Standard Error

For studying standard error first we need to know what is sampling distribution:

Sampling distribution of a statistic:

If we draw a sample of size ‘n’ from the given finite population of size N, the possible samples are ^NC_n  = k (say)

For each of these ‘K’ samples we can compute some statistic t= t(x1,x2….xn), in particular the mean and the the variance are given below.

The set of the values of the statistics so obtained one for each sample constitutes ‘sampling distribution’ of the statistics.
For example the values t₁,t₂,...tn determine the sampling distribution of the statistics ‘t’. In other words, statistics may be regarded as a random variables which can take the values t₁,t₂...tn and  compute the various statistical constants for example mean ,variance, skewness, kurtosis etc. for its distribution.
For example , the mean and variance of the sampling distribution of the statistics ‘t’ is given  by- 

Utility of Standard Error

Standard error plays a very important role in statistics.

Standard error is important in large sample theory and hypothesis testing.

• If we have standard error of several individual quantities we can easily calculate the standard error of some function of the quantities easily in  many cases.
• Having known probability distribution of the value , S.E. is appropriate to use to calculate a good approximation to an exact confidence interval
• When the probability distribution is unknown, relationships like Chebyshev's are be used to calculate a conservative confidence interval.

Thus, if the discrepancy between the observed and expected (hypothetical) value of statistic is greater than Z times its S.E., the null hypothesis is rejected at level of significance.

Hence, if | t - E(t)|  Z * S.E (t),

The deviation is regarded insignificant at 5% level of significance( is  commonly take as 5%) .

The deviation is regarded insignificant at 5% level of significance( is  commonly take as 5%).

The magnitude of the standard error gives an index of the precision of the estimate of the parameter, the reciprocal of the standard error is computed as the measure of reliability or precision of the statistic.

For Example, In Case of Proportion

In other words, the standard error of ‘p’ and ‘’ vary inversely as the square root of the sample size. Thus, in order to double the precision it amounts to reducing the standard error one half, the sample size has to be increased 4 times.

(ii) Standard error helps us to determine the probable limits within which the population parameter is expected to lie.  for example the probable limits for population proportion ‘P’ (‘p’ is estimate of P ) are given by:   

Examples of on Standard Error

A die is thrown 9000 times in which a throw of 3 or 4 is observed 3240 times. Show that Dad cannot be regarded as an unbiased one and find the probable limits between which the probability of a throw of 3 or 4 lies.

If the coming of 3 or 4 is called a success then in usual notations under the null The hypothesis,

n=9000 X = number of successes

Under the null hypothesis H₀ that the die is an unbiased one we get,

P= probability of success=  probability of getting a 3 or 4 = ⅙ + ⅙ = ⅓

Alternate hypothesis will be die is biased. H1: p  1/3
We have, Z = (X - E(X))/ S.E.(X) =  (X - nP)/ ~ N(0,1), since n is large
Now Z = (3240 - 9000 * 1/3)/

Since |Z| > 3 die is not  biased. Which provides evidence against Null hypothesis ,hence we reject null hypothesis and accept alternate hypothesis.

Finding the Limits between which the probability of 3 or 4 lies:

Since, die is not unbiased, P ⅓
=> we take estimate of ‘P’ say ‘p’,

=> Probable limits for ‘P’ are  given by

probability of getting 3 or 4 almost lies between 0.345 and 0.375.

A random sample of 500 pineapples was taken from large consignment of 65 was found to be bad so that the standard error of the population of bad ones in a sample of this size is 0.015 and deduce that the percentage of bad pineapples in the consignment obtained  almost certainly lies between 8.5 and 17.5.

Here we have,

n= the sample size =  500

X = number of bad pineapples in the sample = 65

P= Proportion of bad pineapples in the sample = 65/ 500 = 0.13

since ‘P’ the proportion of bad pineapples in the consignment is not known to us we may (take as in the last example) an estimate of ‘P’ as p.

Hence, p = 0.13, q= 0.87

Thus, the limits for the proportion of bad pineapples in the consignment almost certainly lies between 8.5 and 17.5.

A random sample of 500 apples was taken from a large consignment and 60 were found to be bad. Obtain the 98% confidence limits for the percentage of bad apples in the consignment.

we have,

P = proportion of bad apples in a sample = 60 / 500 = 0.12

Since, significant value of  Z at 98%confidence limit coefficient( 2% level of significance) is 2.33(from normal tables)

=>  98% confidence limits for population proportions are :

Hence, 98% confidence limits are (0.08615,0.15385).

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