Descriptive Statistics

• Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data

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 data points in a set
• Median is the middle value when the data is ordered
• 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 how far each number in the set is from the mean
• Standard deviation is the square root of the variance, indicating the spread of data around the mean
• Descriptive statistics can be visualized through histograms, bar charts, pie charts, and box plots
• Histograms display the distribution of continuous data
• Bar charts compare categorical data
• Pie charts show proportions of a whole
• Box plots display the median, quartiles, and outliers of a dataset

Inferential Statistics

• Inferential statistics involves making inferences and 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
• Hypothesis testing is a method used to evaluate a hypothesis about a population based on sample data
• Null hypothesis is a statement of no effect or no difference
• Alternative hypothesis is a statement that contradicts the null hypothesis
• Significance level (alpha) is the probability of rejecting the null hypothesis when it is true, typically set at 0.05
• 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
• Confidence intervals provide a range of values within which the true population parameter is likely to fall
• A 95% confidence interval means that if the same population were sampled multiple times, 95% of the calculated intervals would contain the true population parameter
• Common inferential tests include t-tests, ANOVA, chi-square tests, and regression analysis
• T-tests compare the means of two groups
• ANOVA compares the means of three or more groups
• Chi-square tests examine the association between categorical variables
• Regression analysis examines the relationship between a dependent variable and one or more independent variables

Types of Data

• Qualitative data is categorical and describes qualities or characteristics
• Examples include color, gender, or type
• Quantitative data is numerical and represents measurable quantities
• Discrete data is quantitative data that can only take on specific values, usually integers
• Examples include the number of students in a class or the number of cars in a parking lot
• Continuous data is quantitative data that can take on any value within a range
• Examples include height, weight, or temperature
• Nominal scale categorizes data into mutually exclusive, unordered categories (e.g., colors, types of fruit)
• Ordinal scale categorizes data into ordered categories (e.g., rankings, satisfaction levels)
• Interval scale has equal intervals between values, but no true zero point (e.g., temperature in Celsius or Fahrenheit)
• Ratio scale has equal intervals between values and a true zero point (e.g., height, weight, temperature in Kelvin)

Sampling Techniques

• Random sampling involves selecting a sample in such a way that every member of the population has an equal chance of being selected
• Simple random sampling is the basic method where each individual is chosen randomly
• Stratified sampling involves dividing the population into subgroups (strata) and then randomly sampling from each stratum
• Ensures representation from each subgroup
• Cluster sampling involves dividing the population into clusters and then randomly selecting entire clusters to sample
• Convenient when the population is geographically dispersed
• Systematic sampling involves selecting every nth member of the population after a random start
• Non-random sampling includes convenience sampling, snowball sampling, and quota sampling
• Convenience sampling involves selecting individuals who are easily accessible
• Snowball sampling involves participants referring other participants
• Quota sampling involves selecting a sample that matches the proportions of certain characteristics in the population

Errors in Statistics

• Sampling error is the difference between the sample statistic and the population parameter due to chance variation
• Non-sampling error includes errors in data collection, data processing, or questionnaire design
• Can lead to systematic biases
• Type I error (false positive) occurs when the null hypothesis is rejected when it is actually true
• Type II error (false negative) occurs when the null hypothesis is not rejected when it is actually false
• Bias is a systematic error that can distort the results of a study
• Selection bias occurs when the sample is not representative of the population
• Measurement bias occurs when the data collection method systematically under- or over-estimates values
• Response bias occurs when respondents provide inaccurate or untruthful answers

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 linear regression involves two or more independent variables
• The regression equation is used to predict the value of the dependent variable based on the values of the independent variables
• The coefficient of determination (R-squared) measures the proportion of variance in the dependent variable that is explained by the independent variables
• Residuals are the differences between the observed values and the values predicted by the regression equation
• Regression analysis assumes linearity, independence of errors, homoscedasticity (constant variance of errors), and normality of errors
• Logistic regression is used when the dependent variable is binary (e.g., yes/no, success/failure)

Analysis of Variance (ANOVA)

• ANOVA is a statistical method used to compare the means of two or more groups
• One-way ANOVA involves one independent variable with multiple levels
• Two-way ANOVA involves two independent variables
• The F-statistic is used to determine if there is a significant difference between the group means
• ANOVA assumes normality, homogeneity of variance, and independence of errors
• Post-hoc tests are used to determine which specific groups differ significantly from each other after a significant ANOVA result

Non-parametric tests

• Non-parametric tests are statistical tests that do not assume that the data follow a specific distribution
• Used when the assumptions of parametric tests are not met
• Examples include the Mann-Whitney U test, Wilcoxon signed-rank test, Kruskal-Wallis test, and Spearman's rank correlation
• Mann-Whitney U test compares two independent groups
• Wilcoxon signed-rank test compares two related groups
• Kruskal-Wallis test compares three or more independent groups
• Spearman's rank correlation measures the monotonic relationship between two variables

Probability

• Probability is the measure of the likelihood that an event will occur
• Probability values range from 0 to 1, where 0 indicates impossibility and 1 indicates certainty
• Independent events are events where the outcome of one does not affect the outcome of the other
• Dependent events are events where the outcome of one affects the outcome of the other
• Conditional probability is the probability of an event occurring given that another event has already occurred
• Bayes' theorem is a formula that describes how to update the probability of a hypothesis based on evidence