Standard Deviation - Formula, Examples & How to Calculate

Standard deviation is a statistical measure that describes how much variation or dispersion there is in a set of data points. It helps us understand how spread out the values in a dataset are compared to the mean (average). A higher standard deviation means the data points are more spread out, while a lower standard deviation means they are closer to the mean.

Standard deviation can be classified as:

• Low standard deviation: The data points are close to the mean, meaning the values are relatively consistent.
• High standard deviation: The data points are spread out over a wider range, meaning there's more variability in the data.

The image below shows the standard deviation along with the formula:

In simple terms, standard deviation tells you how much the numbers in a group are spread out from the average (mean).

• If you have test scores like 90, 91, 92, 93, the standard deviation is small because all the scores are close to each other.
• If you have test scores like 60, 80, 90, 100, 120, the standard deviation is large because the scores are spread out over a wide range.

Here we will learn about standard deviation, population, and sample, their various formulas, steps to calculate them, and much more.

Mathematical Definition

In mathematical terms, the standard deviation is the square root of the variance. Variance is the average of the squared differences from the mean.

• Standard Deviation is defined as the degree of dispersion of the data points from the mean value of the data points.

Mean Deviation = 1/n∑_iⁿ (x_i - x̄)²

Standard deviation is a measure used in statistics to understand how the data points in a set are spread out from the mean value.

It indicates the extent of the data's variation and shows how far individual data points deviate from the average.

Population and Sample

Before learning about the formula, let's explore what population is and what a sample is.

Population

A population is the entire set of individuals or items that you're interested in studying. It represents the whole group about which you want to conclude. The population includes every possible member or data point that fits the criteria of your study.

• Example: If you're studying the average height of all adult women in a country, the population would include every adult woman in that country.

Sample

A sample is a subset of the population. Since it's often impractical or impossible to study the entire population, researchers use a sample, which should ideally be representative of the population. By analyzing the sample, you can make inferences about the population as a whole.

• Example: If you're studying the average height of all adult women in a country, you might only study a random sample of 1,000 adult women instead of everyone in the country.

Standard Deviation Formula

The formula for the standard deviation depends on whether you're working with a sample or an entire population.

For Sample Data

 \bold{s = \sqrt{\frac{\sum_{i=1}^n (x_i - x̄)^2}{n-1}}}

where,

• s is Population Standard Deviation
• x_i is the ith observation
• x̄ is the Sample Mean
• N is the Number of Observations
• ∑ = sum over all the values

For Population Data

 \bold{\sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2}{N}}}

where,

• σ is Population Standard Deviation
• x_i is the i^th Observation
• μ is Population Mean
• N is the Number of Observations
• ∑ = Sum over all the values

It is evident that both formulas look the same and have only slide changes in their denominator. The denominator in the case of the sample is n-1, but in the case of the population is N. Initially, the denominator in the sample standard deviation formula has "n" in its denominator, but the result from this formula was not appropriate. So, a correction was made, and the n is replaced with n-1 formula correction is called Bessel's correction, which in turn produced the most appropriate results.

Related topic:

Population vs Sample in Statistics

Steps to Calculate Standard Deviation

Generally, when we talk about standard deviation, we talk about population standard deviation. The steps to calculate the standard deviation of a given set of values are as follows,

Step 1: Calculate mean of observation using the
(Mean = Sum of Observations/Number of Observations)

Step 2: Calculate squared differences of data values from the mean.
(Data Value - Mean)²

Step 3: Calculate average of squared differences
(Variance = Sum of Squared Differences / Number of Observations)

Step 4: Calculate square root of variance this gives the Standard Deviation
(Standard Deviation = √Variance)

What is Variance?

Variance is a statistical measure that tells us how spread out the values in a data set are from the mean (average).

It is the average of the squared differences from the mean.

Variance shows how much the numbers in your data vary from the average value.

• If the variance is small, the numbers are close to the mean.
• If the variance is large, the numbers are more spread out.

