Standard Deviation (original) (raw)

Last Updated : 26 Dec, 2025

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. It is classified as:

**Example:

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.

\text{Mean Deviation} =\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2

Standard Deviation Formulas

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

population_standard_deviation

**For Sample Data

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

where,

For Population Data

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

where,

It is evident that both formulas look the same and have only slide changes in their denominator.

Historically, the sample standard deviation was first written with **n in the denominator, but the results tended to underestimate the true variability of the population.

To correct this bias, the denominator was changed from **n to **n − 1. This adjustment is known as **Bessel’s correction, and it gives an unbiased estimate of the population variance when working with samples.

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 data
(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)

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.

Variance Formula

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

**Variance (σ 2 ) = Σ [(x - μ) 2 ] / N

Where:

Variance vs Standard Deviation

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

Variance Deviation (Standard Deviation)
The measure of spread in a dataset. Measure of average distance from the mean.
Average of squared differences from the mean. The square root of the variance.
σ^2 (sigma squared) σ (sigma)
Indicates the average squared deviation of data points from the mean. Indicates the average distance of data points from the mean.
Harder to interpret directly (due to squared units) Easier to interpret; shows how spread out the data is

Standard Deviation of Ungrouped Data

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

Standard Deviation by Actual Mean Method

Standard Deviation by the actual mean method uses the basic mean formula to calculate the mean of the given data,and using this mean value, we find 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 = √(∑ i n (x i - x̄) 2 /n)

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

Given,

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 (xi - 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, ..., xn), the assumed mean is A, then the deviation is,

**d i = x i - A

Where,

Standard Deviation formula for the assumed mean method

**σ = √(∑ i n (d i ) 2 /n)

Where,

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, ..., xn), di = xi - 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 i 2 /n) - (∑d i n) 2 ] × i

where,

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:

Formula Based on Discrete Frequency Distribution

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

**σ = √(∑ i n f i (x i - x̄) 2 /n)

where,

**Example: Calculate the standard deviation for the given data

xi fi
10 1
4 3
6 5
8 1

**Solution:

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

n = ∑(fi) = 1 + 3 + 5 + 1 = 10

xi fi (xi - x̄) (xi - x̄)2 fi(xi - x̄)2
10 1 4 16 16
4 3 -2 4 12
6 5 0 0 0
8 1 2 4 4

∑fi​ (xi ​− xˉ)2 = 16 + 12 + 0 + 4 = 32

Now, using standard deviation formula

**σ = √(∑ i n f i (x i - x̄) 2 /n)
⇒ σ = √[(32)/10]
⇒ σ = √(3.2) ≈1.7888

Standard Derivation(σ) = 1.7888

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 i d i ) 2 /n) - (∑f i d i /n) 2 ]

where,

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, ..., xn), di = xi - 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,

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 xi) 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:

**For example: Calculate the standard deviation of continuous grouped data as given in the table.

Class 0-10 10-20 20-30 30-40
Frequency(fi) 2 4 2 2

**Solution:

Class 5-15 15-25 25-35 35-45
xi 10 20 30 40
Frequency(fi) 2 4 2 2

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

xi fi fixi (xi - x̄) (xi - x̄)2 fi(xi - x̄)2
10 2 20 14 196 392
20 4 80 -4 16 64
30 2 60 6 36 72
40 2 80 16 256 512

Now **using standard deviation formula:

**σ = √(∑ i n 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.

Standard Deviation Formula Binomial Distribution

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

**σ = √(npq)

where,

Standard Deviation Formula Poisson Distribution

For a Poisson distribution, the standard deviation is given by

**σ = √λt

where,

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:

**Solved Examples on Standard Deviation

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

xi 5 12 15
fi 2 4 3

**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) = ∑(fi xi)/∑(fi)

⇒ Mean (μ) = 91/8 = 11.375

**using standard deviation formula

σ = √(∑in fi(xi - 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.

Class Frequency
0-10 3
10-20 6
20-30 4
30-40 2
40-50 1

**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 fi(xi - μ)2/n)
⇒ σ = √[(2000)/(16)]
⇒ σ = √(125)
⇒ σ = 11.18

Standard Derivation(σ) = 11.18

Standard Deviation Formula in Excel

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

**Solution:

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