A student scores the following marks in five tests: 45, 54, 41, 57, 43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is:

First, we list the marks already known: $$45,\;54,\;41,\;57,\;43$$. Let the unknown sixth mark be $$x$$.

We are told that the mean (average) score of all six tests is $$48$$. The definition of the mean for $$n$$ observations is

$$\text{Mean} \;=\; \dfrac{\text{Sum of all observations}}{n}.$$

Applying this with $$n=6$$, we have

$$\dfrac{45+54+41+57+43+x}{6}=48.$$

First we add the five known marks:

$$45+54=99,$$ $$99+41=140,$$ $$140+57=197,$$ $$197+43=240.$$

So the equation becomes

$$\dfrac{240+x}{6}=48.$$

Multiplying both sides by $$6$$ gives

$$240+x = 48\times6.$$

Since $$48\times6 = 288,$$ we obtain

$$240+x = 288.$$

Subtracting $$240$$ from both sides, we get

$$x = 288-240 = 48.$$

Hence the sixth mark is $$48$$, and the complete list of marks is $$45,\;54,\;41,\;57,\;43,\;48.$$

Now we calculate the standard deviation. For a set of $$n$$ observations $$x_1,x_2,\ldots,x_n$$ with mean $$\bar{x}$$, the population variance is defined as

$$\sigma^2 = \dfrac{\displaystyle\sum_{i=1}^{n}(x_i-\bar{x})^2}{n}.$$

The standard deviation is the positive square root of the variance:

$$\sigma = \sqrt{\sigma^2}.$$

Our mean is $$\bar{x}=48$$. We form each deviation $$d_i = x_i - 48$$:

$$$ \begin{aligned} 45-48 &= -3,\\ 54-48 &= +6,\\ 41-48 &= -7,\\ 57-48 &= +9,\\ 43-48 &= -5,\\ 48-48 &= 0. \end{aligned} $$$

Next we square each deviation and list the squares:

$$$ (-3)^2=9,\; (+6)^2=36,\; (-7)^2=49,\; (+9)^2=81,\; (-5)^2=25,\; 0^2=0. $$$

Adding these squared deviations, we have

$$$ 9+36=45,\; 45+49=94,\; 94+81=175,\; 175+25=200,\; 200+0=200. $$$

So the sum of squared deviations is $$\sum (x_i-\bar{x})^2 = 200.$$

The variance is therefore

$$\sigma^2 = \dfrac{200}{6} = \dfrac{100}{3}.$$

Taking the square root, we obtain the standard deviation:

$$\sigma = \sqrt{\dfrac{100}{3}} = \dfrac{10}{\sqrt{3}}.$$

Hence, the correct answer is Option 3.