MAT 152 – MATHEMATICS IN THE MODERN WORLD PERIOD 2 REVIEWER Coverage: Data and Data Classification; Data Presentation and Graphs; Ethical Use and Interpretation of Data; Measures of Central Tendency; Measures of Variability; Fundamental Counting Principle and Tree Diagrams; Basic Probability; Z-Scores and Normal Distribution; Pearson Correlation; Linear Regression. I. Data and Data Classification Data are pieces of information collected for observation, analysis, interpretation, or decision-making. Categorical Data Categorical data describe groups, labels, or characteristics. • Examples: favorite fruit, favorite subject, type of transportation, preferred snack. • Memory tip: Categorical = Categories or labels. Numerical Data Numerical data are expressed using numbers and can be used in mathematical calculations. Discrete Data Discrete data are obtained through counting. • Examples: number of students, books, customers, or siblings. • Memory tip: Discrete = Counted. Continuous Data Continuous data are obtained through measurement and can take decimal values. • Examples: height, weight, temperature, distance, and time. • Memory tip: Continuous = Measured. Quick guide: Categorical = categories; Discrete = counted; Continuous = measured. II. Data Presentation and Graphs Tabular Presentation A table organizes information using rows and columns. It is useful when exact values need to be presented and compared. Memory tip: Table = Rows + Columns. Line Graph Useful for showing changes or trends over time. Examples include daily temperature, monthly sales, and attendance over several days. Pie Graph Represents parts or proportions of a whole. It is useful for percentages or distributions among categories.

Histogram Displays the distribution of numerical data, usually grouped into intervals or classes. Scatter Plot Displays the relationship between two numerical variables, such as study hours and test scores or height and weight. Graph guide: Line = change over time; Pie = parts of a whole; Histogram = numerical distribution; Scatter plot = relationship between two variables. III. Ethical Use and Interpretation of Data Data Integrity Data integrity refers to maintaining the accuracy, consistency, and reliability of data. Transparency Transparency means presenting information clearly and providing enough information for others to understand the data. Graphs should use appropriate titles, labels, axes, units, and descriptions. Privacy and Consent • Explain why information is being collected. • Ask for appropriate permission or consent. • Protect personal information. • Avoid unnecessary disclosure. Evaluating Data Sources Before accepting information from social media, websites, or other sources, consider who collected the data, where it came from, whether the source is trustworthy, whether evidence supports it, and whether the data were collected appropriately. Outliers An outlier is an observation unusually far from the other values in a data set. An outlier may represent a genuine unusual event, measurement error, data-entry error, or an unusual individual observation. It should be investigated rather than automatically removed. IV. Measures of Central Tendency Measures of central tendency describe a central or typical value in a data set. Mean x̄ = Σx / n The mean is the arithmetic average. Add all values and divide by the number of values. Weighted Mean x̄w = Σwx / Σw A weighted mean is used when some values have greater importance or frequency than others.

Median The median is the middle value when data are arranged from smallest to largest. If there is an even number of observations, average the two middle values. Mode The mode is the value or category that occurs most frequently. A data set can have one mode, more than one mode, or no mode. Memory guide: Mean = Average; Median = Middle; Mode = Most. V. Measures of Variability Measures of variability describe how spread out the data are. Range Range = Highest − Lowest Sample Variance s² = Σ(x − x̄)² / (n − 1) To find sample variance: (1) find the mean, (2) subtract the mean from each observation, (3) square each deviation, (4) add the squared deviations, and (5) divide by n − 1. Standard Deviation s = √s² Standard deviation is the square root of variance. It describes spread and is expressed in the same units as the original data. Memory guide: Range = Highest − Lowest; Variance = squared spread; Standard deviation = square root of variance. VI. Fundamental Counting Principle and Tree Diagrams Fundamental Counting Principle N = n₁ × n₂ × n₃ × … Use multiplication to determine the total number of possible outcomes when an activity involves several stages or choices. Memory tip: AND → Multiply. Tree Diagrams A tree diagram represents possible outcomes through branches. It is useful for listing outcomes, counting combinations, and solving probability problems. General rule: when choices occur together (AND), multiply. When alternatives are separate (OR), addition may be appropriate.

