sampling error The absolute value of the difference between the sample mean, x̄, and the population mean, μ, written |x̄ − μ|, is called the sampling error. ... The standard deviation of a sampling distribution is called the standard error.
How do you find the population mean from the sample mean?
1:285:04Sample Mean and Population Mean - Statistics - YouTubeYouTubeStart of suggested clipEnd of suggested clipThe population mean represented by the Greek letter mu. Is also the sum divided by n where n is theMoreThe population mean represented by the Greek letter mu. Is also the sum divided by n where n is the number of people in the entire city. This is the entire population.
What is the difference between a sample mean and the population mean called quizlet?
Sampling error is the difference between any sample statistic (the mean, variance, or standard deviation of the sample) and its corresponding population parameter (the mean, variance or standard deviation of the population).
Is there any difference between mean and sample mean?
"Mean" usually refers to the population mean. This is the mean of the entire population of a set. ... The mean of the sample group is called the sample mean.
Is the sample mean always equal to the population mean?
The mean of the distribution of sample means is called the Expected Value of M and is always equal to the population mean μ.
What does the sample mean tell us?
A sample is a set of measurements taken from a larger population. ... The sample mean is simply the average of all the measurements in the sample. If the sample is random, then the sample mean can be used to estimate the population mean.
How do you know if its a sample or population?
A population is the entire group that you want to draw conclusions about. ... A sample is the specific group that you will collect data from. The size of the sample is always less than the total size of the population.
Is the mean of the sampling distribution always equal to the population mean?
The mean of the distribution of sample means is called the Expected Value of M and is always equal to the population mean μ.
What is the mean of the sample means quizlet?
Mean of the Sample Mean. The mean of all possible sample means equals the population mean. Standard Deviation of the Sample Mean. The standard deviation of the sample mean equals the standard deviation of the variable under consideration divided by the square root of the sample size.
Is sample mean and average the same?
A sample mean is an average of a set of data. The sample mean can be used to calculate the central tendency, standard deviation and the variance of a data set. The sample mean can be applied to a variety of uses, including calculating population averages.
What is the difference between a sample and a population?
A population is the entire group that you want to draw conclusions about. A sample is the specific group that you will collect data from. The size of the sample is always less than the total size of the population. In research, a population doesn’t always refer to people. It can mean a group containing elements of anything you want to study, ...
How is the mean of a sample calculated?
“Mean” is the average of all the values in a sample. It can be calculated by adding up all the values and then dividing the sum total by the number of values in the sample. When the provided list represents a statistical population, then the mean is called the population mean.
What does point estimate for two population means mean?
A point estimate for the difference in two population means is simply the difference in the corresponding sample means. In the context of estimating or testing hypotheses concerning two population means, “large” samples means that both samples are large.
Which is an example of sampling distribution of the mean?
The sampling distribution of the mean refers to the pattern of sample means that will occur as samples are drawn from the population at large Example I want to perform a study to determine the number of kilometres the average person in Australia drives a car in one day.
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Someone recently asked me what the difference was between the sample mean and the population mean. This is really a question which goes to the heart of what it means to perform statistical inference. Whatever field we are working in, we are usually interested in answering some kind of question, and often this can be expressed in terms of some numerical quantity, e.g. what is the mean income in the US. This question can be framed mathematically by saying we would like to know the value of a parameter describing some distribution. In the case of the mean US income, the parameter is the mean of the distribution of US incomes. Here the population is the US population, and the population mean is the mean of all the incomes in the US population. For our objective, the population mean is the parameter of interest.
In some (or maybe most) settings, the population is large but finite. However, often the population is so large that we actually assume the population is infinite, to make some of the maths easier. Because the population is large, we usually cannot hope to calculate the parameter of interest (e.g. the population mean) exactly, because to do so we would have to obtain income information from the large population. This is often infeasible due to costs or practicalities.
