On the contrary to that the above-mentioned (conditional) ''probability'' fixes the parameters (hypothesis) and varies observations (data), ''likelihood'' fixes observations (data) and distributionvaries parameters (hypothesis).
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==Standard deviation and Variance==
==Probability, Likelihood==
*[https://sebastianraschka.com/faq/docs/probability-vs-likelihood.html What is the difference between likelihood and probability?]
*[https://rikei-logistics.com/likelihood 確率と尤度の違いは何？]
Statistics is an attempt to estimate population through sample.
A population always follows some kind of distribution, i.e., follows some kind of probability distribution. But parameters of a population – e.g., mean and standard deviation in normal distribution, success probability and number of trials in binary distribution, location and scale in logistic distribution, etc. – are what God knows.
And each distribution has each relevant equation to describe its probability distribution, and the equation is derived from parameters.
===Probability===
But God knows the true parameters of the population, and parameter of the sample always have random error.
An equation relevant to each distribution is derived from the parameters of the sample, and a value the equation makes also has random error.
And that value the equation makes is ''probability''. ''Probability'' is the chance that a given observed data is included in the distribution. Or more specifically, it is conditional probability, because a given observed data (= condition) gives a probability of the existing of the data.
<!--
Chance that a value <math>x_i</math> in the sample is in a range of certain values (e.g., <math>x_i > n</math>) can also be calculated from the dataset. This chance is <math>probability</math>.
:<math>probability = P(x_i\text{ in a range}|\text{parameters})</math>
!-->
===Likelihood===
