Chapter 14 — Front Matter
Purpose and Overview
Algebra is not only a set of procedures for solving equations or simplifying expressions. It is a language for describing patterns, and one of the most important places where patterns appear is in data. Real-world data comes from measurements, surveys, experiments, records, and observations. It comes from science, finance, medicine, sports, engineering, technology, and everyday life. When we collect data, we want to understand it — to describe what is typical, to see how spread out the values are, to find relationships between variables, and to make predictions about the future.
This chapter closes Algebra I by showing how the algebraic tools you have built throughout this course connect directly to data analysis and mathematical modeling. You will use measures of center and spread to describe distributions, scatter plots and lines of best fit to find relationships, slope and intercept to interpret models in context, residuals to measure model error, two-way tables to organize categorical data, and function families to select the model that best fits a situation. By the end of this chapter, algebra should feel not only useful but necessary for reasoning about real situations in the world.
Opening Question
A school is studying whether students who sleep more tend to earn higher grades. A researcher surveys 20 students, recording the average number of hours each student sleeps per night and that student's most recent exam score. When the data is plotted, a pattern appears: students who sleep more tend to score higher, but the points are scattered and do not fall perfectly on a line. How do we describe that pattern? How do we draw the best line through scattered points? How do we use that line to make a prediction for a student who sleeps 7.5 hours? How do we measure how far off our predictions are from the actual values? These are the questions this chapter answers.
14Sections
15Objectives
24Key Terms
12+Worked Examples
Learning Objectives
- 1Explain why data analysis is part of algebraic reasoning.
- 2Calculate and interpret mean, median, and mode.
- 3Calculate and interpret range and interquartile range; describe standard deviation as a concept.
- 4Read and construct dot plots, histograms, and box plots.
- 5Construct and interpret scatter plots.
- 6Describe association and correlation in data.
- 7Draw and use a line of best fit.
- 8Interpret slope and y-intercept in a data context.
- 9Calculate and interpret residuals.
- 10Read and analyze two-way tables including joint, marginal, and conditional frequencies.
- 11Select the most appropriate function model for a data set using differences and ratios.
- 12Apply algebraic models to science and finance situations.
- 13Identify common data interpretation mistakes.
- 14Describe what Algebra II and related MGU courses extend from Algebra I.
Sections 14.1 – 14.4
Describing Data: Center, Spread, and Display
Why Data Matters · Measures of Center · Measures of Spread · Visual Displays
Section 14.1
Why Data Matters in Algebra
Throughout this course, you have worked with equations, functions, graphs, and models. In most of those situations, the numbers were exact — the equations had specific solutions, the functions followed clean rules, the graphs were smooth and predictable. But real-world situations are messier than that. Data rarely falls perfectly on a line or follows an exact equation. Measurements have error. People do not all behave the same way. Natural processes are influenced by many factors at once. Despite this messiness, patterns exist — and algebra gives you tools to find them.
| Data Type | Description | Example |
|---|
| Univariate | Records one variable for many subjects | Test scores for 30 students |
| Bivariate | Records two variables for each subject | Hours studied and exam score, paired |
The Problem-Solving Method in Data Work
Read — What is being measured? What question is being asked?
Represent — Display the data in a table, plot, or graph so patterns become visible.
Solve — Compute statistics, fit a model, or make a prediction.
Check — Does the answer make sense given the data, scale, and context?
Explain — State the result in words, with units, in context. A number without context is not a complete answer.
Section 14.2
Measures of Center
When a data set contains many values, one of the first questions we ask is: what is a typical value? Measures of center summarize a data set by identifying a middle or representative value. The three main measures are the mean, the median, and the mode.
Mean
The mean is the arithmetic average of a data set. Add all values and divide by the number of values. The symbol x̄ (read "x-bar") represents the mean of a sample.
x̄ = (sum of all values) ÷ (number of values)
Example 14.2.1
Finding the Mean
- ProblemHours studied over seven days: 2, 4, 3, 5, 4, 6, 3. Find the mean.
- ThinkAdd all seven values, then divide by 7.
- Solve
x̄ = (2 + 4 + 3 + 5 + 4 + 6 + 3) ÷ 7 = 27 ÷ 7 ≈ 3.86
- CheckValues range from 2 to 6. The mean of approximately 3.86 falls within that range and near the middle.
