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Standard deviation (SD), which is often seen in papers and figures, and standard error (SE) are two similar statistical terms.
In this article, we will explain how the two are used interchangeably.
Consider the function of standard deviation
Let us consider what standard deviation is. As you know, if the degree of scatter in the data obtained from a test of the effects of a drug can be quantified, the drug can be evaluated by comparing its values.
The number that indicates the degree of scatter is called the “standard deviation (SD).” It is important to note that the standard deviation is a measure of the degree of scatter compared with the average value. Is this difference significant? Therefore, if the data distribution is biased, outliers exist, and the mean and median are significantly different, it is not appropriate to use the mean as a representative value for this data. Consequently, the standard deviation will also be strongly influenced by the outliers, resulting in a large value.
Next, let us consider the standard error (SE).
The standard error (SE) is the standard deviation (SD) divided by √n.
As can be seen from the above equation, the standard error (SE) is always smaller than the standard deviation.
What are the measures needed to know the population?
Because the standard error (SE) is always smaller than the standard deviation, it is better to create graphs using the standard error (SE) to make the data appear to have less variation. In fact, the standard error (SE) is used in many medical studies; however, it is never used to improve the appearance of data.
Again, let us take a closer look at the aforementioned equation. A large standard deviation indicates that there are outliers or large variations in the data obtained from the test. Therefore, the smaller the standard deviation, the better.
Therefore, what should we consider for the sample size n? To learn about the population, a larger sample size n is better. Consider the standard error (SE) formula. What standard deviation and n would make the standard error (SE) smaller?
The standard error (SE) is small when the standard deviation is small and n is large. In other words, the standard error (SE) obtained by considering the degree of scatter of the data and the sample size n is a barometer of the population. Therefore, standard error (SE) is important.
Let us take a concrete look at a simple example.
The following figure, “Mean Value Bar Chart for Drug A and Drug B,” is one that you will probably see in many papers.
Look at the error bars shown for the averages for drugs A and B. You know this represents an error, but what does it mean?Error bars indicate the degree of error in the data, such as ± standard deviation, ± standard error (SE), percentile, and 95% confidence interval (CI).
Both figures represent the same data and are therefore not erroneous. However, the error bar lengths differ between (1) and (2). This means that the figures and descriptions in the paper must clearly indicate what the error bars represent.
Let us return to this theme and consider the use of standard deviation and standard error (SE). The reason for using standard deviation for error bars is that standard deviation is an indicator of the variability of the data itself; therefore, when one wants to show or compare purely the variability of the data, standard deviation should be used, for example, when describing a subject’s characteristics. The ± standard deviation can be approximately 2/3 of the data. A ±1.96 x standard deviation would include roughly 95% of the data. These guidelines are fairly accurate when the data follow a normal distribution, but cannot be used when they do not.
As a familiar example, it is well known that the normal range of laboratory values given after a physical examination is an interval that includes 95% of healthy people.
On the other hand, the standard error (SE) is a barometer used to determine the population; therefore, it is an interval estimator of the population mean. Therefore, when estimating or comparing the population mean, we can simply use the standard error (SE). Because it is a population mean, the standard error (SE) is used in many studies.
Calculation of standard deviation
(1) The mean value is subtracted from the individual data.
The value obtained is called “deviation.”
(2) Square the individual deviations. (“deviation × deviation”).
The obtained value is called “squared deviation.”
(3) The obtained n cases of squared deviations are summed.
The total value is called “sum of squared deviation.”
(4) Divide the sum of squared deviation by n numbers (n examples).
The value obtained is called the “variance.”
(5) Calculate the root of variance.
The obtained value is called “standard deviation (SD).”
Calculation of standard error
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