Variance

• Variance (\(Var[X]\)) is a measure of the spread of a distribution relative to its expected value (Dowdy et al. 2004)

  • is the measure of variability in a single variable;
  • is the average squared distance from the mean;
  • is in squared units (has no intuitive meaning);

• Standard Deviation (SD) is the squared root of the variance:

  • \(\text{SD} = \sqrt{\text{Variance}}\)
  • The standard deviation is
    • in the same units of measurement as the measured variable

Variance: population vs. sample

Population variance, \(\sigma^2\)

\[ \sigma^2 = \frac{\sum_{i=1}^N [x_i - \mu]^2 }{N}, \]

where \(\mu\) is the population mean:

\(\mu = {\sum_{i=1}^N x_i }/{N}\)

Only used with the population

Standard Deviation:

\(\sigma = \sqrt{\sigma^2}\)

Sample variance, \(s^2\)

\[ s^2 = \frac{\sum_{i=1}^N [x_i - \bar{x}]^2 }{N-1}, \]

where \(\bar{x}\) is the sample mean:

\(\bar{x} = {\sum_{i=1}^N x_i }/{N}\)

Most commonly used in statistics

Sample Standard Deviation:

\(s = \sqrt{s^2}\)

Standard Deviation

represents the “typical” deviation of observations from the mean.

If the distribution is normal:

• 68.3% obs. are in +-1 SD;
• 95.4% obs. are in +-2 SD;
• 99.7% obs. are in +-3 SD;

The standard deviation is in the same units of measurement as the measured variable.

Not every distribution is normal, but SD can be the same

Note

A good description of distribution should include

• Central tendency;

• SD (Variance);

• Description of the shape;

  • modes

  • symmetry / skew

Diez, David, Mine Cetinkaya-Rundel, and Christopher D Barr. 2022. OpenIntro Statistics: Fourth Edition. OpenIntro. https://www.openintro.org/book/os/.

Dowdy, Shirley, Stanley Weardon, and Daniel Chilko. 2004. Statistics for Research. John Wiley & Sons, Inc. https://doi.org/10.1002/0471477435.