What is Standard Error? Calculation & Interpretation
By busayo.longe
| Last updated:
Mar 12
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7 min read
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Errors are of various types and impact the research process in different ways. Here’s a deep exploration of the standard error, the types, implications, formula, and how to interpret the values
The standard error is a statistical measure that accounts for the extent to which a sample distribution represents the population of interest using standard deviation. You can also think of it as the standard deviation of your sample in relation to your target population.
The standard error allows you to compare two similar measures in your sample data and population. For example, the standard error of the mean measures how far the sample mean (average) of the data is likely to be from the true population mean—the same applies to other types of standard errors.
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First, the standard error of a sample accounts for statistical fluctuation.
Researchers depend on this statistical measure to know how much sampling fluctuation exists in their sample data. In other words, it shows the extent to which a statistical measure varies from sample to population.
In addition, standard error serves as a measure of accuracy. Using standard error, a researcher can estimate the efficiency and consistency of a sample to know precisely how a sampling distribution represents a population.
There are five types of standard error which are:
The standard error of the mean accounts for the difference between the sample mean and the population mean. In other words, it quantifies how much variation is expected to be present in the sample mean that would be computed from every possible sample, of a given size, taken from the population.
How to Find SEM (With Formula)
SEM = Standard Deviation ÷ √n
Where;
n = sample size
Suppose that the standard deviation of observation is 15 with a sample size of 100. Using this formula, we can deduce the standard error of the mean as follows:
SEM = 15 ÷ √100
Standard Error of Mean in 1.5
The standard error of measurement accounts for the consistency of scores within individual subjects in a test or examination.
This means it measures the extent to which estimated test or examination scores are spread around a true score.
A more formal way to look at it is through the 1985 lens of Aera, APA, and NCME. Here, they define a standard error as “the standard deviation of errors of measurement that is associated with the test scores for a specified group of test-takers….”
How to Find Standard Error of Measurement
Where;
rxx is the reliability of the test and is calculated as:
Rxx = S2T / S2X
Where;
S2T = variance of the true scores.
S2X = variance of the observed scores.
Suppose an organization has a reliability score of 0.4 and a standard deviation of 2.56. This means
SEm = 2.56 × √1–0.4 = 1.98
The standard error of the estimate measures the accuracy of predictions in sampling, research, and data collection. Specifically, it measures the distance that the observed values fall from the regression line which is the single line with the smallest overall distance from the line to the points.
How to Find Standard Error of the Estimate
The formula for standard error of the estimate is as follows:
Where;
σest is the standard error of the estimate;
Y is an actual score;
Y’ is a predicted score, and;
N is the number of pairs of scores.
The numerator is the sum of squared differences between the actual scores and the predicted scores.
The standard of error of proportion in an observation is the difference between the sample proportion and the population proportion of your target audience. In more technical terms, this variable is the spread of the sample proportion about the population proportion.
How to Find Standard Error of the Proportion
The formula for calculating the standard error of the proportion is as follows:
Where;
P (hat) is equal to x ÷ n (with number of success x and the total number of observations of n)
Residual standard error accounts for how well a linear regression model fits the observation in a systematic investigation. A linear regression model is simply a linear equation representing the relationship between two variables, and it helps you to predict similar variables.
How to Calculate Residual Standard Error
The formula for residual standard error is as follows:
Residual standard error = √Σ(y – ŷ)2/df
where:
y: The observed value
ŷ: The predicted value
df: The degrees of freedom, calculated as the total number of observations – total number of model parameters.
As you interpret your data, you should note that the smaller the residual standard error, the better a regression model fits a dataset, and vice versa.
The formula for calculating standard error is as follows:
Where
σ – Standard deviation
n – Sample size, i.e., the number of observations in the sample
Here’s how this works in real-time.
Suppose the standard deviation of a sample is 1.5 with 4 as the sample size. This means:
Standard Error = 1.5 ÷ √4
That is; 1.5 ÷2 = 0.75
Alternatively, you can use a standard error calculator to speed up the process for larger data sets.
As stated earlier, researchers use the standard error to measure the reliability of observation. This means it allows you to compare how far a particular variable in the sample data is from the population of interest.
Calculating standard error is just one piece of the puzzle; you need to know how to interpret your data correctly and draw useful insights for your research. Generally, a small standard error is an indication that the sample mean is a more accurate reflection of the actual population mean, while a large standard error means the opposite.
Suppose you need to find the standard error of the mean of a data set using the following information:
Standard Deviation: 1.5
n = 13
Standard Error of the Mean = Standard Deviation ÷ √n
1.5 ÷ √13 = 0.42
After calculating the standard error of your observation, the next thing you should do is present this data as part of the numerous variables affecting your observation. Typically, researchers report the standard error alongside the mean or in a confidence interval to communicate the uncertainty around the mean.
The most common application of standard error are in statistics and economics. In statistics, standard error allows researchers to determine the confidence interval of their data sets, and in some cases, the margin of error. Researchers also use standard error in hypothesis testing and regression analysis.
The major difference between standard deviation and standard error of the mean is how they account for the differences between the sample data and the population of interest.
Researchers use standard deviation to measure the variability or dispersion of a data set to its mean. On the other hand, the standard error of the mean accounts for the difference between the mean of the data sample and that of the target population.
Something else to note here is that the standard error of a sample is always smaller than the corresponding standard deviation.
During calculation, the standard error is represented as σx̅.
No. Margin of Error and standard error are not the same. Researchers use the standard error to measure the preciseness of an estimate of a population, meanwhile margin of error accounts for the degree of error in results received from random sampling surveys.
The standard error is calculated as s / √n where;
s: Sample standard deviation
n: Sample size
On the other hand, margin of error = z*(s/√n) where:
z: Z value that corresponds to a given confidence level
s: Sample standard deviation
n: Sample size
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