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I have answered some statistical questions but all of them gave the distribution and stated whether it is normal or not. But imagine I have taken a completely new sample from a completely new population; so how do I determine the distribution of this sample and population?

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  • $\begingroup$ At the beginning of such a process, I like to look at the histogram and Q–Q (quantile-quantile) plot of the examined feature, which reveals the properties of the distribution of a random variable. $\endgroup$ – Sławomir Jarek Mar 14 at 23:36
  • $\begingroup$ I wonder why you mention the known distributions if they do not relate to the new sample? $\endgroup$ – Lerner Zhang Mar 14 at 23:57
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One thing that helps is to plot a histogram, which gives a visual clue as to outliers, skewness, and so forth. Some packages, e.g., Mathematica, have find distribution routines that test common distributions and mixtures of distributions to identify better fitting candidate distributions, e.g., see FindDistribution. Next, a priori knowledge about the problem type, e.g., the physics and/ or a mathematical proof (e.g., see What is the ratio of a N[0,1] and U[-1/2,1/2]?), may give strong indications as to which distribution(s) are more likely to be correct. A literature search on the topic of the data type is often revealing, and can give strong indications as to which distribution(s) are applicable. A study of data transformation, e.g. taking a log, exponential, reciprocal, square or square root and so on may be strongly suggested by the data or its origin.

There is no single answer, and in some cases, all of the above or combinations of the above may be in agreement. With experience it gets a lot easier to choose a distribution type, but there are times when the distribution is not easily identifiable, and in those cases, an empirical distribution may be useful.

A distribution of a random variable is an example of a density function, specifically, a probability density function, a pdf. (Some people also call a cumulative density function, a CDF, a "distribution".) However, density functions (whose area under the curve is 1, or 100%) may be deterministic, and have no relationship to probability. For example, a concentration density function, could be used to model drug elimination of a non-metabolized drug. That is important because sometimes density functions are created to model deterministic situations and can just as easily be applied to random variables.

One powerful method of creating new density functions is to convolve two more simple density functions. This can be done because the convolution of two density functions is itself a density function, see Why is the sum of two random variables a convolution?. This process, convolution, can be used to solve, or approximate, physical systems or random variables that otherwise seem uninterpretable, or are very poorly explained. For example, see Ans: What are the gamma-Pareto convolutions and how have they been used?, Ans: What is the convolution of a normal distribution with a gamma distribution?, How do gamma distributions add and what would that model?, and What distribution results in adding two Pareto distributions.

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Most often, distributions of real world do not strictly obey any ideal distribution.

You can compare your sample distribution to few relevant ideal distributions and see which gives the closest match. You can do this qualitatively by plotting histogram or quantitatively by using different techniques, such as the Kolmogorov-Smirnov statistic KS by Python.

For example, if your sample gives closest match with normal distribution, you can report that your sample (and thus the population) follows normal distribution with mean X and standard deviation Y.

If you are rigorous, you can do sensitivity analysis: what is level of significance for hypothesis that your sample distribution actually is equal to normal distribution?

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