Univariate Descriptive Statistics

Term Formula Definition
Expected Value Probability-weighted long-run average value of an RV over many repeated trials or occurrences.

- For Discrete --> possible outcomes * probabilities
- For Continuous --> integral over ran
Range \(R = \text{max}(x) - \text{min}(x)\) The difference between the largest and smallest values in the dataset.
Variance (Sample) \(s^2 = \frac{\Sigma (x_i - \bar{x})^2}{n-1}\) The average of the squared differences from the sample mean, divided by \(n-1\).
Variance (Population) \(\sigma^2 = \frac{\Sigma (x_i - \mu)^2}{N}\) The average of the squared differences from the population mean, divided by \(N\).
Standard Deviation (Sample) \(s = \sqrt{\frac{\Sigma (x_i - \bar{x})^2}{n-1}}\) The square root of the sample variance.
Standard Deviation (Population) \(\sigma = \sqrt{\frac{\Sigma (x_i - \mu)^2}{N}}\) The square root of the population variance.
Skewness A measure of the asymmetry of a probability distribution.
Kurtosis A measure of how much mass is contained in the tails of a probability distribution. (3 = normal distribution, anything past 3 is called leptokurtic, or heavy-tailed.)
Moments Shown below

Sam

Calculating Standard Deviation: Applies to both discrete and continuous.

  1. Find the mean:

    • Population: \(\mu = \frac{\sum x_i}{N}\)

    • Sample: \(\bar{x} = \frac{\sum x_i}{n}\)

  2. Compute squared differences: \((x_i - \mu)^2 \text{ (population)} \quad \text{or} \quad (x_i - \bar{x})^2 \text{ (sample)}\)

  3. Sum the squared differences: \(\sum (x_i - \mu)^2 \text{ (population)} \quad \text{or} \quad \sum (x_i - \bar{x})^2 \text{ (sample)}\)

  4. Divide:

    • Population variance: \(\frac{\sum (x_i - \mu)^2}{N}\)

    • Sample variance: \(\frac{\sum (x_i - \bar{x})^2}{n-1}\)

\(n-1\) in sample denominator because \(\bar{x}\) consumes 1 degree of freedom. (called "Bessel's correction")

  1. Take the square root for standard deviation.

Sam

Central Moments

Moment Number Moment Name Moment Equation Moment Explanation
1 Mean (First Moment) \(E(Y - \mu) = 0\) Measures the average difference from the mean; always zero because positive and negative differences cancel out.
2 Variance (Second Moment) \(E((Y - \mu)^2)\) Measures the spread of the data by squaring differences from the mean to make them all positive.
3 Skewness (Third Moment) \(E((Y - \mu)^3)\) Captures the asymmetry of the data; positive skewness means more data above the mean, negative means below.
4 Kurtosis (Fourth Moment) \(E((Y - \mu)^4)\) Measures the "peakedness" of the data; higher values emphasize extreme differences (tails of the distribution).