Answer

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**Hint:**The above question is based on the concept of probability distribution. The main approach towards solving the equation is to apply the formula of range, variance and standard deviation on the given range of values. The formula of all three are dependent on each other.

**Complete step by step solution:**

In general, such terms like range, variance, standard deviation are used in statistical models. These terms help in measuring the data given in terms of range, variance and standard deviation.We first need to know about the term range. Range is defined as the difference between maximum and minimum value. We denote the maximum value or largest value by L and minimum value or smallest value by S. Therefore, range is calculated as $Range = L - S$.Therefore, by applying it on the given data we get,

$Range = L - S = 7 - 1 = 6$

Variance is a measurement of the spread between numbers in a data set. Variance defined as the sum of all the squared distances of each term in the distribution from the mean.

variance is denoted by \[{\sigma ^2}\].

First average or mean needs to be calculated

\[\bar x = \dfrac{1}{7}\sum\limits_{k = 1}^7 {k = \dfrac{{1 + 2 + 3 + 4 + 5 + 6 + 7}}{7} = 4} \]

Variance can be calculated as follows:

\[

{\sigma ^2} = \dfrac{1}{{n - 1}}\sum\limits_{k = 1}^n {{{\left( {{x_k} - \bar x} \right)}^2} \Rightarrow{\sigma ^2} = \dfrac{1}{6}\left( {{{\left( { - 3} \right)}^2} + {{\left( { - 2} \right)}^2} + {{\left( { - 1} \right)}^2} + 0 + {1^2} + {2^2} + {3^2}} \right)} \\

\Rightarrow{\sigma ^2} = \dfrac{{9 + 4 + 1 + 0 + 1 + 4 + 9}}{6} \\

\Rightarrow{\sigma ^2} = \dfrac{{2 + 8 + 18}}{6} \\

\therefore{\sigma ^2} = \dfrac{{14}}{3} \\ \]

**Since the standard deviation is the square root of the variance, now taking square root,\[\sigma = \sqrt {\dfrac{{14}}{3}} \].**

**Note:**An important thing note is that in the formula of variance the value of \[{x_k}\]are the data points given the sequence of data.\[\bar x\]is the average or the mean calculated from all the values and is subtracted with every data point and added together.

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