Answer: The mean study time is approximately 3.86 hours per day.
Median
The median is the middle value when a data set is arranged in order from least to greatest. If the data set has an odd number of values, the median is the single middle one. If even, the median is the average of the two middle values. The median is resistant to outliers.
Example 14.2.2 & 14.2.3
Finding the Median — Odd and Even Counts
- Odd (A)Data: 12, 7, 15, 9, 11. Ordered: 7, 9, 11, 12, 15. The middle value is 11.
- Even (B)Data: 14, 8, 20, 11, 5, 17. Ordered: 5, 8, 11, 14, 17, 20. Average the two middle values: (11 + 14) ÷ 2 = 12.5.
Answers: (A) Median = 11. (B) Median = 12.5.
Mode
The mode is the value that appears most often. A data set may have one mode, more than one mode, or no mode. The mode is most useful for categorical data or when a particular value is especially common.
| Measure | Best Used When | Sensitive to Outliers? |
|---|
| Mean | Data is roughly symmetric, no strong outliers | Yes — pulled toward extremes |
| Median | Data is skewed or contains outliers | No — ignores extreme values |
| Mode | Most common value needed; categorical data | No |
Student Note
Many students assume the mean is always best. It is not. A salary survey that includes a few very high earners will have a mean salary pulled far above what most workers earn. In skewed distributions, the median is a more honest summary.
Section 14.3
Measures of Spread
Two data sets can have the same mean but look completely different in how spread out their values are. Measures of spread describe how much variability exists in the data.
Range
Range = Maximum − Minimum
The range is easy to compute but is affected heavily by outliers, since it depends only on the two most extreme values.
Quartiles and the IQR
The data set can be divided into four equal parts by three values: Q₁ (first quartile, the median of the lower half), Q₂ (the median of the entire data set), and Q₃ (third quartile, the median of the upper half). The interquartile range (IQR) measures the spread of the middle 50 percent of the data and is resistant to outliers.
IQR = Q₃ − Q₁
Example 14.3.2
Finding the IQR
- ProblemFind the IQR for: 4, 7, 9, 12, 14, 18, 22, 25, 27.
- ThinkFind the median, then the medians of the lower and upper halves.
- Solve
Median (Q₂) = 14 (5th of 9 values)
Lower half: 4, 7, 9, 12 → Q₁ = (7 + 9) ÷ 2 = 8
Upper half: 18, 22, 25, 27 → Q₃ = (22 + 25) ÷ 2 = 23.5
IQR = 23.5 − 8 = 15.5
Answer: IQR = 15.5
Standard Deviation — Concept
Standard deviation measures how far values tend to be from the mean on average. A small standard deviation means data clusters closely around the mean; a large one means data is widely scattered. Formal calculation is developed in the MGU Statistics course. At this level, focus on what it means, not the formula.
Section 14.4
Dot Plots, Histograms, and Box Plots
Numbers alone do not show the full picture. Visual displays help you see shape, center, spread, and outliers all at once.
| Display | Best Used For | What It Shows |
|---|
| Dot Plot | Small data sets with individual values | Each value as a dot; easy to spot clusters, gaps, and mode |
| Histogram | Larger data sets; distribution shape | Frequency within equal-width bins; bars touch each other |
| Box Plot | Comparing two or more distributions | Five-number summary; skewness and outliers visible at a glance |
Example 14.4.3
Constructing a Box Plot
- Data12, 15, 20, 24, 25, 28, 30, 34, 38, 40
- Solve
Minimum = 12 Q₁ = 20
Median = (25 + 28) ÷ 2 = 26.5
Q₃ = 34 Maximum = 40
IQR = 34 − 20 = 14
- DiagramDraw a number line from 10 to 45. Draw a rectangle (the box) from 20 to 34. Draw a vertical line at 26.5 inside the box. Extend a whisker left from 20 to 12, and a whisker right from 34 to 40. Label all five summary values.
Five-number summary: 12 | 20 | 26.5 | 34 | 40. IQR = 14.
- Data set: 4, 7, 3, 9, 7, 5, 6, 7, 8. Find the mean, median, and mode.
- Data set: 10, 15, 20, 12, 18, 25, 11, 22. Find the range and IQR.
- Compute the five-number summary for: 3, 5, 7, 8, 10, 12, 14, 18, 20.
