Standard error of the sampling distribution

From Wikipedia, the free encyclopedia

From Wikipedia, the free encyclopedia

For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above and below the actual value.

The standard error (SE)[1] of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution[2] or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM).[1]

The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size.[1] In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.

In regression analysis, the term «standard error» refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals).

Standard error of the sample mean[edit]

Exact value[edit]

Suppose a statistically independent sample of n observations {displaystyle x_{1},x_{2},ldots ,x_{n}} is taken from a statistical population with a standard deviation of sigma . The mean value calculated from the sample, {bar {x}}, will have an associated standard error on the mean, {displaystyle {sigma }_{bar {x}}}, given by:[1]

{displaystyle {sigma }_{bar {x}} ={frac {sigma }{sqrt {n}}}}.

Practically this tells us that when trying to estimate the value of a population mean, due to the factor 1/{sqrt {n}}, reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations.

Estimate[edit]

The standard deviation sigma of the population being sampled is seldom known. Therefore, the standard error of the mean is usually estimated by replacing sigma with the sample standard deviation sigma _{x} instead:

{displaystyle {sigma }_{bar {x}} approx {frac {sigma _{x}}{sqrt {n}}}}.

As this is only an estimator for the true «standard error», it is common to see other notations here such as:

{displaystyle {widehat {sigma }}_{bar {x}}approx {frac {sigma _{x}}{sqrt {n}}}}   or alternately   {displaystyle {s}_{bar {x}} approx {frac {s}{sqrt {n}}}}.

A common source of confusion occurs when failing to distinguish clearly between:

Accuracy of the estimator[edit]

When the sample size is small, using the standard deviation of the sample instead of the true standard deviation of the population will tend to systematically underestimate the population standard deviation, and therefore also the standard error. With n = 2, the underestimate is about 25%, but for n = 6, the underestimate is only 5%. Gurland and Tripathi (1971) provide a correction and equation for this effect.[3] Sokal and Rohlf (1981) give an equation of the correction factor for small samples of n < 20.[4] See unbiased estimation of standard deviation for further discussion.

Derivation[edit]

The standard error on the mean may be derived from the variance of a sum of independent random variables,[5] given the definition of variance and some simple properties thereof. If {displaystyle x_{1},x_{2},ldots ,x_{n}} are n independent samples from a population with mean {bar {x}} and standard deviation sigma , then we can define the total

{displaystyle T=(x_{1}+x_{2}+cdots +x_{n})}

which due to the Bienaymé formula, will have variance

{displaystyle operatorname {Var} (T)approx {big (}operatorname {Var} (x_{1})+operatorname {Var} (x_{2})+cdots +operatorname {Var} (x_{n}){big )}=nsigma ^{2}.}

where we’ve approximated the standard deviations, i.e., the uncertainties, of the measurements themselves with the best value for the standard deviation of the population. The mean of these measurements {bar {x}} is simply given by

{displaystyle {bar {x}}=T/n}.

The variance of the mean is then

{displaystyle operatorname {Var} ({bar {x}})=operatorname {Var} left({frac {T}{n}}right)={frac {1}{n^{2}}}operatorname {Var} (T)={frac {1}{n^{2}}}nsigma ^{2}={frac {sigma ^{2}}{n}}.}

The standard error is, by definition, the standard deviation of {bar {x}} which is simply the square root of the variance:

{displaystyle sigma _{bar {x}}={sqrt {frac {sigma ^{2}}{n}}}={frac {sigma }{sqrt {n}}}}.

For correlated random variables the sample variance needs to be computed according to the Markov chain central limit theorem.

Independent and identically distributed random variables with random sample size[edit]

There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X such that,

{displaystyle operatorname {Var} (T)=operatorname {E} (N)operatorname {Var} (X)+operatorname {Var} (N){big (}operatorname {E} (X){big )}^{2}}[6]

If N has a Poisson distribution, then {displaystyle operatorname {E} (N)=operatorname {Var} (N)} with estimator {displaystyle N=n}. Hence the estimator of {displaystyle operatorname {Var} (T)} becomes {displaystyle nS_{X}^{2}+n{bar {X}}^{2}}, leading the following formula for standard error:

{displaystyle operatorname {Standard~Error} ({bar {X}})={sqrt {frac {S_{X}^{2}+{bar {X}}^{2}}{n}}}}

(since the standard deviation is the square root of the variance)

Student approximation when σ value is unknown[edit]

In many practical applications, the true value of σ is unknown. As a result, we need to use a distribution that takes into account that spread of possible σ’s.
When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. The standard error is the standard deviation of the Student t-distribution. T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. Small samples are somewhat more likely to underestimate the population standard deviation and have a mean that differs from the true population mean, and the Student t-distribution accounts for the probability of these events with somewhat heavier tails compared to a Gaussian. To estimate the standard error of a Student t-distribution it is sufficient to use the sample standard deviation «s» instead of σ, and we could use this value to calculate confidence intervals.

