Key Statistical Concepts

Key Statistical Concepts

• Statistics involves gathering, ordering, and analyzing information to draw conclusions and interpretations.
• Statistical methods are used to analyze collected data for meaningful conclusions.
• A population includes the entire collection of individuals, objects, or items being studied, and it can be finite or infinite.
• The total number of elements in a population is indicated by N.
• Shoes manufactured in a factory on a given day exemplify a finite population.
• Outcomes from flipping a coin repeatedly represent an infinite population.
• A population parameter is a constant value, usually unknown, that describes a measurable aspect of a population, commonly denoted with Greek letters.
• A sample represents a subset of the population that is of interest.
• Samples are used to gather information when it may not be possible or feasible to study the entire population.
• The total number of elements in a sample is given by n.
• A statistic is a number calculated from sample data, describing a measurable sample aspect, commonly represented with Roman letters.
• Raw data refers to unprocessed information, known as the source or primary data, and is initially formed as a dataset.
• Datasets are presented as a matrix with rows (observations) and columns (variables); a sample of n observations and p variables will yield a dataset with n rows and p columns.

Descriptive vs. Inferential Statistics

• Descriptive statistics involves methods to organize and describe the information collected from a sample.
• Inferential statistics involves methods used to make generalizations, predictions, or estimates about the population, using the sampled data.

Random Variables and Data Measurement

• A random variable is an element characteristic of a population (or sample) where observed values vary.
• Variables are denoted by capital letters (e.g., X, Y, Z), while the actual values the random variables assume are denoted by lowercase letters (e.g., x, y, z).
• For example, if X = the height of boys in metres, and three boys are selected at random (n=3) with heights of 1.40m, 1.37m, and 1.41m, then the realisations of the random variable =X are denoted as values of x1 =1.40, x2 = 1.37 and x3 = 1.41.
• Data refers to the collection of all variables measured.
• The unit of measurement of a variable is the standard unit used to express a quantity.
• If time measurements are in whole seconds, the unit of measurement is one second (unit = 1).
• If time measurements are recorded to the tenth of a second, the unit of measurement is 0.1 second (unit = 0.1).

Sigma Notation

• Sigma notation is the standard notation used in mathematics to represent summation.
• It is a compact way to write long sums, denoted by the Greek capital letter sigma, .
• For a random variable X with n observations x1, x2, ..., xn, the sum of all n values is notated as ∑xᵢ (from i=1 to n) or simply ∑x.

Categorical and Numerical Variables

• Variable type can be categorical or numerical; this distinction determines the appropriate analyses performed.
• Categorical variables, also known as qualitative variables, classify data based on characteristics.
• Eye color can be classified as blue, brown, green or grey.
• Categorical variables are often recorded as numerical but have no numerical meaning; they act as labels for categories, such as 1 = Male and 2 = Female.
• Numerical variables, also known as quantitative variables, are measured as numbers.
• Person's height in centimeters.
• Arithmetic operations are directly applicable to these variables.
• Numerical variables are classified further as Discrete (finite number of values obtained by counting; e.g., number of children in a household) and Continuous (infinite number of values obtained by measuring along a real line; e.g., age).

Measurement Scales

• Variables are classified by measurement scale: nominal, ordinal, interval, and ratio.
• Nominal Scale: Used for categorical variables with unordered categories.
• Eye color has no logical ranking.
• Ordinal Scale: Used for categorical variables with ranked categories.
• A person's age classified as young, middle-aged, or old.
• Anxiety ratings on a scale of 1 to 5.
• Interval Scale: Used for numerical variables where values are ordered, and differences are meaningful, but there is no true zero.
• Temperature in Celsius.
• A temperature of 0°C does not indicate the absence of temperature, and while differences are meaningful, ratios are not.
• Ratio Scale: Used for numerical variables with ordered values, meaningful differences, and a true zero point.
• The amount of money in a bank account.
• R0 indicates an absence of money, making ratios meaningful.

Frequency Data

• Frequency Data is raw data in an aggregated format, counts the occurrences of each value/range of values in a dataset.
• The count of the number of times shows how data is distributed across the scale, providing a sampled information overview.
• Univariate frequency data is the counts of a single variable.
• Bivariate frequency data is the counts of the combination of two variables.

Learn about gathering, ordering, and analyzing data to draw conclusions and interpret. Understand population vs sample. Explore the difference between finite and infinite populations.