Standard Error

The standard deviation of a sampling distribution of a statistic is often called its Standard error.

1. Sample Distribution: Means
   Standard Error : \(\sigma_{\bar{X}} =\frac{\sigma}{\sqrt{N}}\)
   If N > 30, sampling distribution is normal even if the underlying population is not normal. Also \(\mu_{\bar{x}}=\mu\), the population mean , in all cases.

2. Sample Distribution: Proportions
   Standard Error : \(\sigma_P =\sqrt{\frac{pq}{N}}\)
   \(\mu_P = p\)

3. Sample Distribution: Standard deviations
   Standard Error : \(\sigma_s =\frac{\sigma}{\sqrt{2N}}\)
   For N >= 100,the sampling distribution of standard deviation is very normal and \(\mu_s = \sigma\). If the population is nonnormal, Standard Error : \(\sigma_s =\sqrt{\frac{\mu_4 - \mu_2{^2}}{4N\mu_2}}\)
   \(\mu_2 = \sigma^2\) and \(\mu_4 = 3\sigma^4\)

4. Sample Distribution: Medians
   Standard Error : \(\sigma_{med} =\sigma\sqrt{\frac{\pi}{2N}}\)
   For N >= 30, the sampling distribution of median is very normal and \(\mu_{med} = \sigma\).The given results holds only if the population is normal.

5. Sample Distribution: First and third quartiles
   Standard Error : \(\sigma_{Q_1} = \sigma_{Q_3}= \frac{1.2533\sigma}{\sqrt{N}}\)
   \(\sigma_{Q_1}\) and \(\sigma_{Q_3}\) are nearly equal to be first and third quartiles of the population. Note that \(\sigma_{Q_2} = \mu\)

6. Sample Distribution: Deciles Standard Error :
   \(\sigma_{D_1} = \sigma_{D_9}= \frac{1.7094\sigma}{\sqrt{N}}\)
   \(\sigma_{D_2} = \sigma_{D_8}= \frac{1.4288\sigma}{\sqrt{N}}\)
   \(\sigma_{D_3} = \sigma_{D_7}= \frac{1.3180\sigma}{\sqrt{N}}\)
   \(\sigma_{D_4} = \sigma_{D_6}= \frac{1.2680\sigma}{\sqrt{N}}\)

7. Sample Distribution: Semi-interquartile ranges Standard Error : \(\sigma_{Q}= \frac{0.7867\sigma}{\sqrt{N}}\)

8. Sample Distribution: Variances
   Standard Error : for N>= 100 \(\sigma_{s^2}= \sigma^2\sqrt{\frac{2a}{N}}\)
   Standard Error : For non normal sample \(\sigma_{s^2}= \sqrt{\frac{\mu_4 - \frac{N-3}{N-1}\mu_{2}{^2}}{N}}\)

9. Sample Distribution: Variances
   Standard Error : for N>= 100 \(\sigma_{s^2}= \sigma^2\sqrt{\frac{2a}{N}}\)
   Standard Error : For non normal sample \(\sigma_{s^2}= \sqrt{\frac{\mu_4 - \frac{N-3}{N-1}\mu_{2}{^2}}{N}}\)

10. Sample Distribution: Coefficient of variation
    Standard Error
    \(\sigma_{V}= \frac{v}{\sqrt{2N}}\sqrt{1+2v^2}\)

Here \(v= \frac{\sigma}{\mu}\) is the population coefficient of variation. This given result holds for normal population and N >=100

## Continuity Correction ##
If P(X=n) use P(n -0.5 < X < n + 0.5)
If P(X > n) use P(X > n + 0.5)
If P(X >= n) use P(X > n - 0.5)
If P(X < n) use P(X > n - 0.5)
If P(X <= n) use P(X > n + 0.5)