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Mathematical Statistics I: Descriptive Statistics

This document provides a comprehensive introduction to descriptive statistics, covering fundamental definitions, methods for summarizing and presenting data, and measures of central location and variability. It is designed to equip students with the foundational knowledge required for understanding and analyzing data in the presence of variability and uncertainty.

Introduction to Descriptive Statistics

Statistics, as a field of study, involves the systematic process of collecting, summarizing, analyzing, and presenting data. The primary goal is to present data in a logical and coherent manner to facilitate informed decision-making about populations, especially when confronted with inherent variability and uncertainty. Statistics employs scientifically approved methods to ensure that information derived from selected objects accurately represents the broader population, thereby minimizing uncertainty and variability.

Definition of Key Terms

Understanding the core terminology is crucial for grasping statistical concepts:

Population: A well-defined collection of all objects of interest. Obtaining information from every object in a population yields the most accurate data. For example, "all adults in a country" constitutes a population.
Sample: A portion or subset of the population of interest. Samples are often used due to practical constraints such as time limitations, high costs associated with measuring every object, or the sheer size (potentially infinite) of populations. While sampling reduces costs and feasibility issues, it's important to note that larger samples generally incur higher costs. An example is "1000 adults selected for a survey in a country."
Parameter: A characteristic of a population. Parameters are actual values that describe the entire population. They can be numerical summaries or measures, such as the population average (mean, denoted by \mu), population proportion (\pi), or population variance (\sigma). For instance, "the average heights of all adults in a country" is a parameter.
Statistic: An estimation of a parameter, calculated from a sample. Statistics are numerical summaries or measures derived from a sample, used to estimate corresponding population parameters. Examples include the sample mean (\bar{x}), sample proportion (p), or sample variance (s).

Notation Convention: Greek letters (e.g., \mu, \pi, \sigma) are typically used to represent population parameters, while Roman letters (e.g., \bar{x}, p, s) denote sample statistics.

Variability and Uncertainty

Variability (Variation): Refers to the differences or fluctuations observed in data, which arise from various reasons or influences. These differences can occur even in repeated measurements and are often unavoidable, stemming from factors like individual differences, measurement processes, and environmental conditions. Variability is mostly a naturally occurring phenomenon.

Sources of Variation

Measurement error: Discrepancies in recorded values due to limitations in measurement tools, human error, or inconsistencies in data collection. For example, two individuals using different scales might report slightly different weights for the same object.
Inherent/natural random variation: Variation that naturally exists within populations or systems due to inherent randomness. An example is the varying heights of all 10-year-olds in a specific province.
Unidentified deterministic component(s): Systematic influences that affect observations but are not explicitly accounted for in a model. For instance, an unknown environmental factor affecting plant growth might introduce systematic variation that appears random but follows an underlying, unidentified pattern.
Sampling variation: Differences that arise because different samples, even when drawn from the same population, will naturally yield different statistics. For example, multiple samples of 1000 adults used to estimate average height will likely produce varying results.
Environmental/Contextual variation: Changes in external conditions that impact observations. Examples include temperature fluctuations affecting reaction times in a chemical experiment or economic conditions influencing consumer behavior in surveys.

Uncertainty: Characterized by a lack of precise knowledge or confidence regarding a value, process, or prediction. It reflects the inability to perfectly measure, predict, or model phenomena, even when variability is present.

Sources of Uncertainty

Model uncertainty: Arises when statistical models are simplifications of reality and fail to capture all relevant factors. No model can perfectly describe complex real-world problems.
Parameter uncertainty: Even if a model is assumed to be correct, the true values of its parameters (e.g., population mean) are often unknown and must be estimated, leading to uncertainty in these estimates.
Prediction uncertainty: The future is inherently uncertain, regardless of the quality of models and data. Predicting next year's average temperature, for example, involves uncertainty due to complex climate interactions.

Variables

A variable is an attribute of an object that can change over time for the same object or vary from one object to another. A random variable is an attribute whose value is not known until it is observed or realized.

Types of Variables

Random variables are broadly categorized into two main types:

Qualitative Variables (Categorical Data): These variables are non-numerical. Values assigned in measurements typically represent categories or attributes of the characteristic being measured. Examples include eye color, gender, age group, and satisfaction levels.
Quantitative Variables (Numerical Data): These variables are numerical. The values assigned are actual observations of the characteristic being measured. Examples include age, height, weight, time, temperature, and the number of children in a family.

Collecting Data

Data collection is a critical aspect of statistics, as the methods used directly influence the types of analyses that can be performed and the accuracy of parameter estimations. Most data dealt with in this context are assumed to have been collected from existing populations.

Sampling Methods

When working with samples, various sampling methods can be employed:

Simple Random Sampling: Every element in the population has an equal chance of being selected. This is the preferred method but often difficult to implement as it requires a complete list of all population elements. Computer-generated random numbers are frequently used.
Systematic Sampling: Easier than simple random sampling, this method involves selecting every k^th element from an ordered list of the population.
Convenience Sampling: The easiest method, but also the least reliable. It involves using readily available data or selecting the first individuals encountered by the surveyor.
Cluster Sampling: The population is divided into groups (clusters), often geographically. Clusters are randomly selected, and all elements within the selected clusters are included in the sample.
Stratified Sampling: The population is divided into groups called strata based on a specific characteristic (e.g., gender, age group), not geography. A sample is then drawn from each stratum using simple random, systematic, or convenience sampling.