VII. Basic Probability P(E) = Favorable outcomes / Total possible outcomes Probability describes how likely an event is to occur. Probability ranges from 0 to 1. For equally likely outcomes, count the favorable outcomes and divide by the total number of possible outcomes. VIII. Z-Scores and Normal Distribution Z-Score z = (x − μ) / σ A z-score indicates how many standard deviations an observation is from the mean. • Positive z-score: the value is above the mean. • Negative z-score: the value is below the mean. • z = 0: the value is exactly at the mean. • The sign gives the direction; the absolute value gives the distance from the mean. Normal Distribution A normal distribution is commonly represented by a bell-shaped curve. Most observations are near the mean, fewer occur farther from the mean, and the distribution is approximately symmetric around the mean. Memory guide: Positive = Above; Negative = Below; Zero = At the mean. IX. Pearson Correlation −1 ≤ r ≤ 1 Pearson's correlation coefficient, r, describes the direction and strength of a linear relationship between two numerical variables. Direction • Positive correlation (r > 0): variables tend to move in the same direction. • Negative correlation (r < 0): variables tend to move in opposite directions. • A value near r = 0 indicates little or no linear relationship. Strength The closer the absolute value of r is to 1, the stronger the linear relationship. • r = 0.95 → very strong/high positive relationship. • r = 0.80 → strong positive relationship. • r = −0.90 → very strong/high negative relationship. • r = −0.75 → strong/high negative relationship. Important: Correlation describes association or relationship. Correlation by itself does not establish causation. Memory guide: Sign = direction; |r| = strength.

X. Linear Regression y = a + bx Regression is used to model a relationship and make predictions. • y = predicted dependent variable • a = intercept • b = slope • x = independent/predictor variable Intercept The intercept a is the predicted value of y when x = 0. Slope The slope b describes the expected change in y for every one-unit increase in x. Using a Regression Equation To predict y: identify x, substitute x into the equation, perform multiplication, then add or subtract as indicated. Example: If y = 10 + 2x and x = 5, then y = 10 + 2(5) = 20. Memory guide: a = intercept; b = slope; x = predictor; y = predicted value. Master Review Table Topic What to Remember Categorical Data Categories/labels Discrete Data Counted Continuous Data Measured Tabular Data Rows and columns Line Graph Change over time Pie Graph Parts of a whole Histogram Numerical distribution Scatter Plot Relationship between variables Data Integrity Accuracy/reliability Transparency Clear presentation Privacy & Consent Protect information and obtain permission Outlier Unusually distant value Mean Average Weighted Mean Mean considering weights Median Middle value Mode Most frequent Range Highest − Lowest Variance Measure of squared spread Standard Deviation Square root of variance Counting Principle Multiply choices Tree Diagram Shows possible outcomes Probability Favorable ÷ Total Positive z Above mean Negative z Below mean z = 0 At the mean Normal Distribution Bell-shaped Positive r Variables move in same direction Negative r Variables move in opposite directions |r| near 1 Stronger relationship

Regression Prediction/modeling y = a + bx Linear regression equation a Intercept b Slope Final Study Checklist ☐ Distinguish categorical, discrete, and continuous data. ☐ Identify the appropriate presentation or graph for a given purpose. ☐ Explain data integrity, transparency, privacy, and consent. ☐ Identify and interpret an outlier. ☐ Calculate the mean and weighted mean. ☐ Find the median and mode. ☐ Calculate range, sample variance, and standard deviation. ☐ Use the Fundamental Counting Principle and interpret tree diagrams. ☐ Calculate basic probability. ☐ Calculate and interpret z-scores. ☐ Recognize the characteristics of a normal distribution. ☐ Interpret positive and negative correlations and their strength. ☐ Distinguish correlation from causation. ☐ Use a linear regression equation for prediction. ☐ Interpret the slope and intercept of a regression equation.