Basic Statistics and Data Analysis

Statistics Lecture Notes, Online MCQs

Tagged: Standard Error

Measure of Dispersion / Sampling and Sampling distributions

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Standard Error of Estimate

Standard error (SE) is a statistical term used to measure the accuracy within a sample taken from population of interest. The standard error of the mean measures the variation in the sampling distribution of the sample mean, usually denoted by $\sigma_\overline{x}$ is calculated as

\[\sigma_\overline{x}=\frac{\sigma}{\sqrt{n}}\]

Drawing (obtaining) different samples from the same population of interest usually results in different values of sample means, indicating that there is a distribution of sampled means having its own mean (average values) and variance. The standard error of the mean is considered as the standard deviation of all those possible sample drawn from the same population.

The size of the standard error is affected by standard deviation of the population and number of observations in a sample called the sample size. The larger the standard deviation of the population ($\sigma$), the larger the standard error will be, indicating that there is more variability in the sample means. However larger the number of observations in a sample smaller will be the standard error of estimate, indicating that there is less variability in the sample means, where by less variability we means that the sample is more representative of the population of interest.

If the sampled my canadian pharmacy population is not very larger, we need to make some adjustment in computing the SE of the sample means. For a finite population, in which total number of objects (observations) is $N$ and the number of objects (observations) in a sample is $n$, then the adjustment will be $\sqrt{\frac{N-n}{N-1}}$. This adjustment is called the finite population correction factor. Then the adjusted standard error will be

\[\frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}\]

The SE is used to:

  1. measure the spread of values of statistic about the expected value of that statistic
  2. construct confidence intervals
  3. test the null hypothesis about population parameter(s)

The standard error is computed from sample statistics. To compute SE for simple random samples, assuming that the size of population ($N$) is at least 20 times larger than that of the sample size ($n$).
\begin{align*}
Sample\, mean, \overline{x} & \Rightarrow SE_{\overline{x}} = \frac{n}{\sqrt{n}}\\
Sample\, proportion, p &\Rightarrow SE_{p} \sqrt{\frac{p(1-p)}{n}}\\
Difference\, b/w \, means, \overline{x}_1 – \overline{x}_2 &\Rightarrow SE_{\overline{x}_1-\overline{x}_2}=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\
Difference\, b/w\, proportions, \overline{p}_1-\overline{p}_2 &\Rightarrow SE_{p_1-p_2}=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}
\end{align*}

The standard error is identical to the standard deviation, except that it uses statistics whereas the standard deviation uses the parameter.

 

For more about SE follow the link Standard Error of Estimate

 

Estimate and Estimation / Sampling and Sampling distributions

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Standard Error: Standard Deviation of the Sampling Distribution

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 $n_2$.