- A data set has a mean of 50 and a median of 35. What does this suggest about the shape of the distribution? Explain.
- For each case, state whether mean or median is a better measure of center. (a) Salaries at a company where the CEO earns 50 times more than others. (b) Heights of students in a single classroom.
Sections 14.5 – 14.9
Bivariate Data: Scatter Plots, Correlation, and Lines of Best Fit
Scatter Plots · Association · Correlation · Lines of Best Fit · Slope and Intercept · Residuals
Section 14.5
Scatter Plots
When you have data about two variables measured on the same subjects, you want to see whether those two variables are related. A scatter plot displays bivariate data by placing one variable on the horizontal axis and the other on the vertical axis, plotting each data pair as a single point.
Example 14.5.1
Constructing and Describing a Scatter Plot
- DataHours studied and exam scores for 8 students:
| Hours Studied (x) | 1 | 2 | 3 | 4 | 5 | 3 | 6 | 4 |
|---|
| Exam Score (y) | 60 | 65 | 70 | 75 | 80 | 68 | 85 | 73 |
- DiagramDraw a coordinate plane. Label the horizontal axis "Hours Studied" (0 to 8). Label the vertical axis "Exam Score" (50 to 100). Plot each ordered pair as a dot.
- DescribeDirection: points rise from left to right — positive association. Form: roughly linear. Strength: fairly tight cluster around an imaginary rising line.
Answer: The scatter plot shows a positive, roughly linear association — more hours studied corresponds to higher exam scores.
Describing a Scatter Plot — Three Questions
Direction — Do points rise (positive) or fall (negative) as x increases?
Form — Do points follow a roughly linear pattern, or do they curve?
Strength — Are points tightly clustered near a line, or widely scattered?
Section 14.6
Association and Correlation
Two variables have an association when the value of one tends to change as the other changes. Correlation refers specifically to linear association and is quantified by the correlation coefficient, denoted r, which always falls between −1 and 1.
| Value of r | Meaning |
|---|
r = 1 | Perfect positive linear correlation — all points on a rising line |
r close to 1 | Strong positive linear correlation |
r close to 0 | Weak or no linear correlation |
r close to −1 | Strong negative linear correlation |
r = −1 | Perfect negative linear correlation — all points on a falling line |
Correlation Is Not Causation
Just because two variables are correlated does not mean one causes the other. Ice cream sales and drowning rates are positively correlated — not because ice cream causes drowning, but because both are influenced by a third variable: hot weather. When you see correlation in data, your job is to describe the pattern, not to declare a cause.
Section 14.7
Lines of Best Fit
When a scatter plot shows a roughly linear pattern, you can draw a line of best fit (also called a trend line) to model the relationship. This line summarizes the overall direction and center of the pattern. To draw one by hand: draw a line that passes through the middle of the cloud of points, with roughly equal numbers of points above and below it. The line does not need to pass through any specific data points.
Example 14.7.3
Finding the Equation of a Line of Best Fit from Two Points
- ProblemThe line of best fit passes through (2, 67) and (5, 82). Find its equation.
- ThinkFind slope using the two points, then apply point-slope form.
- Solve
m = (82 − 67) ÷ (5 − 2) = 15 ÷ 3 = 5
y − 67 = 5(x − 2)
y = 5x + 57
- CheckSubstitute (2, 67): 5(2) + 57 = 67. ✓ Substitute (5, 82): 5(5) + 57 = 82. ✓
Answer: y = 5x + 57
Example 14.7.2
Using the Line for Prediction — Interpolation and Extrapolation
- ProblemUsing y = 5x + 57, predict the score for (A) a student who studies 5 hours, and (B) a student who studies 10 hours.
- Solve A
y = 5(5) + 57 = 25 + 57 = 82 (x = 5 is within the data range — interpolation)
- Solve B
y = 5(10) + 57 = 107 (x = 10 is beyond the data range — extrapolation)
- CheckA score of 107 exceeds 100 — impossible on most exams. Extrapolation produced a nonsensical result.
Answer: (A) Predicted score ≈ 82 (reliable). (B) Model predicts 107, but this is an unreliable extrapolation outside the data range.
MGU Library Connection
- In statistics courses, the line of best fit is computed precisely using least squares regression, which minimizes the sum of squared residuals. This is covered formally in the MGU Statistics course and in Algebra II data sections.
