Descriptive Statistics | Foundations of Statistics

Descriptive statistics organize, summarize, and display data. They help us understand what a sample contains, where its center lies, how widely it varies, whether its distribution is symmetric, and whether unusual observations are present.

Start with the data type#

Data type	Typical summaries	Typical displays
Categorical	Counts and proportions	Frequency table, bar chart
Numerical	Mean and SD, or median and IQR	Histogram, box plot, density plot

Central tendency#

The mode is the most frequent value and may be used for categorical or numerical data. The median is the middle ordered value and is resistant to extreme observations. The arithmetic mean uses every observation and is highly useful for inference, but is more sensitive to outliers.

Median position for odd n

\frac{n+1}{2}

Median for even n

\operatorname{Median}=\frac{x_{(n/2)}+x_{(n/2+1)}}2

Arithmetic mean

\bar x=\frac1n\sum_{i=1}^{n}x_i

Trimmed and geometric means#

A trimmed mean removes a fixed proportion from both tails before averaging, providing a compromise between the mean and median. The geometric mean is appropriate for positive multiplicative quantities such as growth factors, ratios, and some log-normal measurements.

Product form

G=\left(\prod_{i=1}^{n}x_i\right)^{1/n},\quad x_i>0

Log form

G=\exp\!\left(\frac1n\sum_{i=1}^{n}\ln x_i\right)

A logarithm compresses large positive values and may make a right-skewed distribution more symmetric. Ordinary log transformation and the geometric mean are not directly defined when observations include zero or negative values.

Variability: how different are the observations?#

Range

\text{Range}=x_{\max}-x_{\min}

Interquartile range

\operatorname{IQR}=Q_3-Q_1

The range uses only the extremes and is sensitive to outliers. The IQR describes the middle 50% of observations and pairs naturally with the median.

Variance and standard deviation#

Population

\sigma^2=\frac1N\sum_{i=1}^{N}(x_i-\mu)^2,\qquad\sigma=\sqrt{\sigma^2}

Sample

s^2=\frac1{n-1}\sum_{i=1}^{n}(x_i-\bar x)^2,\qquad s=\sqrt{s^2}

Variance averages squared distances from the mean. The standard deviation takes the square root and therefore returns to the original measurement unit. The n−1 denominator in sample variance reflects degrees of freedom and unbiased estimation.

Coefficient of variation#

\operatorname{CV}=\frac{s}{\bar x}\times100\%

The CV expresses standard deviation relative to the mean. It is useful for positive ratio-scale measurements with meaningful zeroes, but is unstable or difficult to interpret when the mean is near zero or negative.

Distribution shape: skewness and kurtosis#

Skewness describes asymmetry. Its sign follows the longer tail: a long left tail gives negative skewness, while a long right tail gives positive skewness.

\gamma_1=E\!\left[\left(\frac{X-\mu}{\sigma}\right)^3\right]=\frac{\mu_3}{\sigma^3}

The third power preserves direction, allowing the sign to distinguish left from right skew.

Kurtosis is better understood as sensitivity to tail weight and extreme observations than simply as the sharpness of a peak.

\beta_2=E\!\left[\left(\frac{X-\mu}{\sigma}\right)^4\right],\qquad\gamma_2=\beta_2-3

Normal-distribution kurtosis β₂ is 3 and excess kurtosis γ₂ is 0. Software may report either definition, so always verify the convention.

A complete descriptive report#

Identify the data type and measurement scale
Report counts and proportions for categorical variables
Report both center and variability for numerical variables
Inspect plots and outliers before choosing mean/SD or median/IQR
Use shape statistics and graphs when helpful, without judging a distribution from one number alone