How to Calculate Standard Error
Understand standard deviation., Know the population mean., Learn to calculate an arithmetic mean., Recognize sample means., Understand normal distribution., Know your basic formula., Calculate the sample mean., Subtract the sample mean from each...
Step-by-Step Guide
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Step 1: Understand standard deviation.
The standard deviation of a sample is a measure of how spread out the numbers are.
A sample standard deviation is generally signified with an s.
The mathematical formula for standard deviation is shown above. -
Step 2: Know the population mean.
The population mean is the mean of a numerical set that includes all the numbers within the entire group – in other words, the average of an entire set of numbers, rather than a sample. , The arithmetic mean is simply an average: the sum of a collection of values divided by the number of values in the collection. , When an arithmetic mean is based on a series of observations obtained by sampling from a statistical population, it is called the “sample mean.” It is the average of a numerical set that includes an average of only a portion of the numbers within a group.
It is denoted as: , Normal distributions, which are the most commonly used of all distributions, are symmetrical, with a single central peak at the mean (or average) of the data.
The shape of the curve is similar to the shape of a bell, with the graph falling off evenly on either side of the mean.
Fifty percent of the distribution lies to the left of the mean, and fifty percent lies to the right.
The spread of a normal distribution is controlled by the standard deviation. , The formula for standard error of a sample mean is shown above. , To find the standard error, first you must determine the standard deviation (because the standard deviation, s, is part of the standard error formula).
Start by finding the average of your sample values.
Sample mean is expressed as the arithmetic mean of measurements x1, x2, . . . xn.
It is calculated with a formula that is shown above.
For example, say you need to calculate the standard error of a sample mean for the weight measurements of five coins, as listed in the table below:
You would calculate the sample mean by plugging the weight values into the formula, like this: , Once you have the sample mean, you can expand your table by subtracting it from each individual measurement, then squaring the result.
In the example above, your expanded table would look like this: , The total deviation is the average of these squared differences from the sample mean.
Add your new values together to determine it.
In the example above, you would calculate as follows:
This equation gives you the total quadratic deviation of measurements from the sample mean.
Note that the sign of the differences is not important. , Once you know the total deviation, you can find the average deviation by dividing by n
-1.
Note that n is equal to the number of measurements.
In the example above, you have five measurements, so n – 1 would equal
4.
You would calculate as follows: , You now have all the necessary values to use the formula for standard deviation, s.
In the example above, you would calculate standard deviation as follows:
Your standard deviation is therefore
0.0071624. , In the example above, you would calculate standard error as follows:
Your standard error (the standard deviation of your sample mean) is therefore
0.0032031 grams. -
Step 3: Learn to calculate an arithmetic mean.
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Step 4: Recognize sample means.
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Step 5: Understand normal distribution.
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Step 6: Know your basic formula.
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Step 7: Calculate the sample mean.
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Step 8: Subtract the sample mean from each measurement and square the value.
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Step 9: Find the total deviation of your measurements from the sample mean.
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Step 10: Calculate the average quadratic deviation of your measurements from the sample mean.
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Step 11: Find the standard deviation.
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Step 12: Use the standard deviation to calculate the standard error
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Step 13: using the basic formula.
Detailed Guide
The standard deviation of a sample is a measure of how spread out the numbers are.
A sample standard deviation is generally signified with an s.
The mathematical formula for standard deviation is shown above.
The population mean is the mean of a numerical set that includes all the numbers within the entire group – in other words, the average of an entire set of numbers, rather than a sample. , The arithmetic mean is simply an average: the sum of a collection of values divided by the number of values in the collection. , When an arithmetic mean is based on a series of observations obtained by sampling from a statistical population, it is called the “sample mean.” It is the average of a numerical set that includes an average of only a portion of the numbers within a group.
It is denoted as: , Normal distributions, which are the most commonly used of all distributions, are symmetrical, with a single central peak at the mean (or average) of the data.
The shape of the curve is similar to the shape of a bell, with the graph falling off evenly on either side of the mean.
Fifty percent of the distribution lies to the left of the mean, and fifty percent lies to the right.
The spread of a normal distribution is controlled by the standard deviation. , The formula for standard error of a sample mean is shown above. , To find the standard error, first you must determine the standard deviation (because the standard deviation, s, is part of the standard error formula).
Start by finding the average of your sample values.
Sample mean is expressed as the arithmetic mean of measurements x1, x2, . . . xn.
It is calculated with a formula that is shown above.
For example, say you need to calculate the standard error of a sample mean for the weight measurements of five coins, as listed in the table below:
You would calculate the sample mean by plugging the weight values into the formula, like this: , Once you have the sample mean, you can expand your table by subtracting it from each individual measurement, then squaring the result.
In the example above, your expanded table would look like this: , The total deviation is the average of these squared differences from the sample mean.
Add your new values together to determine it.
In the example above, you would calculate as follows:
This equation gives you the total quadratic deviation of measurements from the sample mean.
Note that the sign of the differences is not important. , Once you know the total deviation, you can find the average deviation by dividing by n
-1.
Note that n is equal to the number of measurements.
In the example above, you have five measurements, so n – 1 would equal
4.
You would calculate as follows: , You now have all the necessary values to use the formula for standard deviation, s.
In the example above, you would calculate standard deviation as follows:
Your standard deviation is therefore
0.0071624. , In the example above, you would calculate standard error as follows:
Your standard error (the standard deviation of your sample mean) is therefore
0.0032031 grams.
About the Author
Daniel Brown
Creates helpful guides on crafts to inspire and educate readers.
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