Section 14.8
Interpreting Slope and Intercept in Data
When a linear model fits a data set, the slope and intercept have real-world meaning. Interpreting them correctly in context — with units — is one of the most important skills in applied algebra.
How to Interpret Slope in Context
The slope tells you: For each one-unit increase in x, the predicted value of y changes by m units.
Positive slope → both variables increase together. Negative slope → as one increases, the other decreases.
Example 14.8.1 & 14.8.2
Interpreting Slope and y-Intercept
- Model Ay = 5x + 57, where x = hours studied and y = exam score.
- SlopeSlope = 5. For each additional hour studied, the predicted exam score increases by approximately 5 points.
- Intercepty-intercept = 57. A student who studies 0 hours is predicted to score approximately 57. This may represent a baseline from prior knowledge, or it may simply be where the line crosses when extended — interpret with caution.
- Model By = −1.2x + 14, where x = hours of TV daily and y = hours of physical activity per week. Slope = −1.2: for each additional hour of TV watched daily, weekly physical activity decreases by approximately 1.2 hours.
Interpretations require units and context. "Slope = 5" is not a complete answer. "For each additional hour studied, the score increases by 5 points" is.
Section 14.9
Residuals as Model Error
No line of best fit passes through every point perfectly. The vertical distance between each data point and the line represents the error of the model at that point. These distances are called residuals.
Residual Formula
Residual = Observed − Predicted = y − ŷ
Positive residual → actual point is above the line (model underestimated). Negative residual → actual point is below the line (model overestimated). Residual = 0 → point falls exactly on the line.
Example 14.9.1
Computing a Residual
- ProblemModel: y = 5x + 57. A student studied 3 hours and scored 70. Find the residual.
- Solve
ŷ = 5(3) + 57 = 72
Residual = 70 − 72 = −2
- InterpretThe model predicted 72, but the student actually scored 70. The model overestimated by 2 points. The residual is −2.
Answer: Residual = −2. The model over-predicted by 2 points.
Residual Plots
A residual plot graphs residuals on the vertical axis against x-values on the horizontal axis. A well-fitting linear model will show residuals scattered randomly around zero — no clear pattern. If residuals curve or fan out, the linear model may not be the best choice for the data.
- Plot the following data and describe the direction and strength of the association: (x) 1, 2, 3, 4, 5, 6 — (y) 14, 12, 9, 7, 5, 3.
- A scatter plot shows r ≈ 0.9. What does this tell you about the relationship?
- A line of best fit passes through (1, 44) and (7, 74). Find its equation.
- Using your equation from Problem 3, predict y when x = 4. Is this interpolation or extrapolation?
- Model: y = 3x + 10. Data point: (5, 31). Find the residual. Did the model over- or under-predict?
- The line of best fit for advertising spending (thousands of dollars) vs. monthly sales (thousands of units) is y = 3.5x + 12. Interpret the slope and y-intercept in context.
Sections 14.10 – 14.12
Two-Way Tables, Model Selection, and Applications
Categorical Data · Choosing Linear / Quadratic / Exponential Models · Science and Finance
Section 14.10
Two-Way Tables
Not all data is numerical. Sometimes two variables are both categorical — belonging to groups or categories rather than taking numerical values. A two-way table organizes data from two categorical variables by showing the counts for each combination.
Example 14.10.1 & 14.10.2
Reading and Computing Frequencies from a Two-Way Table
- DataSurvey of 100 students: morning vs. evening preference, and alone vs. group preference.
| Alone | With a Group | Row Total |
|---|
| Morning | 30 | 20 | 50 |
| Evening | 15 | 35 | 50 |
| Column Total | 45 | 55 | 100 |
- Joint30 students prefer morning AND studying alone.
- Marginal45 students prefer studying alone, regardless of time of day.
- ConditionalOf the 50 evening students, 35 ÷ 50 = 70% prefer working with a group. Of the 50 morning students, 30 ÷ 50 = 60% prefer working alone. The conditional proportions differ — evidence of an association.
Interpretation: Evening students are more likely to prefer group work than morning students (70% vs. 40%). There appears to be an association between time preference and study style preference.
