Statistical Analysis: Descriptive Statistics

• Statistical analysis involves collecting, analysing, interpreting, presenting, and organizing data
• It is used in various fields to make informed decisions based on empirical evidence

Descriptive Statistics

• Descriptive statistics are used to summarise and describe the main features of a data set
• They provide simple summaries about the sample and the measures

Measures of Central Tendency

• Mean: The average value, calculated by summing all values and dividing by the number of values
• Median: The middle value when the data is arranged in ascending or descending order, useful when data is skewed
• Mode: The most frequently occurring value in the data set

Measures of Dispersion

• Range: The difference between the maximum and minimum values in the data set
• Variance: The average of the squared differences from the mean, indicating the spread of the data
• Standard Deviation: The square root of the variance, providing a more interpretable measure of spread
• Interquartile Range (IQR): The range of the middle 50% of the data, less sensitive to outliers

Measures of Shape

• Skewness indicates the asymmetry of the distribution
  • Positive skew (right skew): The tail on the right side is longer
  • Negative skew (left skew): The tail on the left side is longer
  • Zero skew: Symmetrical distribution
• Kurtosis measures the "tailedness" of the distribution
  • High kurtosis: Heavy tails (more outliers)
  • Low kurtosis: Light tails (fewer outliers)

Inferential Statistics

• Inferential statistics are used to make inferences and generalizations about a population based on a sample of data
• They involve techniques to draw conclusions that extend beyond the immediate data available

Estimation

• Point Estimation: Providing a single value as an estimate of a population parameter
• Confidence Interval: Providing a range of values within which the population parameter is likely to fall, with a certain level of confidence

Hypothesis Testing

• Process of evaluating evidence to support or reject a claim (hypothesis) about a population
• Null Hypothesis (H0): A statement of no effect or no difference
• Alternative Hypothesis (H1 or Ha): A statement that contradicts the null hypothesis, suggesting an effect or difference
• Significance Level (alpha): The probability of rejecting the null hypothesis when it is true (usually 0.05)
• P-value: The probability of obtaining results as extreme as, or more extreme than, the observed results, assuming the null hypothesis is true
• Decision Rule: If the p-value is less than alpha, reject the null hypothesis; otherwise, fail to reject the null hypothesis
• Type I Error: Rejecting the null hypothesis when it is true (false positive)
• Type II Error: Failing to reject the null hypothesis when it is false (false negative)
• Statistical Power: The probability of correctly rejecting a false null hypothesis (1 - probability of Type II error)

Common Inferential Tests

• T-tests are used to compare the means of one or two groups
  • One-sample t-test: Compares the mean of a single sample to a known value
  • Independent samples t-test: Compares the means of two independent groups
  • Paired samples t-test: Compares the means of two related groups (e.g., before and after measurements)
• ANOVA (Analysis of Variance) is used to compare the means of three or more groups
  • One-way ANOVA: Compares means across one independent variable
  • Two-way ANOVA: Compares means across two or more independent variables
• Chi-square tests are used to analyse categorical data
  • Chi-square goodness-of-fit test: Tests if the observed distribution of a categorical variable differs from an expected distribution
  • Chi-square test of independence: Tests if two categorical variables are independent

Regression Analysis

• Regression analysis is used to model the relationship between one or more independent variables (predictors) and a dependent variable (outcome)
• It's used for prediction and understanding the influence of predictors on the outcome

Simple Linear Regression

• Models the relationship between one independent variable and one dependent variable using a linear equation: y = mx + b
  • y is the dependent variable
  • x is the independent variable
  • m is the slope (change in y for a one-unit change in x)
  • b is the y-intercept (value of y when x is zero)

Multiple Linear Regression

• Models the relationship between multiple independent variables and one dependent variable using a linear equation: y = b0 + b1x1 + b2x2 + ... + bnxn
  • y is the dependent variable
  • x1, x2, ..., xn are the independent variables
  • b0 is the y-intercept
  • b1, b2, ..., bn are the coefficients for each independent variable

Assumptions of Linear Regression

• Linearity: The relationship between the independent and dependent variables is linear
• Independence: The residuals (errors) are independent of each other
• Homoscedasticity: The variance of the residuals is constant across all levels of the independent variables
• Normality: The residuals are normally distributed

Evaluating Regression Models

• R-squared: The proportion of variance in the dependent variable that is explained by the independent variables (ranges from 0 to 1)
• Adjusted R-squared: A modified version of R-squared that adjusts for the number of predictors in the model
• Residual Analysis: Examining the residuals to check for violations of the assumptions of linear regression
• P-values: Assessing the statistical significance of each predictor in the model

Hypothesis Testing

• Hypothesis testing is a systematic way to evaluate evidence and make decisions about the validity of claims or hypotheses regarding a population
• It helps determine whether observed effects are likely due to chance or represent a genuine pattern

Steps in Hypothesis Testing

• State the Hypotheses: Formulate the null hypothesis (H0) and the alternative hypothesis (H1 or Ha)
• Choose a Significance Level (alpha): Determine the threshold for rejecting the null hypothesis (commonly 0.05)
• Calculate the Test Statistic: Compute a test statistic based on the sample data (e.g., t-statistic, F-statistic, chi-square statistic)
• Determine the P-value: Find the probability of obtaining results as extreme as, or more extreme than, the observed results, assuming the null hypothesis is true
• Make a Decision
  • If the p-value is less than alpha, reject the null hypothesis in favour of the alternative hypothesis
  • If the p-value is greater than or equal to alpha, fail to reject the null hypothesis

Types of Tests

• Parametric Tests: Assume that the data follows a specific distribution (e.g., normal distribution) and are used when the assumptions are met
• Non-parametric Tests: Do not rely on specific distributional assumptions and are used when the data does not meet the assumptions of parametric tests

Data Visualization

• Data visualization involves the graphical representation of data to facilitate understanding and interpretation
• Effective visualizations can reveal patterns, trends, and outliers that might not be apparent from raw data
• Different types of visualizations are suited to different types of data and analytical goals

Types of Charts and Graphs

• Bar Charts: Used to compare the values of different categories
• Line Graphs: Used to show trends over time or relationships between continuous variables
• Scatter Plots: Used to display the relationship between two continuous variables
• Histograms: Used to show the distribution of a single continuous variable
• Box Plots: Used to display the distribution of a variable, including the median, quartiles, and outliers
• Pie Charts: Used to show the proportion of different categories in a whole (use sparingly, as they can be difficult to interpret accurately)
• Heatmaps: Used to display the relationships between two categorical variables or the magnitude of values in a matrix
• Geographic Maps: Used to display data associated with geographic locations

Principles of Effective Data Visualization

• Clarity: Ensure that the visualization is easy to understand and interpret
• Accuracy: Represent the data accurately and avoid misleading representations
• Efficiency: Use the most appropriate type of visualization for the data and the message being conveyed
• Aesthetics: Design the visualization to be visually appealing and engaging
• Context: Provide sufficient context to understand the visualization, including labels, titles, and legends