Introduction to Sampling Distributions (College Board AP® Statistics): Study Guide
Introduction to sampling distributions
What is the distribution of a population?
The population is all possible individuals that can be sampled
Recall that a sample of size
is taken from a population
You can display the population on a graph
e.g. a relative frequency chart, histogram, boxplot or probability distribution can be drawn
This shows the distribution of a population
Parameters of the population can often be seen from the distribution of the population
e.g. the population mean, population standard deviation, population range, etc.
Examiner Tips and Tricks
Make sure to always talk about population parameters in context, for example replacing "the population mean" with "the mean age of all 500 students in the school".
What is the sampling distribution of a statistic?
Recall that, when taking samples, you need to specify
the sample size,
the sample statistic
This is what it is that you are measuring from the sample
e.g. sample median, sample mean, sample range, etc.
Taking one sample of size
generates one value of the sample statisticbut taking many samples of size
generates many values of the sample statistic
If you could take all possible samples of size
from the population you would have all possible values of the sample statisticThis collection of all possible sample statistics is called the sampling distribution of the statistic
Samples in the sampling distribution of the statistic must all be:
taken from the same population
the same size
Changing the sample size,
, will change the sampling distribution
Sampling distributions are often shown on graphs
e.g. relative frequency charts or histograms
Examiner Tips and Tricks
When commenting on distributions in the exam, make it clear whether you are referring to the distribution of the population or to a sampling distribution for a particular statistic!
How can simulations be used to approximate sampling distributions?
In reality, it is often not possible to take all possible samples of size
to find all possible values of a sample statisticThis means the exact sampling distribution cannot always be generated
Instead, given a population, you can run simulations to approximate the sampling distribution
e.g. use a computer to select a random sample of size 5, find its sample median, then repeat this process 1000 more times
This gives an approximate sampling distribution of sample medians
The more repeats, the better the approximation
Distributions produced by simulations are sometimes called randomization distributions
Worked Example
The ages of viewers in a movie theatre for a particular movie are recorded. A data analyst selects a random sample of 5 viewers and works out their median age.
(a) If the ages in the sample are 13, 38, 25, 50 and 42, calculate the median age.
Answer:
First write the ages in ascending order
13, 25, 38, 42, 50
Then select the middle value
The median age of the sample is 38
(b) Explain how the data analyst could create the sampling distribution of the sample median, for samples of size 5.
Answer:
The data analyst would need to obtain every possible random sample of 5 viewers and compute the median of each sample
The collection of all possible sample medians gives the sampling distribution of the sample median
(c) The sampling distribution of the sample median is shown below. Explain whether or not the median age of the sample in part (a) is unusually old.
Answer:
The median age of 38 from the sample in part (a) is not unusually old
The sampling distribution of the sample median shows that samples with median ages of 38 or greater occur fairly often
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