Reading Two-Way Tables
Make sure to read row totals and column totals correctly. The value 30 in a cell does not mean 30% unless you divide by the appropriate total. Always identify whether you want a joint frequency, a marginal frequency, or a conditional frequency before computing.
Section 14.11
Choosing Linear, Quadratic, or Exponential Models
You have studied three major function families. Each describes a different pattern of change. When you have a data set and want to model it, one of your tasks is selecting the function family that best describes the data. Given a table of equally spaced x-values, examine the differences and ratios of y-values.
| Model Type | Test | Form | Example Situation |
|---|
| Linear | First differences constant | y = mx + b | Constant speed; monthly savings with fixed deposits |
| Quadratic | Second differences constant | y = ax² + bx + c | Height of a thrown ball; area as side length increases |
| Exponential | Ratios of consecutive y-values constant | y = a · bˣ | Population growing at constant rate; radioactive decay |
Example 14.11.1
Identifying the Model Type from a Table
- Table Ax: 0, 1, 2, 3, 4 — y: 3, 7, 11, 15, 19. First differences: 4, 4, 4, 4. Constant first differences → linear model. Equation: y = 4x + 3.
- Table Bx: 0, 1, 2, 3, 4 — y: 1, 3, 9, 19, 33. First differences: 2, 6, 10, 14 (not constant). Second differences: 4, 4, 4 (constant) → quadratic model.
- Table Cx: 0, 1, 2, 3, 4 — y: 2, 6, 18, 54, 162. Ratios: 3, 3, 3, 3. Constant ratios → exponential model. Equation: y = 2 · 3ˣ.
Use differences for linear or quadratic; use ratios for exponential.
Section 14.12
Algebraic Models in Science and Finance
Algebraic models appear in nearly every quantitative field. Here are examples of each function family in applied contexts.
Science Applications
Example — Free Fall (Quadratic)
Finding When an Object Hits the Ground
- ModelAn object dropped from 100 feet: h = −16t² + 100, where h = height in feet and t = time in seconds.
- Solve
Set h = 0: −16t² + 100 = 0
16t² = 100 → t² = 6.25 → t = 2.5 seconds
- CheckAt t = 2.5: h = −16(6.25) + 100 = −100 + 100 = 0. ✓
Answer: The object hits the ground after 2.5 seconds.
Example — Radioactive Decay (Exponential)
Half-Life Model
- ModelA substance starts with 200 grams and has a half-life of 3 years: A = 200 · (1/2)^(t/3).
- SolveAfter 9 years:
A = 200 · (1/2)³ = 200 · (1/8) = 25 grams
Answer: 25 grams remain after 9 years.
Finance Applications
Compound Interest Formula
A = P(1 + r/n)^(nt)
A = amount after t years. P = principal. r = annual rate (decimal). n = compounding periods per year. t = time in years.
Example 14.12.1
Compound Interest
- ProblemDeposit $1,000 at 4% annual interest, compounded annually (n = 1). How much after 5 years?
- Solve
A = 1000(1 + 0.04/1)^(1·5) = 1000(1.04)⁵
(1.04)⁵ ≈ 1.2167
A ≈ 1000 × 1.2167 = $1,216.70
- CheckSimple interest would give $1,000 + 5($40) = $1,200. Compound result is slightly higher. Reasonable. ✓
Answer: After 5 years, the account contains approximately $1,216.70.
MGU Library Connection
- Compound interest is explored in depth in the MGU Personal Finance course
- Logarithms are used to solve for time and rate in exponential models — covered in Algebra II, Chapter 10
- Free-fall and projectile models are extended in MGU Physics
- Radioactive decay is modeled in MGU Chemistry and Earth Science
- A survey table: Library / Alone = 40, Library / Music = 10; Home / Alone = 15, Home / Music = 35 (totals: 55 alone, 45 music, 100 total). (a) What proportion of library students prefer silence? (b) What proportion of home students prefer music? (c) Is there an association? Explain.
- Identify the model type: x = 0, 1, 2, 3, 4 — y = 5, 10, 20, 40, 80. Write an equation for the pattern.
- Identify the model type: x = 0, 1, 2, 3, 4 — y = 3, 7, 11, 15, 19. Write an equation and predict y when x = 10.
- A ball is thrown upward. Heights at t = 0, 0.5, 1.0, 1.5, 2.0 seconds are 4, 24, 36, 40, 36 feet. (a) What type of model fits? (b) What is the maximum height?