Also, Check:

• Difference Between Variance and Standard Deviation
• Mean, Variance and Standard Deviation
• Variance and Standard Deviation

Variance Formula

The formula to calculate the variance of a dataset is as follows:

Variance (σ²) = Σ [(x - μ)²] / N

Where:

• Σ denotes Summation (adding up)
• x represents Each Individual Data Point
• μ is the Mean (Average) of the Dataset
• N is the Total Number of Data Points
• ∑ = sum over all the values

How to Calculate Variance?

The steps to calculate the variance of a dataset:

Step1: Calculate the Mean (Average):
Add up all the values in the dataset and divide by the total number of values. This gives you the mean (μ).

Mean (μ) = (Sum of All Values) / (Total Number of Values)

Step2: Find the Difference from the Mean
Subtract the mean from each value in the data set. This shows how far each value is from the average.
For each data value: xi - Meanx

Step3: Square Each Difference
Square the result from Step 2 to eliminate negative signs and give more weight to larger differences.
Squared Difference for Each Value = (Value - Mean)²

Step4: Calculate the Average of the Squared Differences:
Add up all the squared differences calculated in the previous step, and then divide by the total number of values in the dataset. This gives you the variance (σ^2).

Variance (σ²) = (Sum of all Squared Differences) / (Total Number of Values)
Sample Variance (if you're working with a sample only):
s² = (Sum of all Squared Differences) / N - 1
Population Variance (if you have all data):
σ² = (Sum of all Squared Differences) / N

Variance Vs Standard Deviation

The key difference between variance and standard deviation is given below:

Standard Deviation of Ungrouped Data

For ungrouped data, the standard deviation can be calculated using three methods:

• Actual Mean Method
• Assumed Mean Method
• Step Deviation Method

Standard Deviation by Actual Mean Method

Standard Deviation by actual mean method uses the basic mean formula to calculate the mean of the given data, and using this mean value, we find out the standard deviation of the given data values. We calculate the mean in this method with the formula,

μ = (Sum of Observations)/(Number of Observations)

Standard deviation formula for the Actual mean method

σ = √(∑_iⁿ (x_i - x̄)²/n)

Example: Find the Standard Deviation of the data set, X = {2, 3, 4, 5, 6}

Given,

• n = 5
• x_i = {2, 3, 4, 5, 6}

We know, 

Mean(μ) = (Sum of Observations)/(Number of Observations)
⇒ μ = (2 + 3 + 4 + 5 + 6)/ 5
⇒ μ = 4

using standard deviation formula
σ² = ∑_iⁿ (x_i - x̄)²/n
⇒ σ² = 1/n[(2 - 4)² + (3 - 4)² + (4 - 4)² + (5 - 4)² + (6 - 4)²]
⇒ σ² = 10/5 = 2

Thus, σ = √(2) = 1.414

Standard Deviation by Assumed Mean Method

For very large values of x, finding the mean of the grouped data is a tedious task; therefore, we use an assumed mean method where we assume an arbitrary value (A) as the mean value and then calculate the standard deviation using the normal method. Suppose for the group of n data values ( x₁, x₂, x₃, ..., x_n), the assumed mean is A, then the deviation is,

d_i = x_i - A

Where,

• x_i = data values
• A = assumed mean

Standard Deviation formula for the assumed mean method

σ = √(∑_iⁿ (d_i)²/n)

Where,

• 'n' = Total Number of Data Values
• d_i = x_i - A

Standard Deviation by Step Deviation Method

We can also calculate the standard deviation of the grouped data using the step deviation method. As in the above method, in this method also, we choose some arbitrary data value as the assumed mean (say A). Then we calculate the deviations of all data values (x₁, x₂, x₃, ..., x_n),  d_i = x_i - A

In the next step, we calculate the Step Deviations (d') using

d' = d/i 

where 'i is a Common Factor of all values

Standard Deviation Formula for Step Deviation Method

σ = √[(∑(d_i² /n) - (∑d_in)²] × i

where,

• 'n' = Total Number of Data Values
• d_i = x_i - A

Standard Deviation of Discrete Grouped Data

In grouped data, first, we made a frequency table, and then any further calculation was made. For discrete grouped data, the standard deviation can also be calculated using three methods:

