Question:

The outcome of each of $30$ items was observed; $10$ items gave an outcome $\frac{1}{2} - d$ each, $10$ items gave outcome $\frac{1}{2}$ each and the remaining 10 items gave outcome $\frac{1}{2} + d$ each. If the variance of this outcome data is $\frac{4}{3}$ then |d| equals :

Updated On: Apr 28, 2025

$2$
$\frac{\sqrt{5}}{2}$
$\frac{2}{3}$
$\sqrt{2}$

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The Correct Option is D

Solution and Explanation

Variance is independent of origin. So we shift the given data by $\frac{1}{2}$. so, $\frac{10d^2 + 10 \times 0^2 + 10d^2}{30} -(0)^2 = \frac{4}{3}$ $\Rightarrow \; d^2 = 2 \Rightarrow | d | = \sqrt{2}$

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Concepts Used:

Variance and Standard Deviation

Variance:

According to layman’s words, the variance is a measure of how far a set of data are dispersed out from their mean or average value. It is denoted as ‘σ2’.

Variance Formula:

Standard Deviation:

The spread of statistical data is measured by the standard deviation. Distribution measures the deviation of data from its mean or average position. The degree of dispersion is computed by the method of estimating the deviation of data points. It is denoted by the symbol, ‘σ’.

Types of Standard Deviation:

Standard Deviation for Discrete Frequency distribution
Standard Deviation for Continuous Frequency distribution

Standard Deviation Formulas:

1. Population Standard Deviation

2. Sample Standard Deviation