Standard deviation is statistically measured that shows the dispersion and variation amount in the given data value. It measured how the data spread around the mean and how it is high to tell which lies in which region.
However, the term "standard deviation" was invented by statistician Sir Ronald A. Fisher. Fisher recognized the importance of Pearson's work and popularized the use of standard deviation in statistical preparation. Then, standard deviation has expanded importance as a fundamental tool in inferential statistics allowing scholars to make implications around populations based on sample data.
The concept of standard deviation is closely related to other statistical measures, such as variance, which is the square of the standard deviation. The use of standard deviation expanded across various disciplines, including social sciences, economics, biology, and engineering.
In this article, we will discuss the basic definition of standard deviation, formulas of population and sample, procedure, and its applications.
Standard deviation
Standard deviation is statistically measured that shows the dispersion and variation amount in the given data value. It measured how the data spread around the mean and how it is high to tell which lies in which region.
An advanced standard deviation specifies a superior degree of inconsistency in the data. Simply, a lower standard deviation advises that the data points are closer to the mean. It is determined by attracting the square root of the average of the squared differences between each data point and the mean.
Procedure to find the Standard deviation:
 First, calculate the mean value of the given dataset by summing up all the values and dividing by the total number of values in the data set.
 Calculate the difference between each value and the mean of the data set, and take a square of each value.
 Calculate the variance by summing up the squared differences and dividing by the total number of values minus one.
 The deviation from the mean is equal to the square base of the variance.
Types of standard deviation
The standard deviation is classically considered by the sign “σ” for the population standard deviation and “s” for the sample standard deviation. The formulas for evaluating the population and sample standard deviations are slightly different.
 Sample Standard Deviation
 Population Standard Deviation
Sample Standard Deviation:
The sample standard deviation is used when you have a subsection of the population. The formula is as follows:
Sample Standard Deviation Formula 
In this formula: S.D = s = √Σ (xᵢ  x̄)^{2}/ (n  1)




Standard Deviation for population:
The populationwide standard deviation estimates the populationbased data set's volatility or distribution.
Population Standard Deviation Formula

In this formula: S.D = σ = √Σ (Xᵢ  μ)^{2}/ N




Note:
Both formulas involve squaring the differences between each data point and the mean summing these squared differences by separating them by the suitable sample or population proportions and lastly attracting the square root to gain the standard deviation.
Real life applications
The standard deviation has several applications across various fields.
Numbers variability:
A gauge of the variation or distribution of data points about the mean is provided by the standard deviation. It helps to understand how much the data deviates from the average providing insights into the variability of the dataset.
Quality control:
Standard deviation is utilized in quality control to measure the variability and consistency of product or process outcomes. It helps identify and monitor variations enabling organizations to improve processes and reduce defects.
Statistical conclusion:
The standard deviation is important for concluding statistics. It helps estimate confidence intervals determine the significance of differences between groups and perform hypothesis testing. It provides a measure of the reliability and precision of statistical results.
Data analysis:
Standard deviation is used in data analysis to understand the spread of data and identify outliers. It helps researchers and analysts draw meaningful conclusions identify patterns and make datadriven decisions.
Research and experimentation:
Standard deviation is employed in research studies to assess the consistency and reliability of measurements or observations. It helps evaluate the variability within samples and populations enhancing the validity of research findings.
Example section of standard deviation
In this section, the concept of standard deviation with the help of examples is given.
Example 1: (For population)
Calculate the standard deviation of the population data 10,20,30,45,60,85,100.
Solution:
Step 1:
No 
x_{i} 
(x_{i } μ) 
(x_{i } μ)^{2} 
1 
10 
10 – 50 = 40 
(40)^{2} =1600 
2 
20 
20 – 50 = 30 
(30)^{2} = 900 
3 
30 
30 – 50 = 20 
(20)^{2} = 400 
4 
45 
45 – 50 = 5 
(5)^{2} = 25 
5 
60 
60 – 50 = 10 
(10)^{2} = 100 
6 
85 
85 – 50 = 35 
(35)^{2} = 1225 
7 
100 
100 – 50 = 50 
(50)^{2}= 2500 
N = 7 
μ = Σ(x_{i})/N = 350/7 = 50 
Σ (x_{i } μ)/N = 0/7 = 0 
Σ (x_{i } μ)^{2}/N = 6750/7 = 964.28 
Step 2:
Total terms = N = 7
Mean = μ = Σ(x_{i})/N = 50
Step 3:
For population
⇒ σ = S.D = √Σ (Xᵢ  μ)^{2}/ N
Step 4:
⇒ σ^{2} = variance = Σ (x_{i } μ)^{2}/N = 964.28
step 5:
⇒ S.D = σ = √Σ (Xᵢ  μ)^{2}/ N = √964.28 = 31.05
⇒ σ = 31.05
An SD calculator is an alternate way to find the standard deviation of the given set of data. It will allow you to get the result in seconds without getting involved into lengthy calculations. Here is a example solved through this calculator.
Example 2: (For sample)
Calculate the standard deviation of the sample data 5,15,22,29,64.
Solution:
Step 1:
No 
x_{i} 
(x_{i } X̅) 
(x_{i } X̅)^{2} 
1 
5 
5 – 27 = 22 
(22)^{2 }= 484 
2 
15 
15 – 27 = 12 
(12)^{2} = 144 
3 
22 
22 – 27 = 5 
(5)^{2} = 25 
4 
29 
29 – 27 = 2 
(2)^{2} = 4 
5 
64 
64 – 27 = 37 
(37)^{2} = 1369 
n = 5 
X̅ = Σ(x_{i})/n = 135/5 = 27 
Σ (x_{i } X̅)/n = 0/5 = 0 
Σ (x_{i } X̅)^{2} /(n1) = 2026/4 = 506.5 
Step 2:
Total terms = n = 5
Mean = X̅ = 27
Step 3:
Standard deviation formula for sample
⇒ S.D = s = √Σ (xᵢ  x̄)^{2}/ (n  1)
Step 4:
⇒ Variance = s² = Σ (xᵢ  x̄)^{2}/ (n  1) = 506.5
Step 5:
⇒ S.D = s = √Σ (xᵢ  x̄)^{2}/ (n  1) = √506.5 = 22.50
Standard deviation for sample = s = 22.50
Related question
Question 1:
What is the standard deviation?
Solution:
Standard deviation is a quantity of the total disparity or distribution in a set of values. It quantifies how spread out the data values are from the mean or average
Question 2:
Can a standard deviation have a negative value?
Solution:
No, it can't be negative for the standard deviation. Since it involves squared differences, it is always a nonnegative value or zero.
Summary
In this article, we have discussed the basic definition of standard deviation, the formula of population and standard deviation, applications, and the procedure to determine the standard deviation in detail with the help of examples. To better understand the concept of standard deviation, evaluate different examples using standard deviation.
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