Business Statistics: Summary Measures

Measures of Central Location

• Central location describes how numerical data cluster around a middle value.

• Arithmetic Mean: The primary measure of central location calculated by dividing the sum of observations by the count.

  • Sample Mean: $\bar{x} = \frac{\sum x_i}{n}$

  • Population Mean: $\mu = \frac{\sum x_i}{N}$

  • Sensitive to extreme values or outliers.

• Median: The middle value when observations are sorted in ascending order. Robust to outliers.

  • Odd sample size: Exact middle observation.

  • Even sample size: Average of the two middle observations.

• Mode: The observation that occurs most frequently.

  • Can be unimodal, bimodal, multimodal, or have no mode.

  • Sole meaningful central measure for categorical data.

• Weighted Mean: Account for observations with varying degrees of importance.

  • $\bar{x}_w = \sum (w_i x_i)$, where $\sum w_i = 1$

• Skewness and Central Measures:

  • Symmetric: $\text{Mean} = \text{Median} = \text{Mode}$

  • Positively Skewed (Right-tailed): $\text{Mean} > \text{Median}$

  • Negatively Skewed (Left-tailed): $\text{Mean} < \text{Median}$

Percentiles and Boxplots

• Percentiles: Divide data into 100 equal parts. The $p^{\text{th}}$ percentile has approximately $p\%$ of observations below it and $(100 - p)\%$ above it.

• Quartiles:

  • First Quartile ($Q_1$): 25th percentile

  • Second Quartile ($Q_2$): 50th percentile (Median)

  • Third Quartile ($Q_3$): 75th percentile

• Interquartile Range (IQR): The range of the middle $50\%$ of data.

  • $\text{IQR} = Q_3 - Q_1$

• Five-Number Summary: Minimum, $Q_1$, Median ($Q_2$), $Q_3$, and Maximum.

• Boxplot Structure:

• Box spans from $Q_1$ to $Q_3$ with a line at the median.

• Whiskers extend to the minimum and maximum values within $1.5 \times \text{IQR}$ from the box edges.

• Outliers: Values located further than $1.5 \times \text{IQR}$ from $Q_1$ or $Q_3$.

Measures of Dispersion

• Measures of dispersion evaluate the variability or spread of data.

• Range: Difference between the maximum and minimum values.

  • $\text{Range} = \text{Max} - \text{Min}$

• Mean Absolute Deviation (MAD): Average of absolute deviations from the mean.

  • Sample MAD: $\text{MAD} = \frac{\sum |x_i - \bar{x}|}{n}$

  • Population MAD: $\text{MAD} = \frac{\sum |x_i - \mu|}{N}$

• Variance: Average of squared deviations from the mean.

  • Sample Variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$

  • Population Variance: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$

• Standard Deviation: Positive square root of variance, returning units to the original scale.

  • Sample Standard Deviation: $s = \sqrt{s^2}$

  • Population Standard Deviation: $\sigma = \sqrt{\sigma^2}$

• Coefficient of Variation (CV): Unitless measure of relative risk/variability per unit of mean.

  • Sample CV: $\text{CV} = \frac{s}{\bar{x}}$

  • Population CV: $\text{CV} = \frac{\sigma}{\mu}$

Mean-Variance Analysis and the Sharpe Ratio

• Mean-Variance Analysis: Measures financial asset performance by evaluating reward (mean return) against risk (variance or standard deviation).

• Sharpe Ratio: Measures excess return per unit of risk relative to a risk-free rate ($R_f$).

  • $\text{Sharpe Ratio} = \frac{\bar{x} - R_f}{s}$

  • Higher values indicate superior risk-adjusted return performance.

Analysis of Relative Location

• Chebyshev's Theorem: For any dataset and $k > 1$, the proportion of values within $k$ standard deviations of the mean is at least $1 - \frac{1}{k^2}$.

  • $k = 2$: At least $75\%$ of data lie within $\bar{x} \pm 2s$

  • $k = 3$: At least $89\%$ of data lie within $\bar{x} \pm 3s$

• Empirical Rule: Applies to symmetric, bell-shaped distributions.

• Approximately $68\%$ of observations lie within $\bar{x} \pm 1s$

• Approximately $95\%$ of observations lie within $\bar{x} \pm 2s$

• Almost $100\%$ (roughly $99.7\%$) of observations lie within $\bar{x} \pm 3s$

  • z-Score: Standardized measure of relative position, representing distance from the mean in standard deviation units.

• Sample z-score: $z = \frac{x - \bar{x}}{s}$

• Population z-score: $z = \frac{x - \mu}{\sigma}$

• Values with $|z| > 3$ in bell-shaped distributions indicate potential outliers.

Measures of Association

• Covariance: Quantifies the direction of a linear relationship between two variables.

  • Sample Covariance: $s_{xy} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n - 1}$

  • Population Covariance: $\sigma_{xy} = \frac{\sum (x_i - \mu_x)(y_i - \mu_y)}{N}$

  • Sensitive to measurement units; cannot indicate relationship strength.

• Correlation Coefficient: Unit-free measure of both direction and strength of a linear relationship ($-1 \le r \le 1$).

  • Sample Correlation: $r_{xy} = \frac{s_{xy}}{s_x s_y}$

  • Population Correlation: $\rho_{xy} = \frac{\sigma_{xy}}{\sigma_x \sigma_y}$

  • $r = 1$: Perfect positive linear relationship

  • $r = 0$: No linear relationship

  • $r = -1$: Perfect negative linear relationship

Geometric Mean and Growth Rates

• Geometric Mean Return: Multiplicative average used for compounding multiperiod investment returns.

  • $G_R = \left(((1 + R_1) \times (1 + R_2) \times \dots \times (1 + R_n))\right)^{\frac{1}{n}} - 1$

• Compound Average Growth Rate: Multiplicative average growth rate across $n$ time periods or observations.

  • Using rates: $G_g = \left(((1 + g_1) \times (1 + g_2) \times \dots \times (1 + g_n))\right)^{\frac{1}{n}} - 1$

  • Using end values ($x_n$) and start values ($x_1$): $G_g = \left(\frac{x_n}{x_1}\right)^{\frac{1}{n - 1}} - 1$