- You deposit $2,500 at 5% annual interest, compounded annually. Find the balance after 4 years. Show all work.
Chapter 14 — Back Matter
Common Mistakes and Chapter Summary
Section 14.13
Common Data Interpretation Mistakes
Mistake 1 — Confusing Correlation with Causation
"As ice cream sales increase, so does drowning. Therefore ice cream causes drowning."
Correct: Both are influenced by a third variable — hot weather. Correlation describes a pattern, not a cause. Always say "there is a positive association between x and y," not "x causes y to increase."
Mistake 2 — Extrapolating Too Far
Using y = 5x + 57 to predict a score when x = 40 hours of study.
Correct: The model is only reliable within or near the range of observed data. Extending far beyond it can produce nonsensical results (a score over 100 or a negative population size).
Mistake 3 — Misinterpreting a Residual
"Residual = +15 means the model predicted 15 more than the actual value."
Correct: Residual = Observed − Predicted. A positive residual means the actual value is 15 MORE than the predicted value — the model underestimated, not overestimated.
Mistake 4 — Assuming All Scatter Plots Are Linear
Drawing a straight line through data that curves sharply upward.
Correct: Use difference patterns and ratios to confirm model type before fitting a line. If points curve exponentially, an exponential model will fit far better than a line.
Mistake 5 — Confusing Mean and Median
"Mean = 42, Median = 40, therefore there are no outliers."
Correct: A mean higher than the median suggests right skew, possibly caused by outliers. You cannot conclude there are no outliers from mean and median alone.
Mistake 6 — Misreading a Two-Way Table Cell as a Percentage
"30 is in the Morning / Alone cell, so 30% of morning students prefer working alone."
Correct: 30 ÷ 50 = 60%. You must divide by the appropriate row or column total to find a conditional frequency. Never read a cell count as a percent without dividing.
Chapter Summary
- Sec 14.1Algebra connects to data through patterns. Univariate data is described with center and spread; bivariate data is explored with scatter plots and models. The Read → Represent → Solve → Check → Explain method applies throughout.
- Sec 14.2Mean is the arithmetic average; median is the middle value of ordered data; mode is the most frequent value. Use median for skewed data or data with outliers; mean for symmetric, clean distributions.
- Sec 14.3Range = max − min. IQR = Q₃ − Q₁, measuring the middle 50% of data and resisting outliers. Standard deviation measures how spread values are around the mean — small means clustered, large means dispersed.
- Sec 14.4Dot plots show individual values on a number line. Histograms display frequency within equal-width intervals. Box plots display the five-number summary and are ideal for comparing two or more distributions.
- Sec 14.5A scatter plot plots each (x, y) pair as a point. Describe scatter plots by direction (positive or negative), form (linear or curved), and strength (tight cluster or wide scatter).
- Sec 14.6Association is a general relationship between two variables. Correlation measures linear association by r, which falls between −1 and 1. Strong correlation does not imply causation.
- Sec 14.7A line of best fit summarizes the trend through a scatter plot. Find its equation from two points on the line. Use it for interpolation (reliable) or extrapolation (use cautiously — outside the data range).
- Sec 14.8Slope = rate of change in context: "for each one-unit increase in x, y changes by m units." The y-intercept is the predicted y when x = 0 — meaningful only when x = 0 has a real-world interpretation.
- Sec 14.9Residual = y − ŷ = observed − predicted. Positive residuals mean the model underestimated; negative residuals mean it overestimated. Residual plots reveal whether a linear model is appropriate.
- Sec 14.10Two-way tables organize bivariate categorical data. Joint frequencies are cell counts; marginal frequencies are row and column totals; conditional frequencies are proportions within a row or column. Differing conditional proportions suggest association.
- Sec 14.11Linear: constant first differences. Quadratic: constant second differences. Exponential: constant ratios. Use these tests on tables with equally spaced x-values to identify the model type.
- Sec 14.12Algebraic models describe science and finance. Distance: d = rt (linear). Free fall: h = −16t² + h₀ (quadratic). Decay or growth: exponential. Compound interest: A = P(1 + r/n)^(nt).
