Question

What is the standard deviation of just one number?

Answer 1

We need to find the standard deviation of just one number. Standard deviation, in statistics, is the dispersion of the set of values relative to their mean. Standard deviation is calculated as the square root of variance given by $\sigma =\sqrt{\dfrac{{{\sum{\left( {{x}_{i}}-\mu \right)}}^{2}}}{N}}$.

Complete step-by-step answer:
Variance refers to the variability from the mean. It is calculated by taking the average of squared deviations from the mean.
It is usually denoted by the symbol ${{\sigma }^{2}}$
The formula of variance is given as follows,
$\Rightarrow {{\sigma }^{2}}=\dfrac{\sum{{{\left( {{x}_{i}}-\mu \right)}^{2}}}}{N}$
Here,
${{\sigma }^{2}}$ = variance
${{x}_{i}}$ = value of one observation
$\mu$ = mean
$N$ = number of observations
Standard deviation, in statistics, measures the amount of variation of the set of values relative to their mean.
A low standard deviation indicates that a set of values are close to their mean and a high standard deviation indicates that the set of values are spread out.
It is usually denoted by the symbol $\sigma$
The formula of standard deviation is given as follows,
$\Rightarrow \sigma =\sqrt{\dfrac{{{\sum{\left( {{x}_{i}}-\mu \right)}}^{2}}}{N}}$
Here,
$\sigma$ = Standard deviation
${{x}_{i}}$ = value of one observation
$\mu$ = mean
$N$ = number of observations
According to our question,
If there is just one number in the data set for the calculation of standard deviation, the value of the observation is equal to the mean.
$\Rightarrow {{x}_{i}}=\mu$
Substituting the value in the formula, we get,
$\Rightarrow \sigma =\sqrt{\dfrac{{{\sum{\left( \mu -\mu \right)}}^{2}}}{N}}$
Evaluating the above equation, we get,
$\therefore \sigma =0$
Hence, the standard deviation of just one number is zero.

Note: The value of standard deviation is greater than zero if the values in the data set are unique. The value of standard deviation is equal to zero if we have just one number or numbers that are exactly the same in the data set. The concept of Standard deviation is widely used in finance and opinion polling