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:

We have a total of 30 observed values. Ten of them are $$\frac12-d$$, ten are $$\frac12$$ and the remaining ten are $$\frac12+d$$.

First we find the mean. The mean $$\mu$$ of a data set is defined by the formula

$$\mu=\frac{\text{sum of all observations}}{\text{number of observations}}.$$

So, adding every value once and then dividing by 30, we write

$$ \mu=\frac{10\!\left(\frac12-d\right)+10\!\left(\frac12\right)+10\!\left(\frac12+d\right)}{30}. $$

Now, multiply out each product:

$$ 10\left(\frac12-d\right)=10\cdot\frac12-10d=5-10d, $$

$$ 10\left(\frac12\right)=5, $$

$$ 10\left(\frac12+d\right)=10\cdot\frac12+10d=5+10d. $$

Adding these three partial sums gives

$$ (5-10d)+5+(5+10d)=15. $$

Substituting this back,

$$ \mu=\frac{15}{30}=\frac12. $$

Next, we turn to the variance. The population variance $$\sigma^2$$ is defined by

$$ \sigma^2=\frac1n\sum_{i=1}^n(x_i-\mu)^2, $$

where $$n$$ is the number of observations. Here $$n=30$$.

Let us compute each squared deviation from the mean $$\mu=\frac12$$:

• For every value equal to $$\frac12$$, the deviation is $$\frac12-\frac12=0$$, so the square is 0.
• For every value equal to $$\frac12-d$$, the deviation is $$\left(\frac12-d\right)-\frac12=-d$$, whose square is $$d^2$$.
• For every value equal to $$\frac12+d$$, the deviation is $$\left(\frac12+d\right)-\frac12=+d$$, whose square is again $$d^2$$.

There are ten items of the first kind (square $$0$$), ten items of the second kind (square $$d^2$$) and ten items of the third kind (square $$d^2$$). Hence the total of all squared deviations is

$$ 10\cdot0+10\cdot d^2+10\cdot d^2=20d^2. $$

Applying the variance formula, we divide this sum by the total number of observations (30):

$$ \sigma^2=\frac{20d^2}{30}=\frac23\,d^2. $$

According to the statement of the problem, the variance is given to be $$\frac43$$. Therefore we set

$$ \frac23\,d^2=\frac43. $$

Now, multiply both sides by 3 to clear the denominators:

$$ 2d^2=4. $$

Dividing by 2 gives

$$ d^2=2. $$

Taking absolute value (or, equivalently, the positive square root) yields

$$ |d|=\sqrt2. $$

Hence, the correct answer is Option D.