Note: The Student’s probability distribution is approximated well by the Gaussian distribution when the sample size is over 100. For such samples one can use the latter distribution, which is much simpler.

Assumptions and usage[edit]

An example of how {displaystyle operatorname {SE} } is used is to make confidence intervals of the unknown population mean. If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean. The following expressions can be used to calculate the upper and lower 95% confidence limits, where {bar {x}} is equal to the sample mean, {displaystyle operatorname {SE} } is equal to the standard error for the sample mean, and 1.96 is the approximate value of the 97.5 percentile point of the normal distribution:

Upper 95% limit {displaystyle ={bar {x}}+(operatorname {SE} times 1.96),} and
Lower 95% limit {displaystyle ={bar {x}}-(operatorname {SE} times 1.96).}

In particular, the standard error of a sample statistic (such as sample mean) is the actual or estimated standard deviation of the sample mean in the process by which it was generated. In other words, it is the actual or estimated standard deviation of the sampling distribution of the sample statistic. The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE.

Standard errors provide simple measures of uncertainty in a value and are often used because:

  • in many cases, if the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated;
  • when the probability distribution of the value is known, it can be used to calculate an exact confidence interval;
  • when the probability distribution is unknown, Chebyshev’s or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and
  • as the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.

Standard error of mean versus standard deviation[edit]

In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem.[7]

Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean.[8] If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases.

Extensions[edit]

Finite population correction (FPC)[edit]

The formula given above for the standard error assumes that the population is infinite. Nonetheless, it is often used for finite populations when people are interested in measuring the process that created the existing finite population (this is called an analytic study). Though the above formula is not exactly correct when the population is finite, the difference between the finite- and infinite-population versions will be small when sampling fraction is small (e.g. a small proportion of a finite population is studied). In this case people often do not correct for the finite population, essentially treating it as an «approximately infinite» population.

If one is interested in measuring an existing finite population that will not change over time, then it is necessary to adjust for the population size (called an enumerative study). When the sampling fraction (often termed f) is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a »finite population correction» (a.k.a.: FPC):[9]
[10]

{displaystyle operatorname {FPC} ={sqrt {frac {N-n}{N-1}}}}

which, for large N:

{displaystyle operatorname {FPC} approx {sqrt {1-{frac {n}{N}}}}={sqrt {1-f}}}

to account for the added precision gained by sampling close to a larger percentage of the population. The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N.

This happens in survey methodology when sampling without replacement. If sampling with replacement, then FPC does not come into play.

Correction for correlation in the sample[edit]

Expected error in the mean of A for a sample of n data points with sample bias coefficient ρ. The unbiased standard error plots as the ρ = 0 diagonal line with log-log slope −½.

If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of the mean (actually a correction on the standard deviation part) may be obtained by multiplying the calculated standard error of the sample by the factor f:

f={sqrt {frac {1+rho }{1-rho }}},

where the sample bias coefficient ρ is the widely used Prais–Winsten estimate of the autocorrelation-coefficient (a quantity between −1 and +1) for all sample point pairs. This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Street stock quotes. Moreover, this formula works for positive and negative ρ alike.[11] See also unbiased estimation of standard deviation for more discussion.

See also[edit]

  • Illustration of the central limit theorem
  • Margin of error
  • Probable error
  • Standard error of the weighted mean
  • Sample mean and sample covariance
  • Standard error of the median
  • Variance

References[edit]

