Standard Deviation
Standard deviation is the positive square root of the variance. Standard deviation is one of the basic methods of statistical analysis. Standard deviation is commonly abbreviated as SD and denoted by 'σ’ and it tells about the value that how much it has deviated from the mean value. If we get a low standard deviation then it means that the values tend to be close to the mean whereas a high standard deviation tells us that the values are far from the mean value. Let us learn to calculate the standard deviation of grouped and ungrouped data and the standard deviation of a random variable.
1.  What is Standard Deviation? 
2.  Standard Deviation Formula 
3.  Standard Deviation of Random Variables 
4.  Standard Deviation of Probability Distribution 
5.  FAQs on Standard Deviation 
What is Standard Deviation?
Standard deviation is the degree of dispersion or the scatter of the data points relative to its mean, in descriptive statistics. It tells how the values are spread across the data sample and it is the measure of the variation of the data points from the mean. The standard deviation of a sample, statistical population, random variable, data set, or probability distribution is the square root of its variance.
When we have n number of observations and the observations are \(x_1, x_2, .....x_n\), then the mean deviation of the value from the mean is determined as \(\sum_{i=1}^{n}\left(x_{i}\bar{x}\right)^{2}\). However, the sum of squares of deviations from the mean doesn't seem to be a proper measure of dispersion. If the average of the squared differences from the mean is small, it indicates that the observations \(x_i\) are close to the mean \(\bar x\). This is a lower degree of dispersion. If this sum is large, it indicates that there is a higher degree of dispersion of the observations from the mean \(\bar x\). Thus we conclude that \(\sum_{i=1}^{n}\left(x_{i}\bar{x}\right)^{2}\) is a reasonable indicator of the degree of dispersion or scatter.
We take \(\dfrac{1}{n}\sum_{i=1}^{n}\left(x_{i}\bar{x}\right)^{2}\) as a proper measure of dispersion and this is called the variance(σ^{2}). The square root of the variance is the standard deviation.
Steps to Calculate Standard Deviation
 Find the mean, which is the arithmetic mean of the observations.
 Find the squared differences from the mean. (The data value  mean)^{2}
 Find the average of the squared differences. (Variance = The sum of squared differences ÷ the number of observations)
 Find the square root of variance. (Standard deviation = √Variance)
Standard Deviation Formula
The spread of statistical data is measured by the standard deviation. The degree of dispersion is computed by the method of estimating the deviation of data points. You can read about dispersion in summary statistics. As discussed, the variance of the data set is the average square distance between the mean value and each data value. And standard deviation defines the spread of data values around the mean. Here are two standard deviation formulas that are used to find the standard deviation of sample data and the standard deviation of the given population.
Formula for Calculating Standard Deviation
The population standard deviation formula is given as:
 \(\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N}\left(X_{i}\mu\right)^{2}}\)
Here,
 σ = Population standard deviation
 μ = Assumed mean
Similarly, the sample standard deviation formula is:
 \(s=\sqrt{\frac{1}{n1} \sum_{i=1}^{n}\left(x_{i}\bar{x}\right)^{2}}\)
Here,
s = Sample standard deviation
\(\bar x\) = Arithmetic mean of the observations
Standard Deviation of Ungrouped Data
The calculations for standard deviation differ for different data. Distribution measures the deviation of data from its mean or average position. There are two methods to find the standard deviation.
 actual mean method
 assumed mean method
Standard Deviation by The Actual Mean Method
σ = √(∑\(x\bar x)\)^{2 }/n)
Consider the data observations 3, 2, 5, 6. Here the mean of these data points is 16/4 = 4.
The squared differences from mean = (43)^{2}+(24)^{2 }+(54)^{2 }+(64)^{2}= 10
Variance = Squared differences from mean/ number of data points =10/4 =2.5
Standard deviation = √2.5 = 1.58
Standard deviation by Assumed Mean Method
When the x values are large, an arbitrary value (A) is chosen as the mean. The deviation from this assumed mean is calculated as d = x  A.
σ = √[(∑(d)^{2 }/n)  (∑d/n)^{2}]
Standard Deviation of Grouped Data
When the data points are grouped, we first construct a frequency distribution.
Standard Deviation of Grouped Discrete Frequency Distribution
For n number of observtions, \(x_1, x_2, .....x_n\), and the frequency, \(f_1, f_2, f_3, ...f_n\) the standard deviation is:
\(\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N}f_i \left(X_{i}\bar x\right)^{2}}\). Here N = \(\sum_{i=1}^{N}f_i\)
Example: Let's calculate the standard deviation for the data given below:
\(x_i\)  6  10  12  14  24 

