Variance vs Standard Deviation

Variance and standard deviation are statistical measurements that indicate how data is dispersed from the mean (average) of a dataset. Although both are measures of variability, they differ in many ways.

Measure of Variability

• Variance is a numerical value that indicates how variable the observations are. It is the average of the squared differences between each data point and the mean.
• Standard deviation explains variability by measuring the dispersion of observations within the dataset. It is the square root of the variance. 

For example, in a dataset A = {5, 10, 15, 20, 25}, the variance is 50 and the standard deviation is approximately 7.07

Thus, a higher standard deviation always means a higher variance.

Symbol

Mathematically,

• The variance is typically represented by σ², s², or Var(X)
• The standard deviation is represented by σ or s. 

How to Find

Here are the formulas for calculating standard deviation and variance.

For Calculating Variance

• Population Variance: ${\sigma ^{2}=\dfrac{\sum \left( x_{i}-\mu \right) ^{2}}{N}}$
• Sample Variance: ${s^{2}=\dfrac{\sum \left( x_{i}-\overline{x}\right) ^{2}}{n-1}}$

For Calculating the Standard Deviation

• Population Standard Deviation: ${\sigma =\sqrt{\dfrac{\sum \left( x_{i}-\mu \right) ^{2}}{N}}}$
• Sample Standard Deviation: ${s=\sqrt{\dfrac{\sum \left( x_{i}-\overline{x}\right) ^{2}}{n-1}}}$

We can also get standard deviation by square rooting the variance.

Here,

• N = Total number of data points in the population
• n = Total number of data points in the sample
• x_i = Individual data point
• μ or ${\overline{x}}$ = Mean

Units

• The variance is expressed in squared units of the dataset.
• The standard deviation is expressed in the same units as the original data. 

Applications

• The variance is used in more complex statistical analyses, such as regression, ANOVA, and modeling.
• The standard deviation provides a detailed view of data spread that is used in quality control, finance, and analyzing variability in datasets. 

Summary

Thus, the main differences between standard deviation and variance are listed below: 

Basis	Variance	Standard Deviation
1. Measure of Variability	A numerical value indicating how variable the observations are from the mean	Measures dispersion of observations within the dataset.
2. Mathematical Relation	The average of squared deviations from the mean	The square root of the average squared deviations from the mean
3. Symbols	σ2, s2, or Var(X)	σ or s
4. Formula	Population Variance: ${\sigma ^{2}=\dfrac{\sum \left( x_{i}-\mu \right) ^{2}}{N}}$ Sample Variance: ${s^{2}=\dfrac{\sum \left( x_{i}-\overline{x}\right) ^{2}}{n-1}}$	Population Standard Deviation: ${\sigma =\sqrt{\dfrac{\sum \left( x_{i}-\mu \right) ^{2}}{N}}}$ Sample Standard Deviation: ${s=\sqrt{\dfrac{\sum \left( x_{i}-\overline{x}\right) ^{2}}{n-1}}}$
5. Units	Squared units of the dataset	Same as the dataset
6. Applications	Used in regression, ANOVA, and statistical modeling	Used in descriptive statistics and real-world scenarios

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