Introduction to Statistics

• Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data
• Methods to describe and summarize data are part of descriptive statistics
• Methods for drawing inferences and generalizations based on data are part of inferential statistics
• Statistics is used in science, business, and government

Descriptive Statistics

• Descriptive statistics summarize and present data in a meaningful way
• Measures of central tendency describe the typical or average value of a dataset
• The mean, median, and mode are common measures of central tendency
• The mean is the average of all values in a dataset
• The median is the middle value when the dataset is ordered
• The mode is the most frequent value in a dataset
• Measures of dispersion describe the spread or variability of data
• The range, variance, and standard deviation are common measures of dispersion
• The 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

Inferential Statistics

• Inferential statistics make inferences and generalizations about a population based on a sample
• Hypothesis testing evaluates the validity of a claim or hypothesis about a population
• A null hypothesis states no effect or no difference
• An alternative hypothesis contradicts the null hypothesis
• A significance level (alpha) is the probability of rejecting the null hypothesis when it is true
• A p-value is the probability of observing a result as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true
• The null hypothesis is rejected if the p-value is less than the significance level
• Confidence intervals provide a range of values within which a population parameter is likely to fall
• A confidence level indicates the probability that the confidence interval contains the true population parameter
• Regression analysis models the relationship between a dependent variable and one or more independent variables
• Simple linear regression involves one independent variable
• Multiple regression involves multiple independent variables

Probability

• Probability measures the likelihood that an event will occur
• It is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty
• A random variable's value is a numerical outcome of a random phenomenon
• Discrete random variables have a finite or countably infinite number of possible values
• Continuous random variables can take on any value within a given range
• A probability distribution describes the likelihood of each possible value of a random variable
• The normal distribution, binomial distribution, and Poisson distributions are common probability distributions
• The normal distribution is a continuous distribution that is symmetrical and bell-shaped
• The binomial distribution is a discrete distribution that describes the number of successes in a fixed number of independent trials
• The Poisson distribution is a discrete distribution that describes the number of events occurring in a fixed interval of time or space