Measurement Scales

Data are classified according to four scales of measurement, ordered from lowest to highest specificity:

Nominal Scale: The lowest scale, where only names are meaningful, and order does not matter. Data at this scale are qualitative (e.g., categories like "Car," "Bus," "Bicycle").
Ordinal Scale: The second lowest scale, where the order of names or categories matters, but the differences between values are not meaningful. Data at this scale are also qualitative (e.g., satisfaction levels: "low," "medium," "high").
Interval Scale: The third scale, where both the order of observations and meaningful differences between them matter. However, there is no true zero point, meaning ratios are not meaningful. Data at this scale are quantitative (e.g., temperature in Celsius or Fahrenheit).
Ratio Scale: The highest scale, which includes all properties of the interval scale but also has a true zero point, making ratios meaningful. Data at this scale are quantitative (e.g., height, weight, time).

Descriptive vs. Inferential Statistics

Descriptive Statistics: This branch of statistics focuses on organizing, summarizing, and effectively displaying information from a sample (or population) to describe observed patterns or behaviors. It involves computing statistics to estimate population parameters.
Statistical Inference (Inferential Statistics): This branch uses sample findings to test hypotheses about a population or draw conclusions about population characteristics. Probability is an essential tool for inferential statistics.

Figure 1 illustrates the relationship between Population, Sample, Probability, and Inferential Statistics. The population is the source from which a sample is drawn. Inferential statistics uses the sample data, guided by probability theory, to make conclusions about the population.

Summarising and Presenting Data

Given a dataset of n observations on a variable X, individual observations are denoted as X_1, X_2, X_3, \dots, X_n. The subscript indicates the position of the observation in the dataset, not its magnitude. For instance, X_1 is typically the first observation gathered.

There are two primary methods for summarizing data in Descriptive Statistics:

Graphical summary
Numerical summary

Graphical Summary

Graphical summaries are typically the first step in data analysis, providing an initial visual understanding of the general pattern, trend, and behavior of the characteristic of interest. A well-constructed graph should offer good insight into the characteristic, appropriately represent the data's location, dispersion, association, and tail behavior.

Graphical Methods by Data Type:

Qualitative Variable (Categorical Data): Bar charts, Pie charts.
Quantitative Discrete Variable (Quantitative Discrete Data): Dotplots (for small sample sizes), Stem-and-leaf plots.
Quantitative Continuous Variable (Quantitative Continuous Data): Stem-and-leaf plots, Histograms.

Pie Charts

Pie charts are used to summarize categorical data. They represent counts (frequencies) or proportions (relative frequencies) of different categories as "slices" of a circle. The sum of all proportions in a pie chart must equal 1. To ensure clear representation, it is recommended to limit the number of categories to 5-7, as too many can make the chart cluttered.

Example: Preferred Transportation Mode

A survey of 200 people on their preferred mode of transport yielded the data in Table 1:

Mode	Count
Car	80
Bus	50
Bicycle	30
Walking	20
Train	20

A pie chart for this data would visually represent these proportions: Car (40%), Bus (25%), Bicycle (15%), Walking (10%), and Train (10%). Each slice's size corresponds to its proportion of the total, with different colors distinguishing the modes.

Bar Charts

Bar charts also summarize categorical data, representing counts or proportions using distinct, separate bars for each category. The height of each bar corresponds to the frequency or relative frequency of that category.

Example: Preferred Transportation Mode (Bar Chart)

Using the same transportation preference data, a bar chart would show bars for Car (80), Bus (50), Bicycle (30), Walking (20), and Train (20) on the y-axis (Count) against the Mode of Transport on the x-axis. Each bar would be distinct and separated.

Clustered Bar Graphs

These are an extension of bar charts that allow for sub-grouping data by another category. For example, if the transportation preference data were further grouped by age group (Young vs. Old), a clustered bar graph would display multiple bars for each mode, with each cluster representing an age group.

Example: Preferred Transportation Mode by Age Group

Table 2 shows the counts for preferred transportation mode broken down by age group:

Mode	Age Group	Count
Car	Young	50
Car	Old	30
Bus	Young	30
Bus	Old	20
Bicycle	Young	20
Bicycle	Old	10
Walking	Young	15
Walking	Old	5
Train	Young	10
Train	Old	10

The clustered bar chart would display pairs of bars (one for "Old," one for "Young") for each mode (Car, Bus, Bicycle, Walking, Train), allowing for a visual comparison of preferences across age groups.

Dotplots

Dotplots are used to summarize quantitative discrete data, especially for small datasets or when there are few distinct observations. Observations are grouped, and each occurrence is represented by a dot above its corresponding value on a horizontal measurement scale. When an observation occurs multiple times, the dots are stacked vertically, forming a frequency distribution.

Example: Number of Siblings Reported by 20 Students

Table 3 presents the frequency of siblings reported by 20 first-year students:

Siblings	Frequency
1	1
2	2
3	6
4	3
5	2
6	3
7	1
8	2

A dotplot for this data would show dots stacked above the numbers 1 through 8 on the x-axis, with the height of the stack indicating the frequency. For instance, there would be 6 dots above '3' and 1 dot above '1'.

Stem-and-leaf Plots

Stem-and-leaf plots summarize quantitative data, which can be discrete or continuous, and are ideal for raw data with at least two digits. They separate each observation into a "stem" (leading digits) and a "leaf" (trailing digit).

Steps for Construction:

Select one or more leading digits for the stem values; trailing digits become leaves.
List possible stem values in a vertical column.
Record the leaf for every observation beside its corresponding stem value.
Order the leaves from smallest to largest on each line.
Indicate the units for stems and leaves (legend) somewhere in the display.

Example: Test Completion Times

For a sample of 120 students' test completion times (in minutes), a stem-and-leaf plot would reveal the distribution. For instance, if the data includes times like 27, 35, 37, 35, 31, etc., the stem '2' might have leaves '0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 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