All other things being equal, an increase in Sample Size (n) reduces all types of Sampling Errors, including Alpha and Beta Errors and the Margin of Error.
A Sampling "Error" is not a mistake. It is simply the reduction in accuracy to be expected when one makes an estimate based on a portion – a Sample – of the data in Population or Process. There are several types of Sampling Error.
Two types of Sampling Errors are described in terms of their Probabilities:
All three types of Sampling Error are reduced when the Sample Size is increased.
This makes intuitive sense, because a very small Sample is more likely to not be a good representative of the properties of the larger Population or Process. But, the values of Statistics calculated from a much larger Sample are likely to be much closer to the values of the corresponding Population or Process Parameters
For more on the statistical concepts mentioned here (p, β, MOE, Confidence Intervals, Statistical Errors, Samples and Sampling), please see my book or my YouTube channel -- both are titled Statistics from A to Z -- Confusing Concepts Clarified.
All processes have variation. A process can be said to be "under control", "stable", or "predictable" if the variation is
Such Variation is called Common Cause Variation; it is like random "noise" within an under-control process. Variation which is not Common Cause is called Special Cause Variation. It is a signal that factors outside the process are affecting it.
Any Special Cause Variation must be eliminated before one can attempt to narrow the range of Common Cause Variation.
Until we eliminate Special Cause Variation, we don't have a process that we can improve. There are factors outside the process which affect it, and that changes the actual process that is happening in ways that we don't know.
Once we know that we have Special Cause Variation, we can use various Root Cause Analysis methods to identify the Special Cause, so that we can eliminate it. Only then can we use process/ quality improvement methods like Lean Six Sigma to try to reduce the Common Cause Variation.
Here are some examples of Special Causes of Variation:
Here is an example of a Control Chart. Each point is the Mean of a small Sample of data. The Upper Control Limit (UCL) and the Lower Control Limit (LCL) are usually set at 3 Standard Deviations from the Center Line.
We see that there is one anomalous Sample Mean outside the Control Limits. This is due to Special Cause Variation. So, we need to do some root cause analysis to determine what caused that. And we need to make changes to eliminate it, before we can try to narrow the range of the Control Limits.
In addition to being within Control Chart limits, the data must be random. There are a number of Run Rules which describe patterns which are not random. Some patterns are not always easy to spot by eyeballing charts. Fortunately, the same software which produces Control Charts will usually also identify patterns described by the Run Rules.
Here are some common patterns which indicate non-random (Special Cause) Variation. A Sigma is a Standard Deviation.
Reproduced by permission of John Wiley and Sons, Inc.
from the book, Statistics from A to Z – Confusing Concepts Clarified
I just uploaded a new video:
A larger Test Statistic value (such as that for z, t, F, or Chi-Square) results in a smaller p-value. The p-value is the Probability of an Alpha (False Positive) Error.
And conversely, a smaller Test Statistic value results in a larger value for p.
Here's how it works:
In the close-ups of the right tail, zero is not visible. It is the center of the bell-shaped t curve, and it is out of the picture to the left. So, a larger value of the Test Statistic, t, would be farther to the right. And, the hatched area under the curve representing the p-value would be smaller. This is illustrated in the middle column of the table above.
Conversely, if the Test Statistic is smaller, then it's value is plotted more to the left, closer to zero. And so, the hatched area under the curve representing p would be larger. This is shown in the rightmost column of the table.
In the previous Tip, , we showed how to state the Null Hypothesis as an equation (e.g. H0: μΑ = μΒ).
And the Alternative Hypothesis would be the opposite of that (HA: μA ≠ μB).
These would work for a 2-tailed (2-sided) tests, when we only want to know whether there is a (Statistically Significant) difference between the two Means, not which one may be bigger than the other.
But what about when we do care about the direction of the difference? This would be a 1-tailed (1-sided) test. And the Alternative Hypothesis will tell us whether it's right-tailed or left-tailed. (We need to specify the tail for our statistical calculator or software.)
How does this work? First of all, it's helpful to know that the Alternative Hypothesis is also known as the "Maintained Hypothesis". The Alternative Hypothesis is the Hypothesis which we are maintaining and would like to prove.
We maintain that the Mean lifetime of the lightbulbs we manufacture is more than 1,300 hours. That is, we maintain that µ > 1,300.
This, then becomes our Alternative Hypothesis. HA: µ > 1,300
Note that the comparison symbol of HA points to the right. So, this test is right-tailed.
If, on the other hand, we maintained that the Mean defect rate of a new process is less than the Mean defect rate of the old process, our Maintained/ Alternative Hypothesis would be HA: µ New < µ Old
and the test would be left-tailed.
The concept of Null Hypotheses can be confusing, because it is about nothingness. And the human mind is wired to understand things that exist, not things that don't exist.
I've found that it helps to state the Null Hypothesis as either
So, if we are comparing the Means of Populations A and B, we could say:
Null Hypothesis (H0) "There is no difference between the Mean of Population A and the Mean of Population B."
But, it can be stated even more clearly and succinctly as an equation:
And the Alternative Hypothesis would be the opposite of this:
This is for a 2-tailed test:
It gets more complicated when we get into 1-tailed tests. In those tests, we'd have either
But that would be the subject of another Statistics Tip.
I just uploaded a new video to the the book's channel on YouTube. It's the 3rd of 3 in a playlist on Samples and Sampling. It's called Sample Size Part 2 (of 2) – for Measurements/Continuous Data. https://youtu.be/mxR-Lsc3ikc
For a complete list of videos based on this book that are completed and next in line, please see the videos page of this website.
This was uploaded to my YouTube channel Statistics From A to Z, Confusing Concepts Clarified. It will be part of a playlist on Samples and Sampling.
See the Videos page of this website for the latest status of my statistics videos completed and planned
Confusing language and terminology is a big part of what makes statistics confusing. Each Binomial Trial -- also known as a Bernoulli Trial -- is a random experiment with only 2 possible outcomes, called "success" and "failure". The Probability of success is the same every time the experiment is conducted.
A coin flip illustrates this perfectly. You either get "heads" or "tails", and the Probability of each coin flip (Binomial Trial) is always 50% heads (or 50% tails).
In a Binomial Trial, each trial is counted as either a success or failure. And a success is defined as what we want to count. Let's say we are performing quality control in a manufacturing process. We are counting defects. Every time we find a defect, we add 1 to the count of "successes".
I always found that confusing, so in the book, instead of saying "success" or "failure", I suggest saying "yes" or "no".
I just uploaded a new video, "Samples, Sampling", to my YouTube channel. Here's the link to the video: https://youtu.be/bvglAgAZtXE. It is the first in a playlist on Samples. The other two videos will be about how to determine the Sample Size for Proportions and Count Date, and Sample Size for Measurement/ Continuous Data.
Here are the 6 Keys to Understanding the concept. This 17 minute video explains them in detail.
Andrew A. (Andy) Jawlik is the author of the book, Statistics from A to Z -- Confusing Concepts Clarified, published by Wiley.