Fitting a Theoretical Distribution to Given Data

Basic Concepts

Theoretical distributions are mathematical functions that describe the probabilities of different outcomes.
Fitting a distribution to data involves finding the parameters of the distribution that best explain the data.

The choice of a suitable theoretical distribution to fit the given data depends on the nature of the data. Understanding the characteristics and patterns in the data is crucial.
Histograms and stem-and-leaf plots are useful tools for visualising the shape of the data and guiding the choice of distribution.
If the data is symmetrical, a normal distribution might be an appropriate fit. For positively skewed data, an exponential distribution could be an option.

One way to fit a distribution is the method of least squares, minimising the sum of the squares of the differences between the observed and theoretical values.
Another method is maximum likelihood estimation (MLE), which identifies the parameter values that make the observed data most probable.
Both of these methods require a knowledge of calculus and the ability to solve equations.

A theoretical distribution that has been fitted to data can be assessed by comparing observed values with the expected values from the distribution.
Chi-square tests are commonly used for this purpose, comparing observed and expected frequencies in different categories or bins. A low chi-square value suggests a good fit.
A P-value gives the probability that the differences between observed and expected values arose by chance. A low P-value (typically <0.05) suggests the theoretical distribution is a good fit.

Correlation does not imply causality - so while a theoretical distribution may fit data well, it does not necessarily mean there is a cause-effect relationship.
The process of fitting a theoretical distribution should involve iteration - using the results of an initial fit to inform a second (or further) fit to improve the distribution.
Remember to consider whether the model and its assumptions are reasonable and consistent with what is known about the process that generated the data.