Intro to statistics

• Statistics involves collecting, organizing, summarizing, and analyzing data to facilitate decision-making and prediction.

Key Fields in Statistics

• Descriptive Statistics summarizes and visualizes data using graphs and averages.
• Inferential Statistics draws conclusions from sample data.

Types of Data

• Quantitative (Numerical) Data is represented by numbers, like height or income.
• Discrete data consists of whole numbers.
• Continuous data can take any value within a range.
• Qualitative (Categorical) Data is non-numeric, for example, eye color.

Distributions

• A distribution illustrates the spread of data values.
• Shape can be symmetric, skewed, uniform, or bimodal.
• Center is defined by the mean or median.
• Spread is defined by range, variance, or standard deviation.

Common Graph Types

• Dotplots use dots to represent individual values.
• Histograms group data into bins, which is useful for large datasets.
• Boxplots display the median, quartiles, and potential outliers.
• Scatterplots reveal relationships between two numerical variables.

Measures of Center

• Mean (Average) is the sum of values divided by the number of values, and is sensitive to outliers.
• Median is the middle value in sorted data, and is resistant to outliers.
• Mode is the value that occurs most frequently.

Measures of Spread

• Range is calculated by subtracting the smallest value from the largest value.
• Interquartile Range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1), representing the middle 50% of the data, and is better for skewed data.
• Variance (σ²) is the average squared deviation from the mean.
• Standard Deviation (σ) is the square root of the variance, and measures the typical distance from the mean.

Mean vs. Median

• In a symmetric distribution, Mean ≈ Median.
• In a right-skewed distribution, Mean > Median.
• In a left-skewed distribution, Mean < Median.

Correlation (r)

• Correlation measures the linear relationship between two variables.
• Values range from -1 to +1.
• r = +1 means perfect positive correlation.
• r = -1 means perfect negative correlation.
• r = 0 means no correlation.
• Correlation does not equal causation.

Linear Regression

• Regression Equation: ŷ = a + bx.
• a equals the y-intercept or the predicted y when x = 0.
• b equals the slope, representing the change in y per unit increase in x.
• Interpreting Slope: an example of b = 2.5, means for each additional x, y increases by 2.5.

Coefficient of Determination (r²)

• r² equals the percentage of variability in y explained by x.
• A higher r² indicates a better model fit.

Basic Probability Rules

• Probability ranges between 0 and 1.
• P(A) + P(Aᶜ) = 1 (Complement Rule).
• P(A and B) = P(A) × P(B) for independent events.
• P(A or B) = P(A) + P(B) - P(A and B) (Union Rule).

Conditional Probability

• P(A | B) = P(A and B) / P(B).
• This represents the probability of event A occurring given that event B has occurred.

Normal Distribution (Bell Curve)

• Mean (μ) is the center of the distribution.
• Standard Deviation (σ) indicates the spread.
• Empirical Rule:
• Approximately 68% of data falls within 1σ.
• Approximately 95% of data falls within 2σ.
• Approximately 99.7% of data falls within 3σ.

Binomial Distribution

• Used to count the number of successes in a fixed number of trials.
• Conditions:
• Fixed number of trials (n).
• Only 2 outcomes: success or failure.
• Same probability (p) per trial.
• Trials are independent.
• Mean (Expected Value): μ = n × p.
• Standard Deviation: σ = √(n × p × (1 - p)).