Variance and Standard Deviation
Variance and Standard Deviation are the two important measurements in statistics. Variance is a measure of how data points vary from the mean, whereas standard deviation is the measure of the distribution of statistical data. The basic difference between variance and the standard deviation is in their units. The standard deviation is represented in the same units as the mean of data, while the variance is represented in squared units.
Here we aim to understand the definitions of variance and standard deviation, their properties, and the differences. Also, let us learn here more about both their measurements, formulas along with some examples.
Variance
According to layman’s words, the variance is a measure of how far a set of data are dispersed out from their mean or average value. It is denoted as ‘σ^{2}’.
Properties of Variance
 It is always nonnegative when studied in probability and statistics since each term in the variance sum is squared and therefore the result is either positive or zero.
 Variance always has squared units. For example, the variance of a set of weights estimated in kilograms will be given in kg squared. Since the population variance is squared, we cannot compare it directly with the mean or the data themselves.
Standard Deviation
The spread of statistical data is measured by the standard deviation. Distribution measures the deviation of data from its mean or average position. The degree of dispersion is computed by the method of estimating the deviation of data points. You can read about dispersion in summary statistics. Standard deviation is denoted by the symbol, ‘σ’.
Properties of Standard Deviation
 It describes the square root of the mean of the squares of all values in a data set and is also called the rootmeansquare deviation.
 The smallest value of the standard deviation is 0 since it cannot be negative.
 When the data values of a group are similar, then the standard deviation will be very low or close to zero. But when the data values vary with each other, then the standard variation is high or far from zero.
Variance and Standard Deviation Formula
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.
The formulas for the variance and the standard deviation for both population and sample data set are given below:
Variance Formula:
The population variance formula is given by:
\(\sigma^{2}=\frac{1}{N} \sum_{i=1}^{N}\left(X_{i}\mu\right)^{2}\)
Here,
σ^{2} = Population variance
N = Number of observations in population
Xi = ith observation in the population
μ = Population mean
The sample variance formula is given as:
\(s^{2}=\frac{1}{n1} \sum_{i=1}^{n}\left(x_{i}\bar{x}\right)^{2}\)
Here,
s^{2} = Sample variance
n = Number of observations in sample
xi = ith observation in the sample
x̄ = Sample mean
Standard Deviation Formula
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
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
Variance and Standard Deviation Relationship
Variance is equal to the average squared deviations from the mean, while standard deviation is the number’s square root. Also, the standard deviation is a square root of variance. Both measures exhibit variability in distribution, but their units vary: Standard deviation is expressed in the same units as the original values, whereas the variance is expressed in squared units.
Related Topics
Solved Examples

Example 1: If a die is rolled, then find the variance and standard deviation of the possibilities.
Solution: When a die is rolled, the possible number of outcomes is 6. So the sample space, n = 6 and the data set = { 1;2;3;4;5;6}.
To find the variance, first, we need to calculate the mean of the data set.
Mean, x̅ = (1+2+3+4+5+6)/6 = 3.5
We can put the value of data and mean in the formula to get;
σ^{2} = Σ (xi – x̅)2/n
σ^{2} = ⅙ (6.25+2.25+0.25+0.25+2.25+6.25)
σ^{2} = 2.917
Answer: Therefore the variance is σ2 = 2.917, and standard deviation,σ = √2.917 = 1.708

Example 2: Find the standard deviation of the average temperatures recorded over a fiveday period last winter:
18, 22, 19, 25, 12 (The mean = 19.2)Solution:
This time we will use a table for our calculations.
Mean = 19.2
To find the variance, we divide 51 = 494.8/4 = 23.7
Finally, we find the square root of this variance. √23.7 = 4.9
So the standard deviation for the temperatures recorded is 4.9; the variance is 23.7
Finally, we find the square root of this variance. √23.7 ≈ 4.9
Answer: So the standard deviation for the temperatures recorded is 4.9; the variance is 23.7.
FAQs on Variance and Standard Deviation
What Is the Difference Between Standard Deviation and Variance?
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).
How Do I Calculate the Variance?
The variance can be calculated as:
 Find the mean of the data set. Add all data values and divide by the sample size n.
 Find the squared difference from the mean for each data value. Subtract the mean from each data value and square the result.
 Find the sum of all the squared differences. ...
 Calculate the variance.
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.
Which Is Better Variance or Standard Deviation?
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 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.
Why Is Standard Deviation Used Over Variance?
Standard deviation and variance are closely related descriptive statistics, though the standard deviation is more commonly used because it is more intuitive with respect to units of measurement; the variance is reported in the squared values of units of measurement, whereas standard deviation is reported in the same units
Why Is Variance Important?
Variance is important for two main reasons: For use of Parametric statistical tests, as they are sensitive to variance. The variances of the samples to assess whether the populations they come from differ from each other.
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