One of the requirements for using the Binomial Distribution is that each trial must be independent. One consequence of this is that the Sampling must be With Replacement.

To illustrate this, let's say we are doing a study in a small lake to determine the Proportion of lake trout. Each trial consists of catching and identifying 1 fish. If it's a lake trout, we count 1. The population of the fish is finite. We don't know this, but let's say it's 100 total fish 70 lake trout and 30 other fish. 

Each time we catch a fish, we throw it back before catching another fish. This is called Sampling With Replacement. Then, the Proportion of lake trout is remains at 70%. And the Probability for any one trial is 70% for lake trout. 

If, on the other hand, we keep each fish we catch, then we are Sampling Without Replacement. Let's say that the first 5 fish which we catch (and keep) are lake trout. Then, there are now 95 fish in the lake, of which 65 are lake trout. The percentage of lake trout is now 65/95 =68.4%. This is a change from the original 70%.

So, we don't have the same Probability each time of catching a lake trout. Sampling Without Replacement has caused the trials to not be independent. So, we can't use the Binomial Distribution. We must use the Hypergeometric Distribution instead.

For more on the Binomial Distribution, see my YouTube video.

I just uploaded a new video.  It's the third in a playlist on Regression. To see the current status of my completed and planned videos, please visit the Videos page on this website.  

For more details on ANOVA, I have a 6-video playlist on YouTube.

I just uploaded a new video to YouTube: https://youtu.be/gGkRkDBlICU

For the latest status of my videos completed and planned, see the videos page on this  website.

There are a number of see-saws (aka "teeter-totters" or "totterboards") like this in statistics. Here, we see that, as the Probability of an Alpha Error goes down, the Probability of a Beta Error goes up. Likewise, as the Probability of an Alpha Error goes up, the Probability of a Beta Error goes down.

​This being statistics, it would not be confusing enough if there were just one name for a concept.  So, you may know Alpha and Beta Errors by different names:

• Alpha Error: false positive, type I error, error of the first kind
• Beta Error: false negative, type II error, error of the second kind

The following compare-and-contrast table should help explain the difference:

The see-saw effect is important when we are selecting a value for Alpha (α) as part of a Hypothesis test. Most commonly, α = 0.05 is selected. This gives us a 1 – 0.05 = 0.95  (95%) Probability of avoiding an Alpha Error.

Since the person performing the test is the one who gets to select the value for Alpha, why don't we always select α = 0.000001 or something like that?
​
The answer is, selecting a low value for Alpha comes at price. Reducing the risk of an Alpha Error increases the risk of a Beta Error, and vice versa.

There is an article in the book devoted to further comparing and contrasting these two types of errors. Some time in the future, I hope to get around to adding a video on the subject. (Currently working on a playlist of videos about Regression.) See the  videos page of this website for the latest status of videos completed and planned.

I just uploaded a new video. It's on Covariance, and it is the first in a playlist on Regression.  

​Most users of statistics are familiar with the F-test for Variances. But there is also a Chi-Square Test for the Variance. What's the difference?

​The F-test compares the Variances from 2 different Populations or Processes. It basically divides one Variance by the other and uses the appropriate F Distribution to determine whether there is a Statistically Significant difference.

If you're familiar with t-tests, the F-test is analogous to the 2-Sample t-test. The F-test is a Parametric test. It requires that the data from both the 2 Samples each be roughly Normal.

The following  compare-and-contrast table may help clarify these concepts:

​Chi-Square (like z, t, and F) is a Test Statistic. That is, it has an associated family of Probability Distributions.
 
The Chi-Square Test for the Variance compares the Variance from a Single Population or Process to a Variance that we specify. That specified Variance could be a target value, a historical value, or anything else.
 
Since there is only 1 Sample of data from the single Population or Process, the Chi-Square test is analogous to the 1-Sample t-test.
 
In contrast to the the F-test, the Chi-Square test is Nonparametric. It has no restrictions on the data. 

Videos: I have published the following relevant videos on my YouTube channel, "Statistics from A to Z"

• F Distribution:  https://youtu.be/w1TvaQgoNCY 
• Chi-Square -- the Test Statistic and Its Distributions: https://youtu.be/RJMNkzuxOA4   
• t: the Test Statistic and Its Distributions: youtu.be/3GCJU_RCgoM    

There are 3 categories of numerical properties which describe a Probability Distribution (e.g. the Normal or Binomial Distributions).

• Center: e.g. Mean
• Variation: e.g. Standard Deviation
• Shape: e.g. Skewness 

Skewness is a case in which common usage of a term is the opposite of statistical usage. If the average person saw the Distribution on the left, they would say that it's skewed to the right, because that is where the bulk of the curve is. However, in statistics, it's the opposite. The Skew is in the direction of the long tail.
​
If you can remember these drawings, think of "the tail wagging the dog."

New Video: Standard Error.
See the Videos pages of this website for a listing of available and planned videos.