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Question: How do you find the standard deviation of a set of numbers?...

How do you find the standard deviation of a set of numbers?

Explanation

Solution

To obtain the standard deviation of a set of numbers we use the formula σ=∑(xi−x‾)2n\sigma =\sqrt{\dfrac{{{\sum{\left( {{x}_{i}}-\overline{x} \right)}}^{2}}}{n}}. Firstly we will find the mean of the set given. Then we will subtract the value of mean from each number given separately and square them. Finally we will add all the obtained value and divide it by the number of numbers in the set and get its square root.

Complete step by step answer:
To find the standard deviation of a set of numbers firstly find the mean of the set given as follows:
xi=x1+x2+x3+x4+.....+xnn{{x}_{i}}=\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+.....+{{x}_{n}}}{n}
Where,
x1+x2+x3+x4+.....+xn{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+.....+{{x}_{n}} Are the numbers in the set and n=n= number of numbers in the set
Now, subtract the mean from each numbers in the set and square the difference as:
(x1−xi)2,(x2−xi)2.......(xn−xi)2{{\left( {{x}_{1}}-{{x}_{i}} \right)}^{2}},{{\left( {{x}_{2}}-{{x}_{i}} \right)}^{2}}.......{{\left( {{x}_{n}}-{{x}_{i}} \right)}^{2}}
Next add the terms above and divide it by nn which is the number of numbers in the set.
⇒(x1−xi)2+(x2−xi)2+.......+(xn−xi)2n\Rightarrow \dfrac{{{\left( {{x}_{1}}-{{x}_{i}} \right)}^{2}}+{{\left( {{x}_{2}}-{{x}_{i}} \right)}^{2}}+.......+{{\left( {{x}_{n}}-{{x}_{i}} \right)}^{2}}}{n}
Final step is to get square root of the above term as:
⇒(x1−xi)2+(x2−xi)2+.......+(xn−xi)2n\Rightarrow \sqrt{\dfrac{{{\left( {{x}_{1}}-{{x}_{i}} \right)}^{2}}+{{\left( {{x}_{2}}-{{x}_{i}} \right)}^{2}}+.......+{{\left( {{x}_{n}}-{{x}_{i}} \right)}^{2}}}{n}}
The answer obtain after simplifying above equation is our standard deviation which is written as:
σ=(x1−xi)2+(x2−xi)2+.......+(xn−xi)2n\sigma =\sqrt{\dfrac{{{\left( {{x}_{1}}-{{x}_{i}} \right)}^{2}}+{{\left( {{x}_{2}}-{{x}_{i}} \right)}^{2}}+.......+{{\left( {{x}_{n}}-{{x}_{i}} \right)}^{2}}}{n}}
Hence the standard deviation of a set of numbers is obtained by using the formula below:
σ=(x1−xi)2+(x2−xi)2+.......+(xn−xi)2n\sigma =\sqrt{\dfrac{{{\left( {{x}_{1}}-{{x}_{i}} \right)}^{2}}+{{\left( {{x}_{2}}-{{x}_{i}} \right)}^{2}}+.......+{{\left( {{x}_{n}}-{{x}_{i}} \right)}^{2}}}{n}}
This is also written as σ=∑(xi−x‾)2n\sigma =\sqrt{\dfrac{{{\sum{\left( {{x}_{i}}-\overline{x} \right)}}^{2}}}{n}} in generalized form.

Note: Standard deviation is used to measure the dispersion in a set of values. If the value of standard deviation is low that means the values are close to the mean of the set and if the value of standard deviation is high that means the values of the set is spread out over a wide range. It is often used to compare the model against the real-world data to test the model strength.