Question

If $ {{x}_{1}},{{x}_{2}},......{{x}_{18}} $ are observations such, that $ \sum\limits_{j=1}^{18}{({{x}_{j}}-8)=9} $ and $ \sum\limits_{j=1}^{18}{{{({{x}_{j}}-8)}^{2}}=45,} $ then the standard deviation of these observations is

• A. $ \sqrt{\frac{81}{34}} $
• B. $ 5 $
• C. $ \sqrt{5} $
• D. $ \frac{3}{2} $

Answer 1

The Correct Option is D

Standard deviation $ =\sqrt{\frac{\underset{j=1}{\mathop{\overset{18}{\mathop{\Sigma }}\,}}\,{{({{x}_{j}}-8)}^{2}}}{n}-{{\left( \frac{\underset{j=1}{\mathop{\overset{18}{\mathop{\Sigma }}\,}}\,({{x}_{k}}-8)}{n} \right)}^{2}}} $
$ =\sqrt{\frac{45}{18}-{{\left( \frac{9}{18} \right)}^{2}}} $
$ =\sqrt{\frac{45}{18}-\frac{1}{4}}=\sqrt{\frac{81}{36}}=\frac{9}{6}=\frac{3}{2} $

Answer 2

In statistics, the term "standard deviation" refers to a measurement of the variability in a given set of statistical data. It is also known as data dispersion measurement. Statistics includes both variance and standard deviation.

The positive square root of the variance is the standard deviation, which is one of the fundamental techniques of statistical analysis.

It is denoted by the sign "," which represents the amount by which it deviates from the mean value, and is abbreviated as "S.D."

One of the most crucial considerations for determining the Standard Deviation is locating the data's mean. 

The degree to which values in a distribution deviate from the distribution average is known as dispersion. Certain measurements have been developed to quantify the degree of variance.

They are:

Range

Quartile Deviation

Mean Deviation

Standard Deviation

The level of dispersion is measured by standard deviation. It is the dispersion of the data points with respect to the mean in descriptive statistics.

It displays the distribution of values throughout the sample of data.

It is a measurement of how far the data points vary from the mean.

The variance, random variable, statistical population, data set, or probability distribution of a sample is consequently its square root.

We enlist the aid of a few dispersion methods when the data is dispersed or scattered and estimating the central tendency is challenging.

The dispersion calculated from the data via its mean is referred to as the standard deviation or SD. It may also be thought of as the deviation of a value or group from the mean or average. Standard Deviation is therefore mathematically the positive square root of "Variance," as it reflects the variability of the data. Standard Deviation is denoted by σ. Variance is denoted by (σ)².