\(f_i\)  2  3  4  5  4 
Calculate mean(\(\bar x\)): (6+8 +10+12+ 14)/5 = 10
\(x_i\)  \(f_i\)  \(f_ix_i\)  \(x_i \bar x\)  \((x_i \bar x\))^{2}  \(f_i (x_i \bar x)\)^{2} 

6  2  12  4  16  32 
8  3  24  2  4  12 
10  4  40  0  0  0 
12  5  60  2  4  20 
14  4  56  4  16  64 
18  192  128 
N = 18, ∑\(f_i x_i\) = 192, ∑\(f_i (x_i \bar x\))^{2} = 128
Calculate variance: σ^{2 }= 1/N \(\sum_{i=1}^{N}f_i \left(X_{i}\bar x\right)^{2}\)
= 1/18 × 128 = 7.1
Calculate SD: σ = √Variance = √ 7.1 = 2.66
Standard Deviation of Grouped Continuous Frequency Distribution
If the frequency distribution is continuous, each class is replaced by its midpoint. Then the Standard deviation is calculated by the same technique as in discrete frequency distribution. Consider the following example. \(x_i\) is calculated as the midpoint of each class. Then the same standard deviation formula is applied.
Class  \(f_i\)  \(x_i\) 

010  3  5 
1020  4  15 
2030  6  25 
3040  4  35 
4050  8  40 
Standard Deviation of Random Variables
The measure of spread for the probability distribution of a random variable determines the degree to which the values differ from the expected value. This is a function that assigns a numerical value to each outcome in a sample space. This is denoted by X, Y, or Z, as it is a function. If X is a random variable, the standard deviation is determined by taking the square root of the sum of the product of the squared difference between the random variable, x, and the expected value (š) and the probability associated value of the random variable.
The standard deviation of the probability distribution of X, š = \(\sqrt{(x  š)^2 P(X=x)}\)
This is also equivalent to š = \(\sqrt{E(X)^2[E(X)]^2}\)
Standard Deviation of Probability Distribution
The experimental probability consists of many trials. When the difference between the theoretical probability of an event and its relative frequency get closer to each other, we tend to know the average outcome. This mean is known as the expected value of the experiment denoted by š.
 In a normal distribution, the mean is zero and the standard deviation is 1.
 In a binomial experiment, the number of successes is a random variable. If a random variable has a binomial distribution, its standard deviation is given by: š= √npq, where mean: š = np, n = number of trials, p = probability of success and 1p =q is the probability of failure.
 In a Poisson distribution, the standard deviation is given by š= √λt, where λ is the average number of successes in an interval of time t.
Standard Deviation Tips:
 For n as the sample or the population size, the square root of the average of the squared differences of data observations from the mean is called the standard deviation.
 Standard deviation is the positive square root of variance.
 Standard deviation is the indicator that shows the dispersion of the data points about the mean.
ā Also Check:
Standard Deviation Examples

Example 1: There are 39 plants in the garden. A few plants were selected randomly and their heights in cm were recorded as follows: 51, 38, 79, 46, 57. Calculate the standard deviation of their heights.
Solution:
N = 5
Mean (\(\bar{x}\))= (51+38+79+46+57)/5 = 54.2
Standard Deviation = \( \sqrt{\dfrac{\Sigma (x_i\bar{x})^2}{N1}} \)
= \( \sqrt{\frac{(5154.2)^2 +(3854.2)^2 +(7954.2)^2 +(4654.2)^2 +(5754.2)^2}{4}} \)
= 15.5
Answer: Standard Deviation for this data is 15.5

