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:

1. Variance (σ2sigma^2 or s2s^2) is the average of the squared differences from the mean.
2. Standard Deviation (σsigma or ss) is the square root of the variance.

Mathematical Relationship:

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

Key Differences:

1. 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.

2. 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}{2, 4, 6, 8}

1. Mean (μmu) = 2+4+6+84=5frac{2+4+6+8}{4} = 5
2. Variance (σ2sigma^2) = (2−5)2+(4−5)2+(6−5)2+(8−5)24=9+1+1+94=5frac{(2-5)^2 + (4-5)^2 + (6-5)^2 + (8-5)^2}{4} = frac{9 + 1 + 1 + 9}{4} = 5
3. Standard Deviation (σsigma) = 5≈2.24sqrt{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.

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

what is Variance?

What is standard deviation?

relation between standard deviation and coefficient of variance