Utility of the Concept of Standard Error

The concept of standard error is of great significance in statistical work because of the following reasons:

(i)    It is used as an instrument in testing a given hypothesis. The hypotheses are generally tested at 5 % level of significance. If the difference between observed and expected means is more than 1.96 standard error (S.E.), we say that the result of the experiment does not support the hypothesis at 5 % level, or, in other words, the difference is regarded as significant, i.e., it could not have arisen due to fluctuations of sampling. On the other hand, if the difference between observed and expected results is less than 1.96 S.E., it is not regarded as significant, i.e., it could have arisen due to fluctuations of simple sampling, i.e., we say that the result of the experiment does not provide any evidence against the hypothesis. If the difference is more than 2.58 S.E., it is considered to be significant at 1 % level. In practice, quite often a hypothesis is accepted if the difference is less than 3 S.E., because the probability of a difference greater than 3 S.E., arising by chance, is only about 3 in thousand (0.27) % as 99.73 percent items are covered because mean + 3 σ rule is justified only if n is large. Some people apply a criterion of 2 S.E. in order to determine whether or not the difference could have arisen due to fluctuations of sampling. However, instead of 3 S.E., it is suggested that we should use either 5 % level or 1 % level of significance.

(ii)    Standard error provides an idea about the unreliability of a sample. The greater the standard error. The greater is the departure of actual frequencies from the expected ones and hence the greater the unreliability of the sample. The reciprocal of S.E., i.e., 1/S.E., is a measure of reliability or precious of the sample. The reliability or precision of an observed proportion varies as the square root of the number of items in the sample.

(iii)    With the help of S.E. we can determine the limits within which the parameter values are expected to lie. This is made possible because for large samples, sampling distributions will have their mean values within a range of a population + 1 standard deviation or standard error as it is alternatively called. Similarly a range of mean + 2 S.E. will give 95.45 percent values and mean + 3 S.E. should be taken as the determining limit outside which the value of the parameter probably does not fall. The chance of a value lying outside + 3 S.E. limits is only 0.27 %.

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