Example 2: In a class of 50, 4 students were selected at random and their total marks in the final assessments are recorded, which are: 812, 836, 982, 769. Find the standard deviation of their marks.
Solution:
N = 4
Sample Mean (XĢ) = (812+836+982+769)/4 = 849.75
Variance = \( \dfrac{\sum^{N}_{i=1} (X_i  \bar{X})^2}{N1} \)
=\( \dfrac{\sum^{4}_{i=1} (X_i  849.75)^2}{3} \)
= [(812  849.75)^{2} + (836  849.75)^{2} + (982  849.75)^{2} + (769  849.75)^{2}] /3 = 92.4
Using the standard deviation formula,
Standard Deviation = √92.4 = 2 √23.1 = 9.6
Answer: Standard Deviation for this data is 9.6

Example 3: Find the standard deviation of X which has the probability distribution as shown in the table below.
X P(X) 2 0.2 3 0.3 4 0.5 Solution:
To find the expected value of X, find the product X. P(X) and sum these terms.
X P(X) X. P(X) 4 0.2 0.8 5 0.3 1.5 6 0.4 2.4 E(X) = 0.8+1.5+2.4
=4.7
X P(X) (X  E(X)^{2}) (X  E(X)^{2}). P(X) 4 0.2 4.9 0.98 5 0.3 0.9 0.27 6 0.5 1.69 0.845 Standard Deviation = √(0.98+0.27+0.845)
= √2.095 ≈1.45
Answer:The standard deviation of the probability distribution is 1.45
FAQs on Standard Deviation
What is Standard Deviation?
The standard deviation is the measure of dispersion or the spread of the data about the mean value. It helps us to compare the sets of data that have the same mean but a different range. The sample standard deviation formula is: \(s=\sqrt{\frac{1}{n1} \sum_{i=1}^{n}\left(x_{i}\bar{x}\right)^{2}}\), where \(\bar x\) is the sample mean and \(x_i\) gives the data observations and n denotes the sample size.
How Do You Calculate Standard Deviation?
For n observations in the sample, find the mean of them. Find the difference in mean for each data point and square the differences. Sum them up and find the square root of the average of the squared differences. This is given as \(s=\sqrt{\frac{1}{n1} \sum_{i=1}^{n}\left(x_{i}\bar{x}\right)^{2}}\).
Give an Example of Standard Deviation
If we get a low standard deviation then it means that the values tend to be close to the mean whereas a high standard deviation tells us that the values are far from the mean value. Consider data points 1, 3, 4, 5. The mean is 13/4 = 3.25. The average of mean differences = [(3.251)^{2} + (33.25)^{2}+ (43.25)^{2} + (53.25)^{2}]/4 = 2.06. The standard deviation = √2.06 = 1.43
What Is the Difference Between Standard Deviation Formula and Variance Formula?
Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Both measures reflect variability in distribution, but their units differ: Standard deviation is expressed in the same units as the original values (e.g., minutes or meters). Sample standard deviation formula = \(\sigma=\sqrt{\frac{1}{N1} \sum_{i=1}^{N}\left(X_{i}\mu\right)^{2}}\) and variance formula = σ^{2} = Σ (xi – xĢ )^{2}/(n1)
What Is MeanVariance and Standard Deviation in Statistics?
Variance is the sum of squares of differences between all numbers and means...where μ is Mean, N is the total number of elements or frequency of distribution. Standard Deviation is the square root of variance. It is a measure of the extent to which data varies from the mean. The standard Deviation formula is √variance, where variance = σ^{2} = Σ (xi – xĢ )^{2}/n1
Which Is Better to Use Variance Formula or Standard Deviation Formula?
They each have different purposes. The SD is usually more useful to describe the variability of the data while the variance is usually much more useful mathematically. For example, the sum of uncorrelated distributions (random variables) also has a variance that is the sum of the variances of those distributions.
Why Do We Use Standard Deviation Formula and Variance?
Standard deviation looks at how spread out a group of numbers is from the mean, by looking at the square root of the variance. The variance measures the average degree to which each point differs from the mean—the average of all data points.
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