# Understanding Standard Error in Statistics: The Standard Deviation Explained

Article Summary

When working with statistical data, it is important to have a clear understanding of how the data is distributed. One key measure used in statistics to describe the spread of a distribution is the standard deviation. However, in certain situations, we need to use the standard error instead. Here, we’ll define what standard error (SE) is and how it differs from standard deviation (SD).

## What is standard error?

Standard error is a measure of the variability of sample means around the true population mean. In other words, it represents how much the means of different samples from the same population are likely to vary. This variation is due to random sampling error.

To calculate the standard error, we take the standard deviation of the population and divide it by the square root of the sample size:

SE = SD / sqrt(n)

Where,

• SE: Standard error
• SD: Standard deviation
• n: Sample size

## How standard error differs from standard deviation

While both standard error and standard deviation measure variability, they are not the same thing. Here are the key differences between the two:

• Definition: Standard deviation measures the variability of data points within a sample or population, while standard error measures the variability of sample means.
• Formula: Standard deviation is calculated by taking the square root of the sum of squared deviations from the mean divided by the number of data points, while standard error is calculated by dividing the standard deviation by the square root of the sample size.
• Units: Standard deviation is expressed in the same units as the data points, while standard error is expressed in the units of the mean.
• Purpose: Standard deviation is used to measure the spread of data within a sample or population, while standard error is used to measure the accuracy of the sample mean.

In other words, standard deviation describes how much individual data points deviate from the mean, while standard error describes how much sample means deviate from the true population mean.

## Why standard error is important

Standard error is important because it helps us understand the reliability of our sample means. By calculating the standard error, we can estimate the range within which the true population mean likely falls. This range is known as the confidence interval.

A small standard error indicates that the sample means are tightly clustered around the true population mean, which means we can have high confidence that our sample is representative of the population. Conversely, a large standard error indicates that the sample means are more spread out, which means we have less confidence in our sample.

## When to use standard error

Standard error is typically used when we want to estimate the mean of a population using a sample. By calculating the standard error, we can determine the likely range of the true population mean, which helps us to understand how confident we can be in our estimate.

Standard error is also used when conducting hypothesis tests. In these tests, we compare the mean of a sample to the mean of a population to determine if there is a significant difference between the two. By calculating the standard error, we can determine the likelihood that the difference we observe is due to chance or if it is statistically significant.

## Standard error example

Let’s say we want to estimate the average height of students in a school using a sample of 50 students. We measure the height of each student in the sample and find that the standard deviation of the heights is 3 inches. We can calculate the standard error using the formula:

SE = 3 / sqrt(50) = 0.42 inches

This means that we can be 95% confident that the true population mean height falls within the range of the sample mean height plus or minus 2 standard errors, or:

64.1 inches ± (2 x 0.42 inches) = [63.26 inches, 64.94 inches]

## Limitations of standard error

While standard error is a useful measure for determining the accuracy of sample data, it has some limitations. Some of the limitations of standard error include:

1. Sample size: As mentioned earlier, a larger sample size tends to have a smaller standard error. However, as the sample size increases, the cost of data collection and analysis also increases. Therefore, a researcher may have to strike a balance between the sample size and the cost of data collection.
2. Population distribution: Standard error assumes that the sample data is normally distributed. However, if the population data is not normally distributed, then standard error may not be an appropriate measure of accuracy.
3. Sampling method: The accuracy of the sample data depends on the sampling method used. For instance, a simple random sampling method is more accurate than a convenience sampling method.
4. Outliers: Standard error is sensitive to outliers. Therefore, it may not be an appropriate measure of accuracy when there are outliers in the sample data.

## Standard Error FAQs

### What is the difference between standard error and standard deviation?

Standard deviation measures the variability of a population or sample, while standard error measures the variability of the sample mean. In other words, standard deviation shows how spread out the data is, while standard error indicates how accurate the sample mean is as an estimate of the true population mean.

### What does a larger standard error mean?

A larger standard error means that the sample mean is a less precise estimate of the population mean. This could be due to a smaller sample size or a greater degree of variability within the sample.

### What is the significance of standard error in hypothesis testing?

Standard error is important in hypothesis testing because it helps to determine the likelihood that the difference between two sample means is due to chance. The larger the standard error, the greater the chance that the difference between the means is due to chance alone.

### What are some common misconceptions about standard error?

One common misconception is that standard error tells us about individual data points, when in fact it only pertains to sample means. Another misconception is that a small standard error necessarily means that the sample is representative of the population, when in fact it depends on the size of the sample and the degree of variability within it.

### How can I use standard error in my own research or data analysis?

Standard error can be used to determine the precision of your sample mean as an estimate of the population mean. It can also be used in hypothesis testing to determine whether a difference between two sample means is statistically significant.

## Key takeaways

• The standard error is a measure of the variability of sample means around the true population mean.
• The formula for standard error is the standard deviation of the sample divided by the square root of the sample size.
• Standard error can be used to calculate confidence intervals and test hypotheses about population means.
• It is important to understand the difference between standard error and standard deviation in statistics.
• Standard error plays a crucial role in statistical inference and decision-making.
###### View Article Sources
1. “Why Control for Standard Errors?” – University of Pennsylvania
2. “Standard Errors” – University of Michigan
3. “Standard Error Tables” – National Center for Education Statistics