Question

If \(x_1, x_2, \ldots, x_n\) are 'n' observations and \(\bar{x}\) is their mean. If \[ \sum_{i=1}^{n}(x_i - \bar{x})^2 \text{ is almost zero, then a true statement among the following is} \]

• A. It indicates a higher degree of dispersion of the observations from the mean \(\bar{x}\)
• B. It indicates that there is no dispersion
• C. \(\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2\) is the arithmetic mean of the data
• D. It indicates that each observation xi is very close to the mean bar{x} and hence degree of dispersion is low

Hint:

A near-zero sum of squared deviations indicates the data points are tightly clustered around the mean.

Answer 1

The Correct Option is D

The quantity \[ \sum_{i=1}^{n}(x_i - \bar{x})^2 \] measures the **total squared deviation** from the mean. If it is almost zero, each term \((x_i - \bar{x})^2\) is nearly zero, implying each \(x_i\) is nearly equal to \(\bar{x}\). Therefore, there is **very low dispersion**.