Instead, we take a sample from the population of interest, and calculate the mean of the sample (or, more generally, an estimate of our parameter of interest, based on the sample data), giving the sample mean. Now of course the sample mean will not equal the population mean. But if the sample is a simple random sample, the sample mean is an unbiased estimate of the population mean. This means that the sample mean is not systematically smaller or larger than the population mean. Or put another way, if we were to repeatedly take lots and lots (actually an infinite number) of samples, the mean of the sample means would equal the population mean.
Because the sample mean is not equal to the population mean which we are actually interested in, if we are to use the sample mean in place of the population mean we should always report it with some measure of how precise it is. The most common ways of doing this are to report a standard error or confidence intervals, topics I will return to in later posts.
Often in statistics we’re interested in answering questions like:
- What is the mean household income in a certain city?
- What is the mean weight of a certain species of turtle?
- What is the mean attendance at college football games?
In each scenario, we are interested in answering some question about a population, which represents every possible individual element that we’re interested in measuring.
However, instead of collecting data on every individual in a population we instead collect data on a sample of the population, which represents a portion of the total population.
For example, we might want to know the mean weight of a certain species of turtle that has a total population of 800 turtles.
Since it would take too long to locate and weigh every single turtle in the population, we instead collect a simple random sample of 30 turtles and measure their weights:
We could then use the mean weight of this sample of turtles to estimate the mean weight of all turtles in the population.
How to Calculate the Sample Mean
The formula to calculate the sample mean, often denoted x, is as follows:
x = Σxi / n
where:
- Σ: A fancy Greek symbol that means “sum”
- xi: The value of the ith observation in the dataset
- n: The sample size
For example, suppose we collect a sample of 10 turtles with the following weights (in pounds):
- 70, 80, 80, 85, 90, 95, 110, 120, 140, 150
The sample mean would be calculated as:
- x = (70+ 80+80+85+90+95+110+120+140+150) / 10 = 102
Why the Sample Mean is Unbiased
In statistical jargon, we would say that the sample mean is a statistic while the population mean is a parameter.
Here’s the difference between the two terms:
A statistic is a number that describes some characteristic of a sample.
A parameter is a number that describes some characteristic of a population.
The parameter is the value that we’re actually interested in measuring, but the statistic is the value that we use to estimate the value of the parameter since the statistic is so much easier to obtain.
When we use a method like simple random sampling to obtain a sample, we say that the sample mean is an unbiased estimator of the population mean.
In other words, we have no reason to believe that the sample mean would underestimate or overestimate the true population mean.
The reason is because when we use a method like simple random sampling, every member in the population has an equal chance of being included in the sample, which means the sample is likely to be a “mini version” of the overall population.
We would say that the sample is representative of the overall population, which means that the sample mean should be a good estimate of the population mean, assuming that the sample size is large enough.
On Using Confidence Intervals with the Sample Mean
Although the sample mean provides an unbiased estimate of the population mean, it’s unlikely that the sample mean will exactly match the population mean.
For example, if we want to use a sample of turtles to estimate the mean weight of a population of turtles, it’s possible that we might just happen to pick a sample full of low-weight turtles or perhaps a sample full of heavy turtles.
In order to capture this uncertainty around our estimate of the population mean, we can create a confidence interval.
A confidence interval is a range of values that is likely to contain a population parameter with a certain level of confidence.
For example, we might collect a sample of 30 turtles and find that the mean weight of this sample is 102 pounds. If we then construct a 95% confidence interval, we might find that the interval is as follows:
95% confidence interval = [98.5, 105.5]
We would interpret this to mean there is a 95% chance that the confidence interval of [98.5, 105.5] contains the true population mean weight of turtles.
This confidence interval is more useful than just the sample mean because it gives us a range of values that the true population mean is likely to fall in.
Additional Resources
Population vs. Sample: What’s the Difference?
Statistic vs. Parameter: What’s the Difference?
An Introduction to Confidence Intervals