Statistics is the science of collecting, analyzing, interpreting, presenting, and organizing data to make inferences and decisions in the face of uncertainty.

1. Descriptive Statistics: Methods for summarizing and describing the main features of a dataset.
2. Inferential Statistics: Methods for drawing conclusions about a population based on a sample of data.

• A population is the entire group of individuals or objects that we want to study.
• A sample is a subset of the population that we collect data from.

• A parameter is a numerical value that describes a characteristic of a population.
• A statistic is a numerical value that describes a characteristic of a sample.

Data Types

1. Nominal: Categorical data with no inherent order (e.g., colors, gender).
2. Ordinal: Categorical data with a natural order (e.g., ratings, educational levels).
3. Interval: Numerical data with equal intervals but no true zero (e.g., temperature in Celsius).
4. Ratio: Numerical data with equal intervals and a true zero (e.g., height, weight).

• Discrete data can only take on specific values (e.g., number of children, shoe size).
• Continuous data can take on any value within a range (e.g., height, weight, time).

Descriptive Statistics

1. Mean: The average of all data points.
2. Median: The middle value when data points are ordered.
3. Mode: The most frequent value in the dataset.

1. Range: The difference between the largest and smallest values.
2. Variance: The average squared deviation of each data point from the mean.
3. Standard Deviation: The square root of the variance, representing the average distance of each data point from the mean.

A percentile is a value below which a certain percentage of observations fall.

Probability Distributions

A probability distribution describes the probability of different outcomes in a random event.

• Normal distribution: Bell-shaped curve, symmetric around the mean.
• Binomial distribution: Discrete distribution, representing the number of successes in a fixed number of trials.
• Poisson distribution: Discrete distribution, representing the number of events occurring in a fixed interval of time or space.

The CLT states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.

Inferential Statistics

Hypothesis testing is a statistical procedure for evaluating evidence to decide between two competing hypotheses about a population.

1. Type I error: Rejecting a true null hypothesis (false positive).
2. Type II error: Failing to reject a false null hypothesis (false negative).

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence.

Regression analysis is a statistical method for modeling the relationship between a dependent variable and one or more independent variables.

• Linear regression: Models the relationship as a straight line.
• Logistic regression: Models the probability of a binary outcome.
• Polynomial regression: Models the relationship as a polynomial curve.

Sampling and Experimental Design

• Simple random sampling: Each member of the population has an equal chance of being selected.
• Stratified sampling: The population is divided into groups (strata), and a random sample is taken from each stratum.
• Cluster sampling: The population is divided into groups (clusters), and a random sample of clusters is selected.

• Randomization: Assigning treatments randomly to experimental units to control for extraneous variables.
• Replication: Repeating the experiment multiple times to increase the reliability of the results.
• Control: Holding all other variables constant except for the one being manipulated.

• Observational studies: Researchers observe and measure variables but do not manipulate them.
• Experimental studies: Researchers manipulate one or more variables and measure the effect on other variables.