- Sec 14.13Six recurring errors: confusing correlation with causation; extrapolating beyond the data; misinterpreting residuals; assuming linear when another model fits better; confusing mean and median; reading cell counts as percents.
- Sec 14.14Algebra I is complete. Algebra II extends every major topic — functions, quadratics, systems, polynomials, exponential and logarithmic models, sequences, trigonometry, and formal statistics — to greater depth and generality.
Key Terms Review
association
A relationship between two variables in which the values of one tend to change as the other changes.
bivariate data
Data that records two variables for each subject in the study.
box plot
A visual display of the five-number summary: minimum, Q₁, median, Q₃, and maximum.
correlation
The strength and direction of a linear relationship between two numerical variables.
correlation coefficient (r)
A number between −1 and 1 measuring the strength and direction of linear correlation.
dot plot
A data display that places one dot above a number line for each value in the data set.
extrapolation
Using a model to predict values outside the range of the observed data; less reliable than interpolation.
five-number summary
The set minimum, Q₁, median, Q₃, maximum — the basis for a box plot.
histogram
A display of the frequency of values within equal-width intervals; bars touch each other.
IQR
Interquartile range: Q₃ − Q₁. Measures the spread of the middle 50% of data and resists outliers.
interpolation
Using a model to predict values within the range of observed data; generally reliable.
line of best fit
A line drawn through a scatter plot to model the overall linear trend in bivariate data.
mean
The arithmetic average of a data set: sum of all values divided by the number of values. Sensitive to outliers.
median
The middle value of an ordered data set. Resistant to outliers. Average of two middle values if count is even.
mode
The most frequently occurring value. Most useful for categorical data or when one value clearly dominates.
outlier
A data value that is unusually far from the rest of the data. Can heavily influence the mean and line of best fit.
range
Maximum minus minimum. The simplest measure of spread; sensitive to outliers.
residual
The difference between an observed value and the value predicted by the model: y − ŷ.
scatter plot
A coordinate-plane graph plotting ordered pairs of bivariate numerical data as individual points.
standard deviation
A measure of spread describing how far values tend to be from the mean on average.
two-way table
A table that organizes bivariate categorical data by showing counts for each combination of categories.
univariate data
Data that records one variable for many subjects. Described with measures of center and spread.
Concept Review Questions
- Explain the difference between a measure of center and a measure of spread. Give an example of each.
- When would you choose to report the median instead of the mean? Explain why with an example.
- What does a box plot display? What five values does it require?
- Why is the IQR more resistant to outliers than the range?
- Describe what you would see in a scatter plot that shows a strong negative association.
- What does "correlation does not imply causation" mean? Give an original example.
- What is a residual? What does a positive residual mean? A negative residual?
- Explain the difference between interpolation and extrapolation. Which is more reliable and why?
- How do you use a two-way table to find a conditional frequency?
- How can you tell from a table of values with equally spaced x-values whether the pattern is linear, quadratic, or exponential?
Skill Practice
- Find mean, median, mode for: 8, 13, 5, 9, 13, 11, 4, 13, 7, 10
- Find range and IQR for the data in Problem 1
- Five-number summary for: 2, 5, 7, 9, 11, 14, 18
- Which measure of center is best for a skewed data set and why?
- A data set: mean = 60, median = 45. Describe the likely shape.
- Describe the standard deviation of a data set where all values equal 10.
- Draw a scatter plot for: x = 1,2,3,4,5,6 — y = 42,48,55,58,65,70
- Describe the direction and strength of the association in Part B, Problem 1
- Two points on a line of best fit: (2, 50) and (6, 66). Find the equation.
- Use your equation to predict y when x = 4
- Is the prediction in Problem 4 interpolation or extrapolation?
- Model: y = 4x + 10. Data point: (3, 18). Find and interpret the residual.
- A table: Reading / Morning = 22, Reading / Evening = 10, Videos / Morning = 18, Videos / Evening = 30. (a) What proportion of morning students prefer reading? (b) What proportion of evening students prefer videos? (c) Is there an association?
- x = 0,1,2,3,4 — y = 4, 8, 16, 32, 64. What type of model? Write an equation.
- x = 0,1,2,3,4 — y = 1, 5, 13, 25, 41. Find first and second differences. What type of model?
- You save $200 per month starting with $300. Write a linear model. After how many months do you have $2,500?
