SD - Defining a Sampling Distribution Lesson

Math_Lesson_TopBanner.png SD - Defining a Sampling Distribution Lesson

The Essence of Sampling Distribution Thinking

We have learned how to find stats and probabilities involving a SINGLE randomly selected object.   Now, we will be talking about a randomly selected SAMPLE of size 'n' and doing that OVER and OVER again, in theory.   One could expect the results of those PRETEND samples to have "approximately" the same mean as the population from which they came.

When considering all those POSSIBLE variations of the means we would get a distribution that looks relatively NORMAL...some means would be slightly below or slightly above a central value.

However , the standard deviation is another story.   We cannot use σ , which is the standard deviation for the entire population because we are not dealing with the entire population anymore.   We are dealing with samples of size n FROM that population.   The standard deviation needs to be AMENDED appropriately inside the standardization formula for z to become σ/sqrt n. With this change comes a change in name from standard deviation to standard ERROR. There are two types of sampling distributions:   those for means and those for proportions.   The idea is the same for each but the formulas are slightly different.

Sampling Distribution of the Mean

If we want to form a sampling distribution of the mean we would:

1. Sample repeatedly from the population

2. Calculate the statistic of interest (the mean)

3. Form a distribution based on the set of means obtained from the samples

The set of means obtained will form a NEW distribution, called the SAMPLING distribution...in this case, the sampling distribution of the mean.    

Each sample would have its own mean and standard deviation ( and s respectively...NAMES ARE IMPORTANT).     However, the sampling distribution would have its own mean and standard deviation called mu ( μ ) and sigma ( σ )...sometimes written μ xbar σ xbar .   These would be used to represent (estimate) the total POPULATION parameters.

Sampling Distribution of a Proportion

The proportion obtained from a sample is called phat ( p̂) .   So, when we form a distribution of all the phats they produce a mean called μ phat which is used as an estimator of p (the true POPULATION proportion). The standard deviation of all those sample proportions becomes σ phat using the formula sqrt ( p̂(1 - p̂)/n ).    

Do not forget that all this is related to NORMAL DISTRIBUTIONS, thanks to the CENTRAL LIMIT THEOREM.  The concept of a sampling distribution is often a confusing topic for students and it is a favorite of the College-Board for questions on the AP Stat Exam.   To view a completed example, please download the 2004 AP Stat Exam Question in the sidebar under HAND OUTS.   

More detail and examples are contained in the lesson below:

Sampling Distribution Links to an external site.

Sampling Distributions Review

All You Need to Know about Sampling Distributions 

  • The population is the complete set of items of interest. A sample is a part of a population used to represent the population.  
  • The population mean ( μ ) and the population standard deviation ( σ ) are examples of population parameters.  
  • The sample mean (  ) and the sample standard deviation ( s ) are examples of statistics.   Statistics are used to make inferences about population parameters.  
  • While a population parameter is a fixed quantity, statistics vary depending on the particular sample chosen.  
  • The probability distribution showing how a statistic varies is called a sampling distribution.
  • The sampling distribution is unbiased if its mean is equal to the associated population parameter.

 

 

Remember, the purpose of finding a Statistic from a Sample is to form a conclusion about the Parameter associated with the Population. 

Math_APStatisticsBottomBanner.png

IMAGES CREATED BY GAVS