• Actual Mean Method
• Assumed Mean Method
• Step Deviation Method

Formula Based on Discrete Frequency Distribution

For a given data set, if it has n values (x₁, x₂, x₃, ..., x_n) and the frequency corresponding to them is (f₁, f₂, f₃, ..., f_n), then its standard deviation is calculated using the formula,

σ = √(∑_iⁿ f_i(x_i - x̄)²/n)

where,

• n is Total Frequency (n = f₁ + f₂ + f₃ +...+ f_n )
• x̄ is the Mean of the Data
• x_i Value of data point
• f_i frequency of data points

Example: Calculate the standard deviation for the given data

Solution:

Mean (x̄) = ∑(f_i x_i)/∑(f_i)
⇒ Mean (μ)  = (10×1 + 4×3 + 6×5 + 8×1)/(1 + 3 + 5 + 1)
⇒ Mean (μ) = 60/10 = 6

n = ∑(f_i) = 1 + 3 + 5 + 1 = 10

Now,

using standard deviation formula

σ = √(∑_iⁿ f_i(x_i - x̄)²/n)
⇒ σ = √[(16 + 12 + 0 +8)/10] 
⇒ σ = √(3.6) = 1.897

Standard Derivation(σ) = 1.897

Standard Deviation of Discrete Data by Assumed Mean Method

In grouped data, if the values in the given data set are very large, then we assume a random value (say A) as the mean of the data. Then, the deviation of each value from the assumed mean is calculated as,

d_i = x_i - A

d_i = Deviation of data point from assumed mean

Standard deviation formula for the assumed mean method

σ = √[(∑(f_id_i)² /n) - (∑f_id_i/n)²]

where,

• 'f' is the Frequency of Data Value x
• d_i = Deviation of data point from assumed mean
• 'n' is Total Frequency [n = ∑(f_i)]

Standard Deviation of Discrete Data by Step Deviation Method

We can also use the step deviation method to calculate the standard deviation of the discrete grouped data. As in the above method, in this method also, we choose some arbitrary data value as the assumed mean (say A). Then we calculate the deviations of all data values (x₁, x₂, x₃, ..., x_n),  d_i = x_i - A

In the next step, we calculate the Step Deviations (d') using

d' = d/i 

where 'C' is the Common Factor of all 'd values

Standard deviation formula for the Step Deviation Method

σ = C \sqrt{\frac{\sum f_i d_i^2}{N} - \left(\frac{\sum f_i d_i}{N}\right)^2}

Where,

• σ = Standard Deviation
• C = Common Factor of all 'd values
• ∑f_id_i^2  = Sum total of the squared step deviations multiplied by frequencies
•  ∑f_id_i  =  Sum total of step deviations multiplied by frequencies
• N = Total Number of Data Values

Standard Deviation of Continuous Grouped Data

For the continuous grouped data, we can easily calculate the standard deviation using the Discrete data formulas by replacing each class with its midpoint (as x_i) and then normally calculating the formulas. 

The calculation of each class is calculated using the formula:

x_i (Midpoint) = (Upper Bound + Lower Bound)/2

Standard Deviation Formula for Grouped Data:

s =\sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}}

Where:

• x_i​ = midpoint of each class interval
• f_i​ = frequency of each class interval
• \bar{x}= mean of the grouped data
• s = standard deviation

For example, calculate the standard deviation of continuous grouped data as given in the table.

Solution:

Mean (x̄) = ∑(f_i x_i)/∑(f_i)
⇒ Mean (μ) = (10×2 + 20×4 + 30×2 + 40×2)/(2+4+2+2)
⇒ Mean (μ) = 240/10 = 24
n = ∑(f_i) = 2+4+2+2 = 10

Now using standard deviation formula:

σ = √(∑_iⁿ f_i(x_i - x̄)²/n)
⇒ σ  = √[(392 + 64 + 72 +512)/10] 
⇒ σ  = √(104) = 10.198

Standard Derivation(σ) = 10.198

Standard Deviation of Probability Distribution

In probability of all the possible outcomes are generally equal, and we take many trials to find the experimental probability of the given experiment.

