Data Types, Probability, and Collection Methods

• Data is a collection of facts, figures, objects, symbols, and events gathered from different sources
• Data is organized and structured to provide useful information
• Probability quantifies the likelihood of an event occurring
• Probability is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty

Types of Data

• Qualitative data represents characteristics or attributes
• Qualitative data is non-numerical and describes qualities
• Examples of qualitative data include colors, textures, smells, tastes, appearance, beauty, etc.
• Quantitative data is numerical and represents measurable quantities
• Quantitative data can be discrete or continuous
• Discrete data is countable and has distinct values (e.g., number of students in a class)
• Continuous data can take any value within a range (e.g., height, temperature)

Data Collection Methods

• Surveys involve collecting data from a sample of individuals through questionnaires or interviews
• Experiments involve manipulating variables to observe their effect on other variables
• Observations involve recording data without manipulating any variables
• Simulations involve creating models to mimic real-world processes and generate data
• Data can also be collected from existing sources such as databases, websites, and public records

Data Organization and Representation

• Data can be organized using tables, which present data in rows and columns
• Charts and graphs are visual representations of data, making it easier to understand patterns and trends
• Common types of charts and graphs include bar charts, pie charts, line graphs, scatter plots, and histograms
• Frequency distribution tables summarize the number of times each value or range of values occurs in a dataset

Measures of Central Tendency

• Mean is the average of a set of numbers (sum of values divided by the number of values)
• Median is the middle value in a sorted set of numbers
• Mode is the value that appears most frequently in a set of numbers
• These measures describe the center or typical value of a dataset

Measures of Dispersion

• Range is the difference between the maximum and minimum values in a dataset
• Variance measures the average squared deviation from the mean
• Standard deviation is the square root of the variance and measures the spread of data around the mean
• These measures describe the variability or spread of a dataset

Probability Concepts

• An event is a specific outcome or set of outcomes in a random experiment
• The sample space is the set of all possible outcomes of a random experiment
• Probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes
• Probability is always between 0 and 1

Types of Probability

• Empirical probability is based on observed data from experiments or real-world observations
• Theoretical probability is based on logical reasoning and assumptions about the fairness of the experiment
• Subjective probability is based on personal beliefs or judgments

Basic Probability Rules

• The probability of an event not occurring is 1 minus the probability of the event occurring
• If two events are mutually exclusive (cannot occur at the same time), the probability of either event occurring is the sum of their individual probabilities
• If two events are independent (the outcome of one does not affect the outcome of the other), the probability of both events occurring is the product of their individual probabilities

Conditional Probability

• Conditional probability is the probability of an event occurring given that another event has already occurred
• The conditional probability of event A given event B is denoted as P(A|B)
• P(A|B) = P(A and B) / P(B), where P(B) > 0

Probability Distributions

• A probability distribution describes the probabilities of all possible outcomes of a random variable
• Discrete probability distributions are used for discrete random variables (e.g., binomial distribution, Poisson distribution)
• Continuous probability distributions are used for continuous random variables (e.g., normal distribution, exponential distribution)

Statistical Inference

• Statistical inference involves drawing conclusions about a population based on a sample of data
• Estimation involves using sample data to estimate population parameters (e.g., mean, proportion)
• Hypothesis testing involves using sample data to test claims about population parameters

Sampling Methods

• Random sampling involves selecting a sample from a population in such a way that each member of the population has an equal chance of being selected
• Stratified sampling involves dividing the population into subgroups (strata) and then selecting a random sample from each stratum
• Cluster sampling involves dividing the population into clusters and then randomly selecting some of the clusters to sample
• Convenience sampling involves selecting a sample based on ease of access or availability

Potential data issues

• Bias occurs when sample is not representative of the population
• Insufficient sample size reduces reliability of conclusions
• Measurement errors affect data accuracy
• Confounds can distort associations between variables
• Data dredging involves searching for patterns that do not generalize