Unlocking Statistics: Key Concepts from Samples to Skewness Explained

Sample and Population

In mathematical statistics, the entire set of objects under study is called the population, usually represented by a symbol such as \(N\). Each individual unit in the population is an element. A subset drawn from the population is a sample, denoted by \(X\) with sample size \(n\). The observed value for each element is a sample observation, a fixed numeric value that changes with each draw. Statistics aim to infer the population distribution from these sample observations.

Frequency Tables and Histograms

A set of sample observations may appear chaotic; constructing a frequency table and histogram provides an initial organization and visual description. The data range is divided into intervals, counting the number of observations in each interval (frequency). The frequency table lists these counts, while a histogram plots intervals on the horizontal axis and frequency or relative frequency on the vertical axis as a step‑shaped graph.

Descriptive Statistics

Samples are the starting point for analysis, but we usually work with statistics—functions of the sample that do not involve unknown parameters. Common descriptive statistics include measures of location, dispersion, shape, and association.

Location Measures: Mean and Median

The arithmetic mean (average) represents the central location of the data, denoted by \(\bar{x}\). The median is the middle value after sorting the data; for an even number of observations it is the average of the two middle values.

Dispersion Measures: Standard Deviation, Variance, and Range

The standard deviation measures how far each data point deviates from the mean; variance is the square of the standard deviation. The range is the difference between the maximum and minimum values.

Shape Measures: Skewness and Kurtosis

Skewness reflects the symmetry of the distribution: positive (right‑skewed) when the right tail is longer, negative (left‑skewed) when the left tail is longer, and near zero for symmetric distributions. Kurtosis describes the peakedness; a normal distribution has kurtosis 3, and values greater than 3 indicate heavy tails.

Covariance and Correlation Coefficient

Covariance quantifies the joint variability of two variables, while the correlation coefficient standardizes covariance to a dimensionless measure ranging from –1 to 1.