• For a normal distribution, the expected mean is zero and the Standard Deviation is 1.

Standard Deviation Formula Binomial Distribution

For a binomial distribution, the standard deviation is given by the formula,

σ = √(npq)

where,

• n is the Number of Trials
• P is the Probability of Success of a Trial
• q is Probability of Failure of Trial (q = 1 - p)

Standard Deviation Formula Poisson Distribution

For a Poisson distribution, the standard deviation is given by 

σ = √λt

where,

• λ is the Average Number of Successes
• t is given a time interval

Standard Deviation of Random Variables

Random variables are the numerical values that denote the possible outcomes of the random experiment in the sample space. Calculating the standard deviation of the random variable tells us about the probability distribution of the random variable and the degree of the difference from the expected value.

We use X, Y, and Z as functions, to represent the random variables. The probability of the random variable is denoted as P(X), and the expected mean value is denoted by the μ symbol.

Then the Standard Deviation formula for the standard deviation of a probability distribution is,

σ = \sqrt{(∑ (x_i - μ)^2 × P(x_i))}

where:

• x_i = data points
• p(x_i) = probability of x_i
• μ = Expected mean Value

Articles related to Standard Deviation:

• Mean
• Mode
• Mean Deviation

Standard Deviation Formula Examples

Example 1: Find the Standard Deviation of the following data,

Solution:

First, make the table as follows, so we can calculate the further values easily.

Xi	fi	Xi×fi	Xi-μ	(Xi-μ)2	f×(Xi-μ)2
5	2	10	-6.375	40.64	81.28
12	3	36	0.625	0.39	1.17
15	3	45	3.625	13.14	39.42
Total	8	91			121.87

Mean (μ) = ∑(f_i x_i)/∑(f_i)

⇒ Mean (μ) = 91/8 = 11.375

using standard deviation formula

σ = √(∑_iⁿ f_i(x_i - μ)²/n)
⇒ σ = √[(121.87)/(8)]
⇒ σ = √(15.234)
⇒ σ  = 3.90

Standard Derivation(σ) = 3.90

Example 2: Find the Standard Deviation of the following data table.

Solution:

Class	Xi	fi	f×Xi	Xi - μ	(Xi - μ)2	f×(Xi - μ)2
0-10	5	3	15	-15	225	675
10-20	15	6	90	-5	25	150
20-30	25	4	100	5	25	100
30-40	35	2	70	15	225	450
40-50	45	1	45	25	625	625
Total		16	320			2000

Mean (μ) = ∑(fi xi)/∑(fi)

⇒ Mean (μ) = 320/16 = 20

now, by using standard deviation formula

σ = √(∑_iⁿ f_i(x_i - μ)²/n)
⇒ σ = √[(2000)/(16)]
⇒ σ = √(125)
⇒ σ = 11.18

Standard Derivation(σ) = 11.18

Standard Deviation Formula in Excel

• Use Excel's built-in functions STDEV.P for the entire population or STDEV.S for a sample.
• Step-by-Step Guide: Enter your data set in a single column, then type =STDEV.S(A1:A10) (replace A1:A10 with your data range.) in a new cell to get the standard deviation for a sample.
• Visual Aids: Utilize Excel's chart tools to visually represent data variability alongside standard deviation.

Example: Suppose you have the following numbers in cells A1 to A5:

Solution:

• For population SD: =STDEV.P(A1:A5)
• For sample SD: =STDEV.S(A1:A5)

Read in Detail: Standard Deviation in Excel: How to Calculate, Formulas

Conclusion

The standard deviation is a crucial statistical measure that provides valuable information about the variability or dispersion of data within a dataset. It helps us understand how spread out the data points are from the mean, offering a clearer picture of the data's consistency and reliability. It is widely used in various fields, including statistics, finance, and science, to understand the distribution of data and make informed decisions based on the level of variability present.