In determining which Distribution to use in analyzing Discrete (Count) data, we need to know whether we are interested in Occurrences or Units.
Let's say we are inspecting shirts at the end of the manufacturing line. We may be interested in the number of defective Units – shirts, because any defective shirt is likely to be rejected by our customer. However, one defective shirt can contain more than one defect. So, we are also interested in the Count of individual defects – the Occurrences – because that tells us how much of a quality problem we have in our manufacturing process.
For example, if 1 shirt has 3 defects, that would be 3 Occurrences of a defect, but only 1 Unit counted as defective.
We would use the Poisson Distribution in analyzing Probabilities of Occurrences of defects. To analyze the Probability of Units, we could use the Binomial or the Hypergeometric Distribution.
There is an article in the book focusing on the Poisson Distribution. There is also a video, on my YouTube channel, Statistics from A to Z.
These are all terms used in Correlation and Linear Regression (Simple and Multiple). And some of these terms have several names. I don't know about you, but I get confused trying to keep them all straight. So I wrote this compare-and-contrast table, which should help.
First in a playlist on Statistical Tests. 5 Keys to Understanding and compare-and- contrast tables, help the viewer understand the 3 different types of parametric t-tests. https://youtu.be/ZJlrF_yfiPo. For a complete listing of available and planned videos, please see the Videos page on this website.
In Regression, we attempt to fit a line or curve to the data. Let's say we're doing Simple Linear Regression in which we are trying to fit a straight line to a set of (x,y) data.
We test a number of subjects with dosages from 0 to 3 pills. And we find a straight line relationship, y = 3x, between the number of pills (x) and a measure of health of the subjects. So, we can say this.
But we cannot make a statement like the following:
This is called extrapolating the conclusions of your Regression Model beyond the range of the data used to create it. There is no mathematical basis for doing that, and it can have negative consequences, as this little cartoon from my book illustrates.
In the graphs below, the dots are data points. In the graph on the left, it is clear that there is a linear correlation between the drug dosage (x) and the health outcome (y) for the range we tested, 0 to 3 pills. And we can interpolate between the measured points. For example, we might reasonably expect that 1.5 pills would yield a health outcome halfway between that of 1 pill and 2 pills.
For more on this and other aspects of Regression, you can see the YouTube videos in my playlist on Regression. (See my channel: Statistics from A to Z - Confusing Concepts Clarified.
This is the 9th and final video in my channel on Regression. Residuals represent the error in a Regression Model. That is, Residuals represent the Variation in the outcome Variable y, which is not explained by the Regression Model. Residuals must be analyze several ways to ensure that they are random, and that they do no represent the Variation caused by some unidentified x-factor.
See the videos page in this website for a listing of available and planned videos.
The Binomial Distribution is used with Count data. It displays the Probabilities of Count data from Binomial Experiments. In a Binomial Experiment,
There are many Binomial Distributions. Each one is defined by a pair of values for two Parameters, n and p. n is the number of trials, and p is the Probability of each trial.
The graphs below show the effect of varying n, while keeping the Probability the same at 50%. The Distribution retains its shape as n varies. But obviously, the Mean gets larger.
The effect of varying the Probability, p, is more dramatic.
For small values of p, the bulk of the Distribution is heavier on the left. However, as described in my post of July 25, 2018, statistics describes this as being skewed to the right, that is, having a positive skew. (The skew is in the direction of the long tail.) For large values of p, the skew is to the left, because the bulk of the Distribution is on the right.
New video: Simple Nonlinear Regression.
This is the 7th in a playlist on Regression. For a complete list of my available and planned videos, please see the Videos page on this website.
Variation is also known as "Variability", "Dispersion", "Spread", and "Scatter". (5 names for one thing is one more example why statistics is confusing.) Variation is 1 of 3 major categories of measures describing a Distribution or data set. The others are Center (aka "Central Tendency") with measures like Mean, Mode, and Median and Shape (with measures like Skew and Kurtosis). Variation measures how "spread out" the data is.
There are a number of different measures of Variation. This compare-and-contrast table shows the relative merits of each.
Andrew A. (Andy) Jawlik is the author of the book, Statistics from A to Z -- Confusing Concepts Clarified, published by Wiley.