Understanding Descriptive Statistics

Descriptive Statistics

• Descriptive statistics summarize and describe the main features of a dataset.
• Measures of central tendency include mean, median, and mode.
• Mean is the average of all values, calculated by summing the values and dividing by the number of values.
• Median is the middle value when data is sorted.
• Mode is the most frequently occurring value.
• Measures of dispersion include range, variance, and standard deviation.
• Range is the difference between the maximum and minimum values.
• Variance measures the average squared deviation from the mean.
• Standard deviation is the square root of the variance and indicates the spread of data around the mean.
• Quartiles divide the data into four equal parts; Q1 (25th percentile), Q2 (50th percentile, also the median), and Q3 (75th percentile).
• Interquartile range (IQR) is the difference between Q3 and Q1, representing the spread of the middle 50% of the data.
• Box plots graphically represent the minimum, Q1, median, Q3, and maximum values of a dataset.
• Skewness measures the asymmetry of a distribution.
• Kurtosis measures the "tailedness" of a distribution.

Inferential Statistics

• Inferential statistics involve making inferences or generalizations about a population based on a sample.
• Population is the entire group of individuals or items of interest.
• Sample is a subset of the population used to make inferences about the population.
• Parameter is a numerical value that describes a characteristic of a population.
• Statistic is a numerical value that describes a characteristic of a sample.
• Hypothesis testing is a method for testing a claim or hypothesis about a population.
• Null hypothesis (H0) is a statement of no effect or no difference.
• Alternative hypothesis (H1 or Ha) is a statement that contradicts the null hypothesis.
• Significance level (alpha) is the probability of rejecting the null hypothesis when it is true (Type I error).
• p-value is the probability of obtaining results as extreme as, or more extreme than, the observed results, assuming the null hypothesis is true.
• If the p-value is less than the significance level, the null hypothesis is rejected.
• Type I error (false positive) is rejecting the null hypothesis when it is true.
• Type II error (false negative) is failing to reject the null hypothesis when it is false.
• Confidence intervals provide a range of values within which the population parameter is likely to fall.
• The confidence level is the probability that the confidence interval contains the true population parameter.
• Common confidence levels are 90%, 95%, and 99%.

Probability

• Probability is the measure of the likelihood that an event will occur.
• Probability ranges from 0 (impossible) to 1 (certain).
• Event is a set of outcomes of an experiment.
• Sample space is the set of all possible outcomes of an experiment.
• Independent events are events whose outcomes do not affect each other.
• Dependent events are events whose outcomes affect each other.
• Conditional probability is the probability of an event occurring given that another event has already occurred.
• Bayes' theorem describes how to update the probability of a hypothesis based on new evidence.
• Random variable is a variable whose value is a numerical outcome of a random phenomenon.
• Discrete random variable is a variable that can only take on a finite number of values or a countable number of values.
• Continuous random variable is a variable that can take on any value within a given range.
• Probability distribution describes the likelihood of each possible value of a random variable.
• Expected value is the mean of a probability distribution.

Common Probability Distributions

• Bernoulli distribution models the probability of success or failure of a single trial.
• Binomial distribution models the number of successes in a fixed number of independent trials.
• Poisson distribution models the number of events occurring in a fixed interval of time or space.
• Normal distribution is a continuous probability distribution that is symmetric and bell-shaped.
• Standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
• Exponential distribution models the time until an event occurs.

Sampling Distributions

• Sampling distribution is the probability distribution of a statistic obtained from all possible samples of a specific size taken from a population.
• Standard error is the standard deviation of a sampling distribution.
• Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.

Regression Analysis

• Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables.
• Simple linear regression involves one independent variable.
• Multiple regression involves two or more independent variables.
• Regression equation represents the relationship between the dependent and independent variables.
• Slope represents the change in the dependent variable for a one-unit change in the independent variable.
• Intercept represents the value of the dependent variable when the independent variable is zero.
• R-squared (coefficient of determination) measures the proportion of variance in the dependent variable that is explained by the independent variable(s).
• Residuals are the differences between the observed and predicted values.

ANOVA (Analysis of Variance)

• ANOVA is a statistical method used to compare the means of two or more groups.
• One-way ANOVA involves one independent variable (factor).
• Two-way ANOVA involves two independent variables (factors).
• F-statistic is used to test the null hypothesis that the means of the groups are equal.

Chi-Square Tests

• Chi-square tests are used to analyze categorical data.
• Chi-square goodness-of-fit test is used to test whether a sample distribution fits a hypothesized distribution.
• Chi-square test of independence is used to test whether two categorical variables are independent.

Correlation

• Correlation measures the strength and direction of the linear relationship between two variables.
• Pearson correlation coefficient (r) ranges from -1 to +1.
• A positive correlation indicates that the variables increase together.
• A negative correlation indicates that one variable increases as the other decreases.
• A correlation of 0 indicates no linear relationship.