# Estimate Mean & Variance from Normal Distribution

• Mark J.
In summary, the conversation discusses the process of estimating the mean and variance for a population from a normal distribution. The primary purpose of the project is to investigate the statistical properties of different estimators. Methods such as MLE and moments are used to compare the performance of these estimators. There is also a suggestion to investigate the effects of using non-exact samples on the estimators. The conversation also mentions the importance of having a unique and original approach in order to gain recognition from experts in the field. The suggested approach for comparing the estimators is to simulate batches of samples and analyze the distribution of estimates. The conversation ends with a suggestion to read about the mathematical theory of estimators and study known theoretical results.
Mark J.
Hi..
I have sample data from a normal distribution and I need to estimate mean and variance for the population.
For the sake of study I need to use different methods for estimating these parameters and maybe make a comparison of these methods regarding my case.
Any idea how to proceed in this case more concretely?
Regards

Post #12 in this thread https://www.physicsforums.com/showthread.php?t=616643&highlight=inverse gives three estimators for the variance that you can investigate.

You haven't made your project clear. Is the primary purpose to analyze a particular set of data or is the primary purpose to investigate the statistical properties of various estimators? - i.e. is is this project for a statistics course? - a science project? - a work assignment?

Can you use computer software (or write programs) to do Monte-Carlo simulations?

No the purpose of this science project is to identify to investigate the statistical properties of various estimators for my population of data.
I can use computer software to do Monte-Carlo simulations and actually I was thinking to use R language but need to be a little more clarified on steps to take for this goal.
Thank you Stephen for fast response.
Regards

Investigating the properties of estimators for the mean and variance of a normal distributiion is going down a well-trodden path. (Perhaps that is why someone has sent you down it.) Experts evaluating that sort of project would know the answers in advance, although the average science-fair judge might not. If you do a moderate amount of reading about the subject, you will know the answers in advance too.

A less well known path begins with the fact that you don't have samples from a normal distribution. Real measurements can't be taken exactly; the results are only given to a certain precision. The mathematical ideal of sampling from a normal distribution has no such restriction. Your measurements are not samples from a normal distribution; they are samples from a discrete distribution that you might assume is related to some normal distribution. How does using non-exact samples affect estimators of the mean and variance?

That's a line of research that I find personally interesting. I don't know what your advisors would think.

That's really a very good point.
But mainly what I need was kind of comparing methods (MLE , moments etc) to see which give a better result in estimation of parameters for normal distribution by generating different quantities of random normal distributed variables .
Please tell me if something doesn't fit or if I am going wrong way.
Regards

Mark J. said:
Please tell me if something doesn't fit or if I am going wrong way.

You'll have to describe the scenario better. For example:

Scenario 1:
Dr. Stats Teacher: "Mark, why don't you do your science project on comparing the performance of diferent estimators for the mean and variance of a normal distribution?"

Scenario 2:
Mark: "I've read about estimators. I wonder what the good ones are to use for the mean and variance of a normal distribution."
Mr. Glubbston: "Gosh, that's a good question, Mark. I've forgotten all statistics that I ever took. Maybe you should do a science project on that."

Scenario 3: The judges at Wednesday's science fair will be
Dr. Myron Glevash, chairmain of the statistics department at Oggerland University
Mr. Sydney Dappslie, quality control engineer at Morlin Dynamic Systems

Scenario 4: The judges at Wednesday's science fair will be
Mr. Herman Vipsdale, president of the First National Bank
Dr. Edward Laxtilly, M.D.

Well that was very nice reading )))
Scenario 1 fits better for my needs.
Regards

Mark J. said:
Scenario 1 fits better for my needs.

If (as in scenario 1) an expert in statistics has suggested the path of your investigations then you should take it. The expert is correct that your path of investigation will produce useful results.

These are considerations that I see. 1) The results of such investigations are already well-known to experts, so experts evaluating your work will know the results that you "should" get and will criticize any omissions. 2) You can read enough about the topic to know more than any non-statistical experts who ask you questions 3) Experts won't give you any points for originality unless you also investigate some side-line to the commonly used estimators. (Perhaps your advisor already has such a side-line in mind and will reveal it once you've become familar with more well-known results.)

Please how to proceed on comparing the performance of different estimators for the mean and variance of a normal distribution?
Best regards

Simulate batches of N samples from a know normal distribution, apply the estimators to each batch, histogram how estimates are distributed. See which estimators tend to cluster most tightly around the true value of the parameter being estimated.

Then you should start reading about the mathematical theory of estimators and study the known theoretical results if you have a sufficient mathematical background.

## 1. How do you estimate the mean of a normal distribution?

To estimate the mean of a normal distribution, you can use the sample mean, which is the average of all the data points in your sample. This is an unbiased estimator of the population mean and becomes more precise as the sample size increases.

## 2. What is the formula for estimating the variance of a normal distribution?

The formula for estimating the variance of a normal distribution is the sum of squared differences between each data point and the sample mean, divided by the sample size minus one. This is known as the sample variance and is an unbiased estimator of the population variance.

## 3. How does the sample size affect the accuracy of the estimated mean and variance?

The larger the sample size, the more accurate the estimated mean and variance will be. This is because as the sample size increases, the sample mean and variance will approach the true population mean and variance. In other words, larger sample sizes provide more precise estimates of the population parameters.

## 4. Can you estimate the mean and variance from a non-normal distribution?

Yes, you can still estimate the mean and variance from a non-normal distribution. However, the estimates may not be as accurate as they would be for a normal distribution. It is important to consider the distribution of the data when choosing appropriate methods for estimating the mean and variance.

## 5. What are some potential limitations of estimating the mean and variance from a normal distribution?

Estimating the mean and variance from a normal distribution assumes that the data is normally distributed. If the data is not normally distributed, the estimates may not accurately reflect the true population parameters. Additionally, if the sample size is small, the estimates may not be as reliable. It is important to consider the assumptions and limitations when using these estimates.

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