Central Measures

Measures of central tendencies are nothing but the measures to describe the "central" values of a collected sample. For an ungrouped set of data these measures are: the mean, the median, and the mode.

Standard Error of Mean

Once we calculated one of the measures of the central tendency, say the sample mean, one might like to know how good is our estimated sample mean?

We answer this by calculating a confidence interval for the sample mean. For this purpose we need to calculate the standard error of the mean (SEM). The standard error is the error in estimating the population mean using a sample drawn from that population.

By definition the standard error of the sample mean is nothing but the standard deviation of the sample mean. What it means in theory is that we will have to repeat our sampling procedure several times to make, say, m samples. Then calculate the sample mean for each sample, resulting in m sample means.

The standard deviation of the m sample means about the super mean (the mean of all sample means) is known as the standard error of the sample mean (SEM).

However, in practice SEM can be calculated from the sample standard deviation (S) as:

SEM = S / sqrt(N)

where N - number of observations; and

Confidence Interval for Mean

A standard deviation or a standard error has little practical use in itself. But it becomes meaningful when we use it to calculate confidence intervals.

We can do this easily by multiplying a standard error by a t-value obtained from the table of t-distribution. The confidence intervals show us the range within which 95% or 99% or 99.9% of observations could be expected to lie. For example, for a large sample size (i.e., size greater than 30), the t-value for the 95% confidence level is 1.96.

Therefore the 95% confidence interval is given by the interval:

( SEM*1.96, + SEM*1.96)

Measures of Central Tendencies play an integral part of the Statistical Information Techniques.

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