Fundamental Statistical Concepts and Data Measurement

Statistical Attributes and Variables

Attributes are qualitative characters. The different categories of an attribute are called modalities. These conditions should be exhaustive and mutually exclusive.

Variables are quantitative traits. Variables can also be expressed as attributes. The measurement of a variable results in a value, which can be discrete or continuous.

Defining Population and Samples

• Population: A collective of individuals or elements that have certain common characteristics and are the object of observation and statistical study.
• Subpopulation: A subset of elements of the population that has a certain characteristic which differentiates them from others.
• Sample: A subset of elements of the population that aims to represent the entire population. It is essentially a small-scale population and can be random or non-random.
• Characters: Properties, features, or qualities presented by the elements of a population.

Censuses vs. Polls

Censuses are statistical research methods where all elements of the population are observed (comprehensive observation). They study structural and static population characters and are conducted at intervals of 5 to 10 years.

Polls are statistical research methods in which a sample is used (partial observation). They study more dynamic phenomena and are conducted more frequently than censuses, such as quarterly or yearly.

Understanding Quantiles

Quantiles are variable values determined by ordering observations in increasing order. They use the same unit of measurement as the variable. Frequently used quantiles include:

• Quartiles
• Deciles
• Percentiles

Measures of Dispersion and Variation

Variance and Coefficient of Variation

Variance is the arithmetic mean of the squared deviations of observed values from the arithmetic mean of the distribution. It is a measure of dispersion expressed in the unit of measurement of the variable, squared.

The Coefficient of Variation (CV) represents the number of times that the standard deviation contains the mean. A higher rate indicates the mean is less representative. The CV is dimensionless.

• Minimum Value: 0 (when the standard deviation is zero).
• Maximum Value: Infinite (in which case the coefficient loses significance).
• Interpretation: If CV < 0.5, the average is considered representative. If CV > 1, the average is not representative. If CV > 5, its representativeness must be questioned.

Standardized Variables

A standardized (typified) variable is a transformation of the variable x. This involves a shift of origin (subtracting the average of x) and a change of scale (multiplying by 1/s, where s is the standard deviation). The new variable is thus standardized.