Maths vs Statistics: Concepts & Variables

• Maths and statistics are related but distinct disciplines
• Mathematics is concerned with abstract structures and relationships, while statistics is focused on the collection, analysis, interpretation, and presentation of data
• Statistics uses mathematical tools, but it is also concerned with the practical application of these tools to real-world problems
• Probability theory is the branch of mathematics that provides the foundation for statistical inference

Basic Statistical Concepts

• Population: The entire group of individuals, objects, or events of interest in a study
• Sample: A subset of the population that is selected for study
• Variable: A characteristic that can take on different values
• Data: The values of the variable that are collected from the sample
• Descriptive statistics: Methods for summarizing and presenting data
• Inferential statistics: Methods for drawing conclusions about a population based on a sample

Types of Variables

• Categorical (Qualitative): Variables that represent categories or labels
• Nominal: Categories have no inherent order e.g., colors, types of fruit
• Ordinal: Categories have a meaningful order e.g., education level (high school, bachelor's, master's), satisfaction ratings (very dissatisfied, dissatisfied, neutral, satisfied, very satisfied)
• Numerical (Quantitative): Variables that represent numerical values
• Discrete: Variables that can only take on a finite number of values or a countable number of values e.g., number of children, number of cars
• Continuous: Variables that can take on any value within a given range e.g., height, temperature

Descriptive Statistics

• Measures of Central Tendency:
• Mean: The average of a set of numbers, calculated by summing all the values and dividing by the number of values
• Median: The middle value in a sorted set of numbers
• Mode: The value that appears most frequently in a set of numbers
• Measures of Dispersion (Variability):
• Range: The difference between the maximum and minimum values in a set of numbers
• Variance: A measure of how spread out the data is from the mean; it is the average of the squared differences from the mean
• Standard Deviation: The square root of the variance; it measures the typical distance of data points from the mean
• Other Descriptive Statistics:
• Percentiles: Values that divide the data into 100 equal parts e.g., the 25th percentile is the value below which 25% of the data falls
• Quartiles: Values that divide the data into four equal parts; the 25th percentile is the first quartile (Q1), the 50th percentile is the second quartile (Q2, also the median), and the 75th percentile is the third quartile (Q3)

Probability

• Probability is a measure of the likelihood that an event will occur
• It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty
• Basic Concepts:
• Experiment: A process that results in an outcome
• Sample Space: The set of all possible outcomes of an experiment
• Event: A subset of the sample space
• Types of Probability:
• Classical Probability: Assumes all outcomes in the sample space are equally likely; the probability of an event is the number of outcomes in the event divided by the total number of outcomes in the sample space
• Empirical Probability: Based on observed data; the probability of an event is the number of times the event occurs divided by the total number of observations
• Subjective Probability: Based on personal beliefs or opinions
• Probability Rules:
• Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)
• Multiplication Rule: P(A and B) = P(A) * P(B|A), where P(B|A) is the conditional probability of B given A
• Complement Rule: P(A') = 1 - P(A), where A' is the complement of A
• Conditional Probability: The probability of an event A, given that another event B has already occurred, denoted by P(A|B) = P(A and B) / P(B)

Probability Distributions

• A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can assume
• Discrete Probability Distributions:
• Bernoulli Distribution: Represents the probability of success or failure of a single trial
• Binomial Distribution: Represents the probability of obtaining a certain number of successes in a fixed number of independent trials
• Poisson Distribution: Represents the probability of a certain number of events occurring in a fixed interval of time or space
• Continuous Probability Distributions:
• Normal Distribution: A symmetric, bell-shaped distribution characterized by its mean and standard deviation; many natural phenomena follow a normal distribution
• Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1
• Exponential Distribution: Represents the time until an event occurs

Inferential Statistics

• Estimation:
• Point Estimate: A single value that is used to estimate a population parameter
• Confidence Interval: A range of values that is likely to contain the population parameter with a certain level of confidence
• Hypothesis Testing:
• Null Hypothesis: A statement about the population parameter that is assumed to be true
• Alternative Hypothesis: A statement that contradicts the null hypothesis
• Test Statistic: A value calculated from the sample data that is used to determine whether to reject the null hypothesis
• P-value: The probability of obtaining a test statistic as extreme as or more extreme than the one observed, assuming the null hypothesis is true
• Significance Level (alpha): The probability of rejecting the null hypothesis when it is actually true
• Decision Rule: If the p-value is less than or equal to the significance level, reject the null hypothesis

Common Statistical Tests

• t-tests: Used to compare the means of two groups
• One-sample t-test: Compares the mean of a single sample to a known value
• Independent samples t-test: Compares the means of two independent groups
• Paired samples t-test: Compares the means of two related groups
• Analysis of Variance (ANOVA): Used to compare the means of three or more groups
• Chi-Square Tests: Used to analyze categorical data
• Chi-square test of independence: Tests whether two categorical variables are independent
• Chi-square goodness-of-fit test: Tests whether a sample distribution fits a hypothesized distribution
• Regression Analysis: Used to model the relationship between two or more variables
• Linear Regression: Models the relationship between a dependent variable and one or more independent variables using a linear equation

Correlation

• Correlation measures the strength and direction of the linear relationship between two variables
• Values range from -1 to +1
• +1 indicates a perfect positive correlation (as one variable increases, the other increases)
• -1 indicates a perfect negative correlation (as one variable increases, the other decreases)
• 0 indicates no linear correlation
• Common correlation coefficients:
• Pearson correlation: Measures the linear relationship between two continuous variables
• Spearman correlation: Measures the monotonic relationship between two variables, regardless of whether the relationship is linear

Sampling Methods

• Simple Random Sampling: Each member of the population has an equal chance of being selected
• Stratified Sampling: The population is divided into subgroups (strata) based on shared characteristics, and a random sample is taken from each stratum
• Cluster Sampling: The population is divided into clusters, and a random sample of clusters is selected; all members of the selected clusters are included in the sample
• Systematic Sampling: Members of the population are selected at regular intervals

Common Statistical Software

• R
• Python (with libraries like NumPy, SciPy, Pandas, and Statsmodels)
• SAS
• SPSS
• Excel