# Standard Error – How To Calculate And Report It

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In statistics, the standard error is used to demonstrate the potential difference between the average within a population compared to the average of a given sample of that population. It is mostly used in hypothesis testing, confidence interval construction, and inferential statistics, serving as a measure for the reliability and accuracy of results. In this article, you will find out how to use it correctly, what it can be applied to and why it is often used when reporting data from samples.

## Standard Error – In a Nutshell

• Standard error is used to measure the accuracy of a sample distribution.
• It makes use of standard deviation to represent a population.
• The standard error of the mean is used to describe this variation.
• It represents the approximate standard deviation of a sample population.
• It allows you to estimate how representative their sample data is.
• The greater number of data points, the lower the standard error.

## Definition: Standard error

The standard error of a set of data, or an estimate of a statistical parameter, is defined as the standard deviation of the given sampling distribution. If the statistic in question is a mean average, then it is referred to as the standard error of the mean, or SEM. In many cases, the term standard error is used to mean SEM. However, it is important to note that other forms of standard error, or SE, can be used to describe deviations of mean averages and proportions.

SE – or SEM – can be used in statistical models that are built on samples. Hypothesis testing is one area where SE is widely used because it indicates how accurate a smaller sample might be when compared to a wider population. Therefore, it is widely used in quantitative research, such as political polling.

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## The importance of standard error

Showing the SE of a sample is important in statistics because it means someone reading the data is able to gain a clear understanding of how representative – or otherwise – that sample might be compared to the wider population the sample has been taken from. When you collect a sample randomly, it may be a close representation of the wider population or, conversely, it may not.

The SE shows how much sampling error there might be in a set of data. So, an SE calculation will help to show whether conclusions drawn from a sample are likely to be accurate. Statisticians can help to lower the SE of their data by taking larger samples, thereby minimizing bias.

 High Standard Error Low Standard Error • The sample data does not match the population • Hypotheses drawn from the sample are not valid • The sample data matches the population closely • Hypotheses drawn from the sample are more valid

## Standard error vs. standard deviation

In some cases, standard error and standard deviation may be confused. As such, it is important to have a clear distinction between the two despite the fact that they both describe variability.

• Standard deviation measures the amount of variance – or dispersion – of data points from the average within a single sample.
• SE is used to calculate the variability across multiple samples, thereby normalizing standard deviation within a given population.

Example

A random sample of 100 men’s shoe sizes could reveal the mean average size is 6 ½. However, based on other surveys, it may be known that the average men’s shoe size is 9.

• Finding the square of each data point’s distance to the mean for standard deviation
• Add them all up and subsequently, divide by the number of data points
• This provides an idea of how much the average data point in the sample differs from 9
• The standard error tells you the difference between the sample mean of 6 ½ and 9

## Standard error formula

More than one formula is used to calculate the standard error depending on whether or not the population parameters are known. Both formulae only work when the sample group is made up of at least 20 data points.

### Population parameters are available

When population parameters are already known, the standard error is calculated by dividing the standard deviation of the population by the square root of the number of elements in the sample. Use the following formula:

### Population parameters are unavailable

When population parameters are not known, the standard error is calculated by dividing the standard deviation of the sample by the square root of the number of elements in the sample. For this, use the following formula:

This approach means using the sample’s standard deviation as a point estimate to get an approximation of the SE. As such, the resulting SE will only be an estimation based on the available, limited data.

Example

In a survey of 50 women being asked about their height, the sample standard deviation is found to be 15 cm, that is to say, the average difference between all the data points in the sample and the mean average of the sample is 15.

Therefore:

## Reporting the standard error

Typically, SE is reported after the mean average of a set of data is given with a plus or minus figure.

Example

The official estimates of personal income in the UK are given with standard errors. Therefore, from a sample of different incomes, you might express the average annual earnings to be £28,500 ± £900 (SE).

In addition, SE can be expressed with a confidence interval. In the above example, the confidence interval of average earnings from the sample would be £27,600 to £29,400, accounting for an SE of plus or minus £900. The latter is considered better for non-technical readers since it doesn’t rely on them doing any calculations.

Note: The confidence interval is a range of values. These represent where an average figure from an unknown population parameter would be expected to fall even if new random samples were added to the known data set.

Confidence intervals reveal the confidence level that can be assigned to a given set of data. Standard errors that are ± 1.96 of the sample mean can be said to have a high confidence interval level of 95%. In other words, the true population parameter could be said to be within the given range with 95% confidence, based on a randomized sample.

A confidence interval of 0.95 – 95% – would be the sample mean ± (1.96 multiplied by the standard error).

Therefore, a confidence interval of this level would be possible with a sample mean of 35 if the sample’s standard deviation was 10 based on 100 data points in the sample.

Example

In this example, it would be possible to say that the SE’s lower and upper bounds would be 33.04 and 36.96, or 35 ± 1.96, at 95% confidence.

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## Other types of standard error

As previously mentioned, SEM is not the only form of SE. It is merely the most common. Other types include:

 Definition SE of the Estimate The difference between the actual value of the dependent variable and its predicted value based on the multiple regression model. SE of Measurement The assessment of how much test scores that are measurable deviate from a known, so-called, 'true' measurement.

## FAQs

#### What is the standard error?

SE is a measure of statistical accuracy that is equal to the standard deviation of the theoretical distribution of such estimates.

#### What does the standard error tell us?

It helps to gain a rapid understanding of how representative a sample might be.

#### What's a good SE?

The lower the number, the more reliable the sample data is likely to be.

#### Are SE and SEM the same?

No, SE can mean SEM but SEM is just one common type of standard error.

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