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Sampling Distribution of Arithmetic Mean

Dr. T. T. Kachwala

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Sampling Distribution of Arithmetic Mean

If we select a number of independent random samples of similar size from a given population and calculate some statistical measures like Arithmetic Mean for each sample, we shall get a series of values of which need not be the same and most probably would vary from sample to sample.

According to Central Limit Theorem in statistics, if the sample selected are random, independent and of a sufficiently large size, then the sample means will form a distribution that is approximately normal. Such a probability distribution of all the possible Arithmetic means of the different samples is referred by statisticians as Sampling distribution of the mean.

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Concept of Standard Error

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A normal distribution is characterised by µ (Mean) and (S.D). Conceptually, standard deviation measures the average deviation of the variable 'X' from its A.M. µ (i.e. It measures the spread of the distribution).

On similar lines, Sampling distribution of Arithmetic Mean is characterised by (mean of the sampling distribution of Arithmetic means) and (standard deviation of the sampling distribution of Arithmetic mean).

This standard deviation of the sampling distribution of mean is referred by statisticians as Standard Error of the Mean.

Xμ

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Concept of Standard Error

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Standard Error of Mean measures the deviation of the sample statistic from the mean of sampling distribution i.e. it measures the spread of the sampling distribution of mean.

A distribution of sample mean that is less spread out (that has a small standard error) is a better estimator of the population mean than a distribution of sample mean that has a wider dispersion (i.e. larger standard error)

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Relationship between the Population Distribution and the Sampling Distribution of Arithmetic Mean

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'' is the mean of the population distribution & ' ' is the Standard Deviation of the population distribution. is the mean of the sampling distribution of the mean & is the standard error of the Arithmetic Mean.

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The following formula summarizes the relation:

μμ X

05.0Nnsize population infinite

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X σσ

05.0 Nn size population finite

1 - N

n - N

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σσ

Graphical Relationship between the Population Distribution and the Sampling Distribution of Arithmetic Mean

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The standard deviation of the sampling distribution of mean would be less than the standard deviation of the individual items in the population as indicated graphically below:

Sampling distribution of Mean

Population Distribution

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Standardizing the Sample Mean

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In normal distribution to convert random variable 'X' to standard random variable 'z' we apply z - transformation;

σμ -X

zSNV

On similar lines, one can standardize the sample mean

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x μμ -X

Xz

σσ

Where, is the Sample Mean

is the Population Mean

is the Standard Error of Mean

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Sampling from Non Normal Populations

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Consider the following population distribution (skewed distribution)

According to the central limit theorem in statistics, the sampling distribution of Arithmetic Mean even for Non Normal Distribution will tend to a Normal Distribution as the sample size increases as indicated in the diagram in the next slide

Sampling from Non Normal Populations

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xμ

μμ X

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σσ

Statisticians use the normal distribution as an approximation to the sampling distribution whenever the sample size is at least '30'.

Significance of the Central limit theorem

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The significance of the central limit theorem is that it permits us to use sample statistics to make inferences about population parameters without knowing anything about the shape of the population distribution.

The central limit theorem is one of the most powerful concepts in statistics. What it really says is that the distribution of sample means tends to be normally distributed regardless of the shape of the population distribution from which samples were taken.

Relationship between Sample Size and Standard Error

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Standard error is a measure of dispersion of the sample means around the population mean.

If the dispersion decreases (if becomes smaller) then the values taken by the sample mean tend to cluster more closely around .

Conversely, if the dispersion increases (if becomes larger), the values taken by the sample mean tend to cluster less closely around ''.

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Concept of “Diminishing Return in Sampling"

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Because varies inversely with , there is a diminishing return in sampling".

It is true that sampling more items will decrease the standard error, but this "benefit" may not be worth the "cost"

That is why, smart managers focus on the concept of the 'right' sample size i.e. the sample size which is a good balance between both the cost and the worth of the precision desired.

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Thanks and Good Luck

Have a nice Day

Dr. T. T. Kachwala