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In simple terms, the sample mean standard error is an estimate of the variance between the sample mean and the population mean, while the sample standard deviation is the degree to which people in the sample differ from the sample mean.

The terms "standard error" and "standard deviation" are often confused. ^{ 1 } The contrast between the two reflects an important difference between describing data and inference, which should be appreciated by all researchers.

## How do you interpret the standard error of the mean?

The standard error of the mean allows the researcher to establish a confidence interval in which the mean of the population is likely to fall. Formula (1-P) (usually P <0.05) represents the probability that the population mean is within the calculated range (usually 95%). Standard deviation (often SD) is a measure of variability. When we calculate the sample standard deviation, we use it as an estimate of the variability of the population from which the sample was taken. For data with a normally distributed ^{ 2 }, about 95% of people have values within 2 standard deviations of the mean, and the remaining 5% are evenly distributed above and below these values. limits. Contrary to popular misconception, standard deviation is an acceptable measure of variability regardless of distribution. About 95% of distribution observations usually fall within 2 standard deviation limits, although they may all be at the same end. However, we can choose a different summary statistic if the data is skewed. ^{ 3 }

When we compute the mean of a sample, we usually don't care about the mean of that sample.a specific sample, but rather an average for people of this type, in terms of statistics, for the population from which the sample was taken. We usually collect data to summarize them, so we use the sample mean as an estimate of the mean for the entire population. Now the sample mean varies from sample to sample. How this change occurs is described by the "sample distribution" of the mean. We can estimate how much the sample mean differs from the standard deviation of this sample distribution, which we call the standard error (SE) of the mean estimate. Since the standard error is a kind of standard deviation, the confusion is understandable. Another way to account for standard error is to measure the accuracy of the sample mean.

The standard error of the sample mean depends on both the standard deviation and the sample size by a simple relationship SE = SD / √ (sample size). The standard error decreases as the sample size increases as the amount of random variation decreases. This idea is used, for example, dTo calculate the sample size for a controlled experiment. In contrast, the standard deviation does not change as our sample size increases.

Therefore, when we want to determine the degree of dispersion of some indicators, we use the standard deviation. When we want to report the uncertainty in the estimate of the mean, we report the standard error of the mean. Standard error is most useful for calculating the confidence interval. For a large sample, the 95% confidence interval is obtained as 1.96 × SE on either side of the mean. We will discuss confidence intervals in more detail in the statistical note below. The standard error is also used to calculate R. values in many cases.

The principle of distribution of samples is applied to other quantities that we can estimate from a sample, such as B. proportion or regression coefficient, and for contrasts between two samples, for example. B. the risk ratio or the difference between two funds or stocks. All these quantities have uncertainties due to sample variability, and for such an estimate,A standard error can be calculated to indicate the level of uncertainty.

## What is the standard error of the proportion?

The standard error of proportion is defined as the spread of the proportion of the sample over the proportion of the population. Specifically, standard error is an estimate of the standard deviation of a statistic. It has a similar nature with standard deviation as both are measures of variance. Many publications use the ± sign to relate the standard deviation (SD) or standard error (SE) to the observed mean - for example, 69.4 ± 9.3 kg. This designation does not indicate that the second number is a standard deviation or standard error (or actually something else). A review of 88 articles published in 2002 found that 12 (14%) did not indicate which measure of variance was reported (and three did not report any measure of variance). ^{ 4 } Policy BMJ and many other journals should remove the ± marks and ask authors to clearly state whether a standard deviation or standard error is being reported. All journals must follow this practice.

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