Content

• Steps
• Part 1 of 3: The Basics
• Part 2 of 3: Calculating the standard deviation
• Part 3 of 3: Finding the Standard Error
• Tips

The standard error is a value that characterizes the standard (root-mean-square) deviation of the sample mean. In other words, this value can be used to estimate the accuracy of the sample mean. Many applications of standard error assume a normal distribution by default. If you need to calculate the standard error, go to step 1.

Steps

Part 1 of 3: The Basics

1. 1 Remember the definition of standard deviation. Sample standard deviation is a measure of the dispersion of a value. The sample standard deviation is usually indicated by the letter s. The mathematical formula for the standard deviation is given above.
2. 2 Find out what the true mean is. The true mean is the average of a group of numbers that includes all the numbers in the entire group — in other words, it is the average of the entire group of numbers, not a sample.
3. 3 Learn to calculate the arithmetic mean. Arithmetic mean simply means the average: the sum of the values ​​of the collected data divided by the number of values ​​of that data.
4. 4 Find out what a sample mean is. When the arithmetic mean is based on a series of observations obtained from samples from a statistical population, it is called the “sample mean”. This is the average of a sample of numbers, which describes the average of only a fraction of the numbers from the entire group. It is designated as:
5. 5 Understand the concept of a normal distribution. Normal distributions, which are used more often than other distributions, are symmetric, with a single maximum in the center - on the mean of the data. The shape of the curve is similar to the shape of a bell, with the graph dipping evenly on either side of the mean. Fifty percent of the distribution lies to the left of the mean, and the other fifty percent lies to the right of it. The scattering of the values ​​of the normal distribution is described by the standard deviation.
6. 6 Remember the basic formula. The formula for calculating the standard error is given above.

Part 2 of 3: Calculating the standard deviation

1. 1 Calculate the sample mean. To find the standard error, you first need to determine the standard deviation (since the standard deviation s is included in the formula for calculating the standard error). Start by finding averages. The sample mean is expressed as the arithmetic mean of the measurements x1, x2,. ... ... , xn. It is calculated using the formula above.
   • Let's say, for example, that you need to calculate the standard error of the sample mean of the measurements of the mass of the five coins shown in the table:
     You can calculate the sample mean by substituting the mass values ​​into the formula:
2. 2 Subtract the sample mean from each measurement and square the resulting value. Once you get the sample mean, you can expand your spreadsheet by subtracting it from each dimension and squaring the result.
   • For our example, the extended table will look like this:
3. 3 Find the total deviation of your measurements from the sample mean. The total deviation is the sum of the squared differences from the sample mean. Add up your new values ​​to determine it.
   • In our example, you will need to perform the following calculation:
     This equation gives the sum of the squares of the deviations of the measurements from the sample mean.
4. 4 Calculate the standard deviation of your measurements from the sample mean. Once you know the total deviation, you can find the mean deviation by dividing the answer by n -1. Note that n is equal to the number of dimensions.
   • In our example, 5 measurements were made, therefore n - 1 will be equal to 4. The calculation should be carried out as follows:
5. 5 Find the standard deviation. Now you have all the values ​​you need to use the formula to find the standard deviation s.
   • In our example, you will calculate the standard deviation as follows:
     Therefore, the standard deviation is 0.0071624.

Part 3 of 3: Finding the Standard Error

1. 1 Use the basic standard deviation formula to calculate the standard error.
   • In our example, you will be able to calculate the standard error as follows:
     Thus, in our example, the standard error (standard deviation of the sample mean) is 0.0032031 grams.

Tips

• Standard error and standard deviation are often confused. Note that the standard error describes the standard deviation of the sampled distribution of statistical data, not the distribution of individual values.
• In scientific journals, the concepts of standard error and standard deviation are somewhat blurred. The ± sign is used to combine the two values.