  1. ^ a b c d Altman, Douglas G; Bland, J Martin (2005-10-15). «Standard deviations and standard errors». BMJ: British Medical Journal. 331 (7521): 903. doi:10.1136/bmj.331.7521.903. ISSN 0959-8138. PMC 1255808. PMID 16223828.
  2. ^ Everitt, B. S. (2003). The Cambridge Dictionary of Statistics. CUP. ISBN 978-0-521-81099-9.
  3. ^ Gurland, J; Tripathi RC (1971). «A simple approximation for unbiased estimation of the standard deviation». American Statistician. 25 (4): 30–32. doi:10.2307/2682923. JSTOR 2682923.
  4. ^ Sokal; Rohlf (1981). Biometry: Principles and Practice of Statistics in Biological Research (2nd ed.). p. 53. ISBN 978-0-7167-1254-1.
  5. ^ Hutchinson, T. P. (1993). Essentials of Statistical Methods, in 41 pages. Adelaide: Rumsby. ISBN 978-0-646-12621-0.
  6. ^ Cornell, J R, and Benjamin, C A, Probability, Statistics, and Decisions for Civil Engineers, McGraw-Hill, NY, 1970, ISBN 0486796094, pp. 178–9.
  7. ^ Barde, M. (2012). «What to use to express the variability of data: Standard deviation or standard error of mean?». Perspect. Clin. Res. 3 (3): 113–116. doi:10.4103/2229-3485.100662. PMC 3487226. PMID 23125963.
  8. ^ Wassertheil-Smoller, Sylvia (1995). Biostatistics and Epidemiology : A Primer for Health Professionals (Second ed.). New York: Springer. pp. 40–43. ISBN 0-387-94388-9.
  9. ^ Isserlis, L. (1918). «On the value of a mean as calculated from a sample». Journal of the Royal Statistical Society. 81 (1): 75–81. doi:10.2307/2340569. JSTOR 2340569. (Equation 1)
  10. ^ Bondy, Warren; Zlot, William (1976). «The Standard Error of the Mean and the Difference Between Means for Finite Populations». The American Statistician. 30 (2): 96–97. doi:10.1080/00031305.1976.10479149. JSTOR 2683803. (Equation 2)
  11. ^ Bence, James R. (1995). «Analysis of Short Time Series: Correcting for Autocorrelation». Ecology. 76 (2): 628–639. doi:10.2307/1941218. JSTOR 1941218.
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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

What is a Standard Error? 

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. 

Explore: Survey Errors To Avoid: Types, Sources, Examples, Mitigation

Why is Standard Error Important? 

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. 

How Many Types of Standard Error Exist?

There are five types of standard error which are:

  1. Standard error of the mean
  2. Standard error of measurement
  3. Standard error of the proportion
  4. Standard error of estimate
  5. Residual Standard Error

1. Standard Error of the Mean (SEM) 

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 

2. Standard Error of Measurement

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….” 

Read: 7 Types of Data Measurement Scales in Research

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 

3. Standard Error of the Estimate 

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.

4. Standard Error of Proportion 

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) 

5. Residual Standard Error 

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. 

How Do You Calculate Standard Error? 

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. 

How to Interpret Standard Error Values

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. 

Standard Error Example

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

How Should You Report the Standard Error?

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. 

Applications of Standard Error

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. 

FAQs About Standard Error

  • What Is the Difference Between Standard Deviation and Standard Error of the Mean? 

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. 

  • What Is The Symbol for Standard Error?

During calculation, the standard error is represented as σx̅. 

  • Is Standard Error the Same as Margin of Error?

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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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.

Источник

The standard error of a statistic is actually the standard deviation of the sampling distribution of that statistic. Standard errors reflect how much sampling fluctuation a statistic will show. The inferential statistics (deductive statistics) involved in the construction of confidence intervals and significance testing are based on standard errors. Increasing the sample size, the Standard Error decreases.

In practical applications, the true value of the standard deviation of the error is unknown. As a result, the term standard error is often used to refer to an estimate of this unknown quantity.

The size of the standard error is affected by two values.

  1. The Standard Deviation of the population which affects the standard error. Larger the population’s standard deviation (σ), larger is standard error i.e. $frac{sigma}{sqrt{n}}$. If the population is homogeneous (which results in small population standard deviation), the standard error will also be small.
  2. The standard error is affected by the number of observations in a sample. A large sample will result in a small standard error of estimate (indicates less variability in the sample means)

Application

Standard errors are used in different statistical tests such as

  • to measure the distribution of the sample means
  • to build confidence intervals for means, proportions, differences between means, etc., for cases when population standard deviation is known or unknown.
  • to determine the sample size
  • in control charts for control limits for means
  • in comparisons test such as z-test, t-test, Analysis of Variance, Correlation and Regression Analysis (standard error of regression), etc.

