Which of the following refers to the standard deviation of a sampling distribution?

Sample statistics are random variables because they vary from sample to sample. As a result, sample statistics have a distribution called the sampling distribution. The video below demonstrates the construction of a sampling distribution for a known population proportion using StatKey (http://www.lock5stat.com/StatKey/index.html). StatKey is a free online application that we will be using throughout the course.

An important aspect of a sampling distribution is the standard error (SE). The standard error is the standard deviation of a sampling distribution.  For a single categorical variable this may be referred to as the standard error of the proportion. For a single quantitative variable this may be referred to as the standard error of the mean. If a sampling distribution is constructed using data from a population, the mean of the sampling distribution will be approximately equal to the population parameter.

Sampling Distribution Distribution of sample statistics with a mean approximately equal to the mean in the original distribution and a standard deviation known as the standard error

Standard Error Standard deviation of a sampling distribution

Using StatKey to Construct a Sampling Distribution Given a Known Population Proportion

Note that this method of constructing a sampling distribution requires that we have population data. In most cases we do not know all of the population values. If we did, then we wouldn't need to construct a confidence interval to estimate the population parameter! In those cases we can use bootstrapping methods which you will see in the next section. 

As you look through the following examples, note that when the sample size is large the sampling distribution is approximately symmetrical and centered at the population parameter. 

What Is the Standard Error?

The standard error (SE) of a statistic is the approximate standard deviation of a statistical sample population.

The standard error is a statistical term that measures the accuracy with which a sample distribution represents a population by using standard deviation. In statistics, a sample mean deviates from the actual mean of a population; this deviation is the standard error of the mean.

Key Takeaways

• The standard error (SE) is the approximate standard deviation of a statistical sample population.
• The standard error describes the variation between the calculated mean of the population and one which is considered known, or accepted as accurate.
• The more data points involved in the calculations of the mean, the smaller the standard error tends to be.

Standard Error

Understanding Standard Error

The term "standard error" is used to refer to the standard deviation of various sample statistics, such as the mean or median. For example, the "standard error of the mean" refers to the standard deviation of the distribution of sample means taken from a population. The smaller the standard error, the more representative the sample will be of the overall population.

The relationship between the standard error and the standard deviation is such that, for a given sample size, the standard error equals the standard deviation divided by the square root of the sample size. The standard error is also inversely proportional to the sample size; the larger the sample size, the smaller the standard error because the statistic will approach the actual value.

The standard error is considered part of inferential statistics. It represents the standard deviation of the mean within a dataset. This serves as a measure of variation for random variables, providing a measurement for the spread. The smaller the spread, the more accurate the dataset.

Standard error and standard deviation are measures of variability, while central tendency measures include mean, median, etc.

Formula and Calculation of Standard Error

The standard error of an estimate can be calculated as the standard deviation divided by the square root of the sample size:

SE = σ / √n

where

• σ = the population standard deviation
• √n = the square root of the sample size

If the population standard deviation is not known, you can substitute the sample standard deviation, s, in the numerator to approximate the standard error.

Requirements for Standard Error 

When a population is sampled, the mean, or average, is generally calculated. The standard error can include the variation between the calculated mean of the population and one which is considered known, or accepted as accurate. This helps compensate for any incidental inaccuracies related to the gathering of the sample.

In cases where multiple samples are collected, the mean of each sample may vary slightly from the others, creating a spread among the variables. This spread is most often measured as the standard error, accounting for the differences between the means across the datasets.

The more data points involved in the calculations of the mean, the smaller the standard error tends to be. When the standard error is small, the data is said to be more representative of the true mean. In cases where the standard error is large, the data may have some notable irregularities.

The standard deviation is a representation of the spread of each of the data points. The standard deviation is used to help determine the validity of the data based on the number of data points displayed at each level of standard deviation. Standard errors function more as a way to determine the accuracy of the sample or the accuracy of multiple samples by analyzing deviation within the means.

Standard Error vs. Standard Deviation

The standard error normalizes the standard deviation relative to the sample size used in an analysis. Standard deviation measures the amount of variance or dispersion of the data spread around the mean. The standard error can be thought of as the dispersion of the sample mean estimations around the true population mean. As the sample size becomes larger, the standard error will become smaller, indicating that the estimated sample mean value better approximates the population mean.

Example of Standard Error

Say that an analyst has looked at a random sample of 50 companies in the S&P 500 to understand the association between a stock's P/E ratio and subsequent 12-month performance in the market. Assume that the resulting estimate is -0.20, indicating that for every 1.0 point in the P/E ratio, stocks return 0.2% poorer relative performance. In the sample of 50, the standard deviation was found to be 1.0.

The standard error is thus:

SE = 1.0/√50 = 1/7.07 = 0.141

Therefore, we would report the estimate as -0.20% ± 0.14, giving us a confidence interval of (-0.34 - -0.06). The true mean value of the association of the P/E on returns of the S&P 500 would therefore fall within that range with a high degree of probability.

Say now that we increase the sample of stocks to 100 and find that the estimate changes slightly from -0.20 to -0.25, and the standard deviation falls to 0.90. The new standard error would thus be:

SE = 0.90/√100 = 0.90/10 = 0.09.

The resulting confidence interval becomes -0.25 ± 0.09 = (-0.34 - -0.16), which is a tighter range of values.

What Is Meant by Standard Error?

Standard error is intuitively the standard deviation of the sampling distribution. In other words, it depicts how much disparity there is likely to be in a point estimate obtained from a sample relative to the true population mean.

What Is a Good Standard Error?

Standard error measures the amount of discrepancy that can be expected in a sample estimate compared to the true value in the population. Therefore, the smaller the standard error the better. In fact, a standard error of zero (or close to it) would indicate that the estimated value is exactly the true value.

How Do You Find the Standard Error?

The standard error takes the standard deviation and divides it by the square root of the sample size. Many statistical software packages automatically compute standard errors.

The Bottom Line

The standard error (SE) measures the dispersion of estimated values obtained from a sample around the true value to be found in the population. Statistical analysis and inference often involves drawing samples and running statistical tests to determine associations and correlations between variables. The standard error thus tells us with what degree of confidence we can expect the estimated value to approximate the population value.

Which of the following refers to the standard deviation of sampling distribution?

Which of the following refers to the standard deviation of a sampling distribution A It is the sum of squares?

Which of the following is the standard deviation of the sampling distribution of the sample mean when the population is finite?

Which of the following properties describes the sampling distribution of the sample mean?