Standard error and calculation of standard error of different statistics

Standard error.

It is defined as the standard deviation of the sampling distribution of the statistic is called the standard error of the statistic.

The standard error measures the reliability in sampling due to chance. The standard error of a statistic gives an index of the reliability or precision of the estimate of the parameter. Greater the standard error, greater is the departure of actual varies fro the expected ones. Hence, smaller the value of the standard error, greater the uniformity of sampling distribution and hence, greater the reliability of the estimate. Standard error plays an important role in statistical hypothesis testing and interval. Also, standard error decreases when the sample size is increased. The standard error of some important statistics is given a belowThe standard error of the mean sample.

1. The standard error of the mean sample.

$$Sample\,mean \overline{X}=\frac{1}{n}\sum_{i=1}^nX_i$$

$$Standard\,error=\frac{σ}{\sqrt n}$$

$$Where\,σ=Population\,standard\,deviation$$

2. The standard error of the proportion.

$$=\sqrt{\frac{PQ}{n}}$$

3. For random sample drawn from a normal population with s.d (σ)

$$S.E\,of\,sample(s)=\sqrt{\frac{σ^2}{2n}}$$

4. If$\overline{X_1}$ and$\overline{X_2}$denotes the mean calculation from independent random samples of sizes n₁ and n₂ and drawn from two normal population with s.d (σ₁) and (σ₂) respectively, then.

$$S.E\,\overline{X_1}-\overline{X_2}=standard\,error\,of\,difference\,beteen\,two\,means$$

$$=\sqrt{{\frac{σ_1^2}{n_1}}+{\frac{σ_2^2}{n_2}}}$$

In particular, if the population have the same s.dσ(i,eσ₁=σ₂=σ)

$$S.E\,\overline{X_1}-\overline{X_2}=\sqrt{{σ^2\frac{1}{n_1}}+{\frac{1}{n_2}}}$$

5. . If P₁ and P₂ denote the sample proportions calculated from independent random samples of size n₁ and n₂ and drawn from two populations with proportions P₁ and P₂ respectively, then the standard error of a difference between proportions is given by.

$$=P_1-P_2\sqrt{{\frac{P_1Q_1}{n_1}}+{\frac{P_2Q_2}{n_2}}}$$

In particular, if it is assumed that the two populations proportions P₁ and P₂ are equal, say P₁=P₂=P then,

$$=P_1-P_2=\sqrt{{PQ\frac{1}{n_1}}+{\frac{1}{n_2}}}$$

6. If s₁ and s₂ and sample s.d calculated from independent random samples of size n₁ and n₂ and drawn from two populations with s.d (σ₁²) and (σ²₂) respectively, then the standard error of a difference between two s.d is given by.

$$S.E(S_1-S_2)=\sqrt{{\frac{σ_1^2}{2n_1}}+{\frac{σ_2^2}{2n_2}}}$$

Standard error plays the key role to form the basis of the estimation and the testing the hypothesis.

It is used as.

1. To determine the precisions or reliability of the sample of the sample estimates of some populations parameter.

$$Precision \,of\,\overline{X}=\frac{1}{S.E,\overline{X}}$$

2. To test if the sample statistic differs significantly from the correspondings hypothesis value in the population I,e to test the significant of the difference ($\overline{X-µ}$).

For large samples, the z-statistic is

$$Z=\frac{\overline{X-µ}}{{S.E,\overline{X}}}$$

1. To obtain points estimates of the population parameters.
2. To obtain interval estimates of the population parameter I,e to obtain probable limits between which the true value of the parameter may be expected to lie.
3. Probable limits between which the true value of the parameter may be expected to lie.
4. To test the significance of the difference between two independent sample estimates of the same population parameter.

Reference.

Kerlinger, F.N. Foundation of Behavioural Research. New Delhi: Surjeet Publication, 2000.

Kothari, C.R. Research Methodology. India: Vishwa Prakashan, 1990.

Singh, M.L. and J.M Singh. Understanding Research Methodology. 1998.

Singh, Mrigendra Lal. Understanding Research Methodology. Nepal: National Book centre, 2013.