(1) Standard Error of Means

The standard error for the mean or standard deviation of the sampling distribution of the mean measures the deviation/ variation in the sampling distribution of the sample mean, denoted by $sigma_{bar{x}}$ and calculated as the function of the standard deviation of the population and respective size of the sample i.e

$sigma_{bar{x}}=frac{sigma}{sqrt{n}}$                      (used when population is finite)

If the population size is infinite then ${sigma_{bar{x}}=frac{sigma}{sqrt{n}} times sqrt{frac{N-n}{N}}}$ because $sqrt{frac{N-n}{N}}$ tends towards 1 as N tends to infinity.

When standard deviation (σ) of the population is unknown, we estimate it from the sample standard deviation. In this case standard error formula is $sigma_{bar{x}}=frac{S}{sqrt{n}}$

(2) Standard Error for Proportion

Standard error for proportion can also be calculated in same manner as we calculated standard error of mean, denoted by $sigma_p$ and calculated as $sigma_p=frac{sigma}{sqrt{n}}sqrt{frac{N-n}{N}}$.

In case of finite population $sigma_p=frac{sigma}{sqrt{n}}$
in case of infinite population $sigma=sqrt{p(1-p)}=sqrt{pq}$, where p is the probability that an element possesses the studied trait and q=1-p is the probability that it does not.

(3) Standard Error for Difference between Means

Standard error for difference between two independent quantities is the square root of the of the sum of the squared standard errors of the both quantities i.e $sigma_{bar{x}_1+bar{x}_2}=sqrt{frac{sigma_1^2}{n_1}+frac{sigma_2^2}{n_2}}$, where $sigma_1^2$ and $sigma_2^2$ are the respective variances of the two independent population to be compared and $n_1+n_2$ are the respective sizes of the two samples drawn from their respective populations.

Unknown Population Variances
If the variances of the two populations are unknown, we estimate them from the two samples i.e. $sigma_{bar{x}_1+bar{x}_2}=sqrt{frac{S_1^2}{n_1}+frac{S_2^2}{n_2}}$, where $S_1^2$ and $S_2^2$ are the respective variances of the two samples drawn from their respective population.

Equal Variances are assumed
In case when it is assumed that the variance of the two populations are equal, we can estimate the value of these variances with a pooled variance $S_p^2$ calculated as a function of $S_1^2$ and $S_2^2$ i.e

[S_p^2=frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}]
[sigma_{bar{x}_1}+{bar{x}_2}=S_p sqrt{frac{1}{n_1}+frac{1}{n_2}}]

(4) Standard Error for Difference between Proportions

The standard error of the difference between two proportions is calculated in the same way as the standard error of the difference between means is calculated i.e.
begin{eqnarray*}
sigma_{p_1-p_2}&=&sqrt{sigma_{p_1}^2+sigma_{p_2}^2}\
&=& sqrt{frac{p_1(1-p_1)}{n_1}+frac{p_2(1-p_2)}{n_2}}
end{eqnarray*}
where $p_1$ and $p_2$ are the proportion for infinite population calculated for the two samples of sizes $n_1$ and $n_2$.

Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan.
Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan.
l like Applied Statistics, Mathematics, and Statistical Computing.
Statistical and Mathematical software used is SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel.
Like to use type-setting LaTeX for composing Articles, thesis, etc.

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Standard error of the mean measures how spread out the means of the sample can be from the actual population mean. Standard error allows you to build a relationship between a sample statistic (computed from a smaller sample of the population) and the population’s actual parameter.

Standard Error – A practical guide with examples. Photo by Sergio.

What is Standard Error?

The sample error serves as a means to understand the actual population parameter (like population mean) without actually estimating it.

Consider the following scenario:

A researcher ‘X’ is collecting data from a large population of voters. For practical reasons he can’t reach out to each and every voter. So, only a small randomized sample (of voters) is selected for data collection.

Once the data for the sample is collected, you calculate the mean (or any statistic) of that sample. But then, this mean you just computed is only the sample mean. It cannot be considered the entire population’s mean. You can however expect it to be somewhere close to population’s mean.

So how can you know the actual population mean?

While its not possible to compute the exactly value, you can use standard error to estimate how far the sample mean may spread from the actual population mean.

To be more precise,

The Standard Error of the Mean describes how far a sample mean may vary from the population mean.

In this post, you will understand clearly:

  1. What Standard Error Tells Us?
  2. What is the Sample Error Formula?
  3. How to calculate Standard Error?
  4. How to use standard error to compute confidence interval?
  5. Example Problem and solution

How to understand Standard Error?

Let’s first clearly understand the intuition behind and the need for standard error.