- An account earns 3% interest compounded annually. You deposit $5,000. Find the balance after 6 years.
Error Analysis
Each problem shows a student's work or conclusion. Identify the error and write the correct reasoning or answer.
Problem 1
"Positive correlation between hours of TV and lower grades proves watching TV causes bad grades."
What is wrong? What is the correct way to describe this relationship?
Problem 2
Model: y = 2.1x + 8 (data range x = 1 to 10). A student predicts y when x = 80 and calls it reliable.
Identify and explain the error in the student's reasoning.
Problem 3
Residual = +15 → "The model predicted 15 more than the actual value."
Identify the error. Write the correct interpretation of a residual of +15.
Problem 4
Data table: first differences are 4, 8, 4, 8 alternating. Student says: "Second differences are 4, so this is quadratic."
Identify and explain the error. What is the correct conclusion from these differences?
Problem 5
Two-way table cell value is 30. Student says "30% of that group prefer this option."
What is missing from this reasoning? What step must the student complete first?
Word Problems
Show all work. Define variables. Give answers with units and context.
- Science: A cup of tea starts at 180°F and cools by approximately 12% every 5 minutes. Write an exponential model for temperature after t five-minute intervals. Predict the temperature after 20 minutes.
- Finance: You want to save $5,000. You have $800 and save $150 per month. (a) Write a linear equation for total savings S after m months. (b) How many months until you reach $5,000?
- Health: Daily step counts for 7 days: 4200, 6100, 5800, 7300, 6900, 8200, 7500. (a) Find the mean and median. (b) On day 8, the patient walks 200 steps. Recalculate the mean. Which measure changed more dramatically, and what does this suggest?
- Sports: A coach models a player's shot percentage P after s practice sessions with P = 2.4s + 48. (a) Predict the percentage after 15 sessions. (b) Is it reasonable to use this model for s = 40 sessions? Explain.
- Economics: Monthly sales data (units) for 5 months: 200, 220, 242, 266, 293. Find the ratio between consecutive months. What model fits? Write an equation. Predict sales in month 8.
- Science: A researcher models a plant's height h (cm) over d days with h = −0.05d² + 2d + 5. (a) Find the height at d = 10 and d = 20. (b) What type of function is this? (c) When does the plant reach maximum height? (Hint: use the vertex formula from Chapter 12.)
Chapter 14 Checkpoint
Chapter 14 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 14
Part 1 — Center and Spread
- Data set: 8, 13, 5, 9, 13, 11, 4, 13, 7, 10. Find the mean, median, mode, and range.
- Use the data in Question 1 to find Q₁, Q₃, and the IQR.
- Construct a five-number summary and describe the box plot you would draw.
Part 2 — Lines and Residuals
- The equation of a line of best fit is
y = −4x + 100, where x = number of absences and y = student grade. (a) Interpret the slope. (b) Interpret the y-intercept. (c) Predict the grade for a student with 6 absences.
- Using the model from Question 4, a student has 5 absences and a grade of 75. Compute the residual and explain what it means.
Part 3 — Model Identification
- Table: x = 1, 2, 3, 4, 5 — y = 3, 12, 27, 48, 75. (a) Find the first differences. (b) Find the second differences. (c) What type of model fits?
- Table: x = 0, 1, 2, 3, 4 — y = 6, 12, 24, 48, 96. (a) Compute ratios of consecutive y-values. (b) What type of model? (c) Write its equation.
Part 4 — Two-Way Tables
- Survey of 80 students: Morning / Reading = 22, Morning / Videos = 18; Evening / Reading = 10, Evening / Videos = 30. (a) What proportion of morning students prefer reading? (b) What proportion of evening students prefer videos? (c) Is there an association between time of day and content preference? Explain.
Part 5 — Applications
- You invest $2,500 in an account earning 5% annual interest, compounded annually. (a) Write the compound interest formula for this account. (b) Find the balance after 4 years. Show your work.
- A scatter plot shows a moderate negative association between daily screen time (hours) and hours of sleep per night. (a) Describe what this association means in context. (b) Explain what "correlation does not imply causation" means in this specific example.
Part 6 — Error Analysis
- A student says: "Ice cream sales and sunburn cases are positively correlated, so eating ice cream causes sunburns." Identify and explain the error.
- A student calculates a residual and gets +8. The student says: "The model predicted