“Standard error” refers to the standard deviation of the statistical sample distribution. In other words, it can be used to measure the accuracy of the sample mean. Many uses of standard error implicitly assume a normal distribution. To calculate the standard error, scroll down to Step 1.
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Part 1 of 3: Understanding the Basics
Step 1. Understand the standard deviation
The sample standard deviation is a measure of how spread out the numbers are. The sample standard deviation is generally indicated by s. The mathematical formula for the standard deviation is shown above.
Step 2. Find the population mean
The population mean is the mean of a set of numbers that includes all the numbers in the entire group - in other words, the average of the entire set of numbers and not the sample.
Step 3. Find out how to calculate the arithmetic mean
The arithmetic mean is the average: the number of collections of values divided by the number of values in the collection.
Step 4. Identify the sample mean
When the arithmetic mean is based on a series of observations obtained by sampling from a statistical population, it is called the “sample mean”. This is the average of a set of numbers that includes the average of some of the numbers in a group. It is denoted as:
Step 5. Understand the normal distribution
The normal distribution, the most commonly used of all distributions, is symmetrical, with a single central peak being at the mean (or mean) of the data. The shape of the curve is similar to that of a bell, with the graph falling evenly on both sides of the mean. Fifty percent of the distribution lies to the left of the mean, and fifty percent lies to the right. The normal distribution is controlled by the standard deviation.
Step 6. Know the basic formula
The formula for the sample mean standard error is shown above.
Part 2 of 3: Calculating Standard Deviation
Step 1. Calculate the sample mean
To find the standard error, you must first determine the standard deviation (because the standard deviation, s, is part of the standard error formula). Start by finding the average of the sample values. The sample mean is expressed as the arithmetic mean of the measurements x1, x2,… xn. It is calculated by the formula as shown above.
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For example, suppose you want to calculate the standard error of the sample mean for a measurement of the weight of five coins, as listed in the table below:
You'll calculate the sample mean by plugging the weight values into the formula, like this:
Step 2. Subtract the sample mean from each measurement and then square the values
Once you have the sample mean, you can expand the table by subtracting it from each individual measurement, and then squaring the result.
In the example above, the expanded table would look like this:
Step 3. Find the total measurement deviation from the sample mean
The total deviation is the average of the differences in the squares of the sample mean. Add the new values together to define them.
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In the example above, the calculation is as follows:
This equation gives the total squared deviation of the measurement from the sample mean. Note that the sign of the difference is not important.
Step 4. Calculate the mean squared deviation of the sample mean
Once you know the total deviation, find the average deviation by dividing by n-1. Note that n is equal to the number of measurements.
In the example above, there are five measurements, so n-1 equals 4. Calculate as follows:
Step 5. Find the standard deviation
Now you have all the values needed to use the standard deviation formula, s.
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In the example above, you would calculate the standard deviation as follows:
Your standard deviation is 0.0071624.
Part 3 of 3: Finding the Standard Error
Step 1. Use the standard deviation to calculate the standard error, using the basic formula
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In the example above, calculate the standard error as follows:
your standard error (standard deviation from the sample mean) is 0.0032031 grams.
Tips
- Standard error and standard deviation are often confused. Note that the standard error represents the standard deviation of the statistical sample distribution, not the distribution of individual values.
- In scientific journals, standard error and standard deviation are sometimes blurred. The ± sign is used to combine these two measurements.
