Which of the following statements is true?

Question

Question

This question was previously asked in

SSC Selection Post 2024 (Matriculation Level) Official Paper (Held On: 25 Jun, 2024 Shift 1)

The sum of the squared deviations of the items from the arithmetic mean is maximum.
Arithmetic mean is the sum of deviations from the mid value.
The sum of deviations of the items from the arithmetic mean is always one.
The sum of deviations of the items from the arithmetic mean is always zero.

Answer 1

Formula used:

The sum of deviations of the items from the arithmetic mean is always zero.

Calculation:

Let \(x_1, x_2, ..., x_n\) be a set of n observations and \(\bar{x}\) be the arithmetic mean.

Sum of deviations from the mean: \(\sum_{i=1}^{n} (x_i - \bar{x})\)

⇒ \(\sum_{i=1}^{n} x_i - \sum_{i=1}^{n} \bar{x}\)

⇒ \(\sum_{i=1}^{n} x_i - n \bar{x}\)

Since \(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\)

⇒ \(\sum_{i=1}^{n} x_i - n \left(\frac{\sum_{i=1}^{n} x_i}{n}\right)\)

⇒ \(\sum_{i=1}^{n} x_i - \sum_{i=1}^{n} x_i\)

⇒ 0

∴ The correct answer is option 4.

Download Solution PDF