Variance and standard deviation these two terms comes from statistics. there is an amazing relation between variance and standard deviation. Hence, the relation between variance and standard deviation is standard deviation is always equal to the square root of variance for a given set of data. so the formula of relation between variance and standard deviation is σ = √ 1/n ✕ ∑ (xi – x)2.

Relationship between variance and standard deviation

The relationship between variance and standard deviation is straightforward, as both are measures of the spread or dispersion of data in a dataset. They are closely related mathematically:

Variance ( $sigma^2$ or $s^2$ ) is the average of the squared differences from the mean.
Standard Deviation ( $sigma$ or $s$ ) is the square root of the variance.

Mathematical Relationship:

$text{Variance} = (text{Standard Deviation})^2$
$text{Standard Deviation} = sqrt{text{Variance}}$

Key Differences:

Units:
- Variance is expressed in squared units of the data (e.g., if the data is in meters, the variance is in square meters).
- Standard deviation is expressed in the same units as the data (e.g., meters), making it more interpretable.
Interpretability:
- Standard deviation is more commonly used because it is directly comparable to the data values.
- Variance is helpful for theoretical purposes, such as in statistical modeling or calculations involving probability distributions.

Example:

Consider the dataset: ${2, 4, 6, 8}$

Mean ( $mu$ ) = $frac{2+4+6+8}{4} = 5$
Variance ( $sigma^2$ ) = $frac{(2-5)^2 + (4-5)^2 + (6-5)^2 + (8-5)^2}{4} = frac{9 + 1 + 1 + 9}{4} = 5$
Standard Deviation ( $sigma$ ) = $sqrt{5} approx 2.24$

This shows how standard deviation is derived from variance and is easier to interpret in the context of the original data.

basically relation between mean variance and standard deviation give a unique formula that is σ = √ variance. where sigma is standard deviation. for example,

Relation b/w variance and standard deviation for a sample data set

find the mean, variance and standard deviation for the following data: 5, 9, 8, 12, 6, 10, 6, 8.

Hence, n = 8.

Mean = x = 1/8 (5+9+8+12+6+10+6+8) = 64/8 = 8.

the value of (xi – x) are,

-3, 1, 0, 4, -2, 2, -2, 0.

therefore, ∑ (xi – x)2 = (9+1+0+16+4+4+4+0) = 38.

variance (σ2) = 1/n ✕ ∑ (xi – x)2 = 38/8 = 19/4 = 4.75

and standard deviation = σ = √ 4.75 = 2.17.

To understand the mathematical and theoretical relation between variance and standard deviation, we have to understand the meaning of Variance and Standard deviation.

Before explain the terms variance and standard deviation we should know where this terms works. so variance and standard deviation has many works. it helps to understand the exact value in calculation of average numbers. for example suppose we have a group of people having different height then standard deviation helps us to know the exact average value of the height. this will clear in explanation of standard deviation.

what is Variance?

Variance is defined as the squares of the deviations from the mean is called the variance. it is denoted by σ2 read as sigma square. the formula of variance for n observations x1, x2, x3, ……….. xn is given by σ2 = 1/n ✕ ∑ (xi – x)2.

please note that for finding the variance we have to calculate mean.

What is standard deviation?

Standard deviation is defined as the positive square root of variance and it is denoted by σ. the formula for standard deviation is given by σ = √ 1/n ✕ ∑ (xi – x)2.

relation between standard deviation and coefficient of variance

standard deviation is defined as the positive square root of variance. on the other hand coefficient of variance is defined as the mean and standard deviation of the given data be x and σ respectively. then, the coefficient of variation is given as,

Hence, by this explanation we got the relation b/w standard deviation and coefficient of variance. so the relationship is coefficient of variance is equal to the standard deviation multiplied and divided by 100 and mean x. 100 as numerator and mean as denominator.