Now, let’s suppose you are working in agriculture domain and you want to know the annual yield of a particular variety of coconut trees. While the entire population of coconut trees has a certain mean (and standard deviation) of annual yield, it is not practical to take measurements of each and every tree out there.

So, what do you do?

To estimate this you collect samples of coconut yield (number of nuts per tree per year) from different trees. And to keep your findings unbiased, you collect samples across different places.

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Let’s say, you collected data from approx ~5 trees per sample from different places and the numbers are shown below.

# Annual yield of coconut
sample1 = [400, 420, 470, 510, 590]

sample2 = [430, 500, 570, 620, 710, 800, 900]

sample3 = [360, 410, 490, 550, 640]

In above data, the variables sample1, sample2 and sample3 contain the samples of annual yield values collected, where each number represents the yield of one individual tree.

Observe that the yield varies not just across the trees, but also across the different samples.

Image showing sampling distribution to explain standard error

Although we compute means of the samples, we are actually not interested in the means of the sample, but the overall mean annual yield of coconut of this variety.

Now, you may ask: ‘Why can’t we just put the values from all these samples in one bucket and simply compute the mean and standard deviation and consider that as the population’s parameter?‘

Well, the problem is, if you do that practically what happens is, as you receive few more samples, the real population’s parameter begins to come out which is likely to be (slightly) different from the parameter you computed earlier.

Below is a code demo.

from statistics import mean, stdev

# Overall mean from the first two samples
sample1 = [400, 420, 470, 510, 590]
sample2 = [430, 500, 570, 620, 710, 800, 900]
print("Mean of first two samples: ", mean(sample1 + sample2))

# Overall mean after introducing 3rd sample
sample3 = [360, 410, 490, 550, 640]
print("Mean after including 3rd sample: ", mean(sample1 + sample2 + sample3))

Output:

Mean of first two samples:  576.6666666666666
Mean after including 3rd sample:  551.1764705882352

As you add more and more samples, the computed parameters keep changing.

So how to tackle this?

If you notice, each sample has its own mean that varies between a particular range. This mean (of the sample) has its own standard deviation. This measure of standard deviation of the mean is called the standard error of the mean.

Its important to note, it is different from the standard deviation of the data. The difference is, while standard deviation tells you how the overall data is distributed around the mean, the standard error tells you how the mean itself is distributed.

This way, it can be used to generalize the sample mean so it can be used as an estimate of the whole population.

In fact, standard error can be generalized to any statistic like standard deviation, median etc. For example, if you compute the standard deviation of the standard deviations (of the samples), it is called, standard error of the standard deviation. Feels like a tongue twister. But most commonly, when someone mention ‘Standard error’ it typically refers to the ‘Standard error of of the mean’.

What is the Formula?

To calculate standard error, you simply divide the standard deviation of a given sample by the square root of the total number of items in the sample.

$$SE_{bar{x}} = frac{sigma}{sqrt{n}}$$

where, $SE_{bar{x}}$ is the standard error of the mean, $sigma$ is the standard deviation of the sample and n is the number of items in sample.

Do not confuse this with standard deviation. Because standard error of the sample statistic (like mean) is typically much smaller than the population standard deviation.

Notice few things here:

  1. The Standard error depends on the number of items in the sample. As you increase the number of items in the sample, lower will be the standard error and more certain you will be about the estimates.
  2. It uses statistics (standard deviation and number of items) computed from the sample itself, and not of the population. That is, you don’t need to know the population parameters beforehand to compute standard error. This makes it pretty convenient.
  3. Standard error can also be used as an estimate of how representative a given sample is of a population. The smaller the value, more representative is the sample of the whole population.

Below is a computation for the standard error of the mean:

# Compute Standard Error
sample1 = [400, 420, 470, 510, 590]
se = stdev(sample1)/len(sample1)
print('Standard Error: ', round(se, 2))

Standard Error:  15.19

How to calculate standard error?

Problem Statement

A school aptitude test for 15 year old students studying in a particular territory’s curriculum, is designed to have a mean score of 80 units and a standard deviation of 10 units. A sample of 15 answer papers has a mean score of 85. Can we assume that these 15 scores come from the designated population?

Solution

Our task is to determine if this sample comes from the above mentioned population.

How to solve this?

We approach this problem by computing the standard error of the sample means and use it to compute the confidence interval between which the sample means are expected to fall.

If the given sample mean falls inside this interval, then its safe to assume that the sample comes from the given population.

Time to get into the math.

Using standard error to compute confidence interval

Standard error is often used to compute confidence intervals

We know, n = 15, x_bar = 85, σ = 10

$$SE_bar{x} = frac{sigma}{sqrt{n}} = frac{10}{sqrt{15}} = 2.581$$

From a property of normal distribution, we can say with 95% confidence level that the sample means are expected to lie within a confidence interval of plus or minus two standard errors of the sample statistic from the population parameter.

But where did ‘normal distribution’ come from? You may wonder how we can directly assume a normal distribution is followed in this case. Or rather shouldn’t we test if the sample follows a normal distribution first before computing the confidence intervals.

Well, that’s NOT required. Because, the Central Limit Theorem tells us that even if a population is not normally distributed, a collection of sample means from that population will infact follow a normal distribution. So, its a valid assumption.

Back to the problem, let’s compute the confidence intervals for 95% Confidence Level.

  • Lower Limit : 80 - (2*2.581) = 74.838
  • Upper Limit : 80 + (2*2.581) = 85.162

So, 95% of our 15 item sample means are expected to fall between 74.838 and 85.162.

Since the sample mean of 85.0 lies within the computed range, there is no reason to believe that the sample does not belong to the population.

Conclusion

Standard error is a commonly used term that we sometimes ignore to fully understand its significance. I hope the concept is clear and you can now relate how you can use standard error in appropriate situations.

Next topic: Confidence Interval

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The standard error (SE) of the sample mean refers to the standard deviation of the distribution of the sample means. It gives analysts an estimate of the variability they would expect if they were to draw multiple samples from the same population. While the standard deviation measures the variability obtained within one sample, the standard error gives an estimate of the variability between many samples.

Provided the population standard deviation, σ, is known, analysts use the following formula to estimate the standard error of the sample mean, denoted as σx:

$$ sigma_x=cfrac {sigma}{sqrt n} $$

Where n is the sample size.

However, the population standard deviation, σ, is usually unknown. In such a case, the following formula is used to estimate the standard error of the sample mean, also denoted as Sx:

$$ S_x =cfrac {S}{sqrt n} $$

Where S is the sample standard deviation; and (S^2 =cfrac {sum left(X_i- X right)^2}{n- 1}  ).

Breaking Down the Standard Error of the Sample Mean

The standard error of the sample mean gives analysts an idea of how precisely the sample mean estimates the population mean. A lower value of the standard error indicates a more precise estimation of the population mean. On the other hand, a larger value of the standard error indicates a less precise estimate of the population mean.

It’s also important to note that the standard error becomes smaller as the sample size increases. This happens because increasing the sample size ultimately brings the sample mean closer to the true value of the population mean.

Example: Calculating the Standard Error of the Sample Mean When the Population Standard Deviation is Known

In a certain property investment company with an international presence, workers have a mean hourly wage of $12 with a population standard deviation of $3. Given a sample size of 30, estimate and interpret the SE of the sample mean:

$$ begin{align*} sigma_x & =cfrac {sigma}{sqrt n} \ & =cfrac {3}{sqrt {30}} \ & = $0.55 \ end{align*} $$

Interpretation: if we were to draw several samples of size 30 from the employee population and construct a sampling distribution of the sample means, we would end up with a mean of $12 and a standard error of $0.55.

Example: Calculating the Standard Error of the Sample Mean When σ is Unknown

A sample of 30 latest returns on XYZ stock reveals a mean return of $4 with a sample standard deviation of $0.13. Estimate the SE of the sample mean.

$$ begin{align*} S_x & =cfrac {S}{sqrt n} \ & =cfrac {0.13}{sqrt {30}} \ & = $0.02 \ end{align*} $$

Interpretation: If we were to draw more samples from the population of yearly returns on XYZ stock and construct a sample mean distribution, we would end up with a mean of $4 and a standard error of $0.02.

Question

Assume that we have increased the sample size to 80 in the example above and derived similar values for the mean and standard deviation of returns. Estimate the standard error of the sample mean.

A. 0.01

B. 0.02

C. 0.08

Solution

The correct answer is A.

$$ begin{align*} S_x & =cfrac {S}{sqrt n} \ & = cfrac {0.13}{sqrt {80}} \ & = $0.01 \ end{align*} $$

This clearly proves that increasing the sample size reduces the SE of the sample mean.

Reading 10 LOS 10f:

Calculate and interpret the standard error of the sample mean

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