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Sampling Distribution

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The statistics is a very important and a vast branch of mathematics that in higher classes becomes a separate subject. It is the study of collection, organization, management, interpretation and calculation over the big numerical data. Statistics plays a vital role in performing various researches and surveys. In statistics, there are two types of data - population and sample.

Population data includes the whole field of study which is suitable for the requirements of a research. Sample data is the data that is chosen from the population. It is the representative of the population. Since population data is very big in size, it is difficult to deal with it. Therefore, the sample is chosen from population using suitable sample methods. In this way, it becomes easier to make the calculations.

Sample distribution or sampling distribution is another important concept that is very useful in mathematics. In this article, we are going to go ahead and focus on sampling distribution, its types and examples based on it.

Definition

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The sampling distribution is defined as a random variable that is the distribution of the given statistic which is to be derived from a random sample having sample size of "n". Sampling distribution is considered as the distribution for the possible samples taken from the given population of a particular size.
It depends upon few factors:
1) The distribution of the given population
2) The statistic that is chosen.
3) The method of sampling.
4) The sample size.
The sampling distribution of a given data population is actually the distribution of the frequencies for the various predictable outcomes which could possibly happen to occur in a statistic of that population.

Types

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The types of sampling distribution are as follows:

1) Sampling Distribution of the Mean
Sampling distribution of means of a population data is defined as the theoretical probability distribution of the sample means which are obtained by extracting all the possible samples  having the same size from the given population.

2) Sampling Distribution of the Proportion
Sampling distribution of the proportion is found when the sample proportion and proportion of successes are given.

3) T Distribution
T distribution is said to be the sampling distribution derived from two variances i.e. squared standard deviations.

4) F distribution
F distribution is defined as the sampling distribution of means possessing an estimated standard deviation.

Formula

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Let us suppose that the random samples of size n (where, n is large enough) are drawn from a population. Also assume that the population does have finite standard deviation ($\sigma$) and mean ($\mu$). The mean of the sample be $\bar{x}$. Then, according to the Central Limit Theorem, the formula for the mean of sampling distribution of the sample mean is equal to the mean of the population. i.e.
$\mu_{\bar{x}}=\mu$
Also, the formula for standard deviation of sampling distribution is given below :
$\sigma_{\bar{x}}=$$\frac{\sigma}{\sqrt{n}}$

Simulation

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The simulation of sampling distribution indicates the imitation or application of sampling distribution in real and practical world which may include the following points:

1)
To find how the sample size affects the rate of sampling distribution tending towards normal distribution.

2) In order to estimate the effect of number of samples onto the speed of approaching of the sampling distribution to the normal distribution.

3) To calculate the effect of the central limit theorem for other parameter estimates for example mean, Standard deviation.

4) To find the effect of the shape of distribution on the speed by which the sampling distribution converges to the normal one.

Sampling Distribution of Proportion

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In statistics, sometimes we come across the sampling distribution of proportion. Let us assume the a population with size K and standard deviation $\sigma$ is given. The probability of happening (or success) of an event be denoted by P; while the probability of not happening (or failure) of that event is Q. The samples having sample size n are drawn from the given population. Now, the proportion of successes be p and that of failures be q within each of these samples.

The mean of sampling distribution of such proportion (i.e. $\mu_{p}$) will be equal to the success probability in population. Such as:
$\mu_{p} = P$
Also, the standard error of sampling distribution of proportion (i.e. $\sigma_{p}$) is calculated by the means of population standard deviation, size of population and size of sample as given below :
$\sigma_{p}= \sigma $$\sqrt{\frac{1}{n}-\frac{1}{K}}$$= $$\sqrt{\frac{PQ}{n}-\frac{PQ}{K}}$

Standard Deviation of Sampling Distribution

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The standard deviation, in general, is defined as the deviation of the statistical data from its central tendency. In the reference of sampling distribution, the standard deviation of a sampling sampling distribution for a given statistic is obtained by dividing the standard deviation of the population by square root of sample size. It is calculated by the following formula:

$\sigma_{\bar{x}}$=$\frac{\sigma}{\sqrt{n}}$Where, $\bar{x}$ is the sample mean.
$\sigma_{\bar{x}}$ is the standard deviation of the sampling distribution.
$\sigma$ is the standard deviation of the population.
n is equal to the size of the samples drawn from population.

Standard Error

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In statistics, standard error is a term that indicates the accuracy by which a sample is the representative of a population. Usually, the sample mean is deviated from the actual population mean. This kind of deviation is known as standard error. It is utilized in order to refer to standard deviation of different sample statistics.

For Example: mean or median. The smaller standard error indicates the sample would be more representative of the whole population. Also, the standard error is inversely proportional to size of the sample. It means that the larger sample size causes the smaller standard error.

The standard error of a sampling distribution is equal to the standard deviation of given sample distribution. In other words, the standard error of the sampling distribution is the standard deviation of all the possible samples having a particular size, drawn from the given population.

Examples

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The simple examples based on sampling distribution are as follows:

Example 1:
In a school, there are total 3000 students. It has day scholars as well as hostlers. The average day scholars in each class are 30, while their standard deviation is 6. If the samples of 25 students are observed, then determine the mean and standard deviation of each sample.

Solution : Given that $\mu$ = 30, $\sigma$ = 6 and n = 25
Then, Mean of the sample is given by the formula -
$\mu_{x} = \mu$ = 30

And standard deviation is calculated by:

$\sigma_{x}$=$\frac{\sigma}{\sqrt{n}}$

= $\frac{6}{\sqrt{25}}$

= $\frac{6}{5}$

= 1.2

Example 2: During a research, it is found that the total number of births in a particular city are 33,556 in a month. On an average, the total number of baby girls in each county of the city are 15,310 with the non normal standard deviation be 50. If all possible samples with sample size 400 are drawn out of the population, find the mean and standard deviation of the samples.

Solution : Given that $\mu$ = 15,310, $\sigma$ = 50 and n = 400
The mean is given by:
$\mu_{x} = \mu$ = 15,310

The standard deviation is estimated by the formula:

$\sigma_{x}$=$\frac{\sigma}{\sqrt{n}}$

= $\frac{50}{\sqrt{400}}$

= $\frac{50}{20}$

= 2.5