Answer: Standard deviation in statistics is calculated using formula:
s=\frac{\sqrt{\sum({x_i-\bar{x}})^2}}{N}
Statistics is the study of data gathering, organization, interpretation, analysis, and presentation in mathematics. The primary goal of statistics is to plan the obtained data in terms of experimental designs and statistical surveys. Statistics is regarded as a mathematical science that deals with numerical data. In brief, statistics is a critical process that assists in making data-driven decisions. An example of statistical analysis is determining the percentage of a town's population who watched a movie out of the entire population. The small group of persons picked from the population is referred to as the sample here.
What is Standard Deviation?
Standard deviation is a metric that represents the amount to which various values of a statistical series tend to fluctuate or disperse from its mean or median. It describes how the values are distributed over the data sample and is a measure of the data points' deviation from the mean. The square root of the variance of a sample, statistical population, random variable, data collection, or probability distribution is its standard deviation.
Standard Deviation Formula
The formula for sample standard deviation(s) is as follows:
s=\frac{\sqrt{\sum({x_i-\bar{x}})^2}}{N} where,
xi = data values in the set
x̄ = mean of the data
N = number of data values
Example: Find the standard deviation for the data: 42, 38, 35, 26, 45, 52, 48.
Solution:
We have, N = 7.
Mean (x̄) = (42+38+35+26+45+52+48)/7 = 40.85
Standard deviation =
\frac{\sqrt{\sum({x_i-\bar{x}})^2}}{N} =
\frac{\sqrt{(40.85-42)^2+(40.85-38)^2+(40.85-35)^2+(40.85-26)^2+(40.85-45)^2+(40.85-52)^2+(40.85-48)^2}}{7} = 8.07
Sample Problems
Question 1. Find the standard deviation for the data: 41, 28, 69, 36, 47.
Solution:
We have, N = 5.
Mean (x̄) = (41+28+69+36+47)/5 = 44.2
Standard deviation =
\frac{\sqrt{\sum({x_i-\bar{x}})^2}}{N} =
\frac{\sqrt{(44.2-41)^2+(44.2-28)^2+(44.2-69)^2+(44.2-36)^2+(44.2-47)^2}}{5} = 13.87
Question 2. Find the standard deviation for the data: 3, 2, 7, 1, 4.
Solution:
We have, N = 5.
Mean (x̄) = (3+2+7+1+4)/5 = 3.4
Standard deviation =
\frac{\sqrt{\sum({x_i-\bar{x}})^2}}{N} =
\frac{\sqrt{(3.4-3)^2+(3.4-2)^2+(3.4-7)^2+(3.4-1)^2+(3.4-4)^2}}{5} = 2.059
Question 3. Find the standard deviation for the data: 2, 4, 6, 8, 10.
Solution:
We have, N = 5.
Mean (x̄) = (2+4+6+8+10)/5 = 6
Standard deviation =
\frac{\sqrt{\sum({x_i-\bar{x}})^2}}{N} =
\frac{\sqrt{(6-2)^2+(6-4)^2+(6-6)^2+(6-8)^2+(6-10)^2}}{5} = 2.82
Question 4. Find the standard deviation for the data: 1, 3, 5, 7, 9.
Solution:
We have, N = 5.
Mean (x̄) = (1+3+5+7+9)/5 = 5
Standard deviation =
\frac{\sqrt{\sum({x_i-\bar{x}})^2}}{N} =
\frac{\sqrt{(5-1)^2+(5-3)^2+(5-5)^2+(5-7)^2+(5-9)^2}}{5} = 2.82
Question 5. Find the standard deviation for the data: 18, 32, 9, 20, 15.
Solution:
We have, N = 5.
Mean (x̄) = (18+32+9+20+15)/5 = 18.8
Standard deviation = \frac{\sqrt{\sum({x_i-\bar{x}})^2}}{N}
=
\frac{\sqrt{(18.8-18)^2+(18.8-32)^2+(18.8-9)^2+(18.8-20)^2+(18.8-15)^2}}{5} = 7.57
Question 6. Find the standard deviation for the data: 16, 21, -3, 12, 20.
Solution:
We have, N = 5.
Mean (x̄) = (16+21-3+12+20)/5 = 13.2
Standard deviation = \frac{\sqrt{\sum({x_i-\bar{x}})^2}}{N}
=
\frac{\sqrt{(13.2-16)^2+(13.2-21)^2+(13.2+3)^2+(13.2-12)^2+(13.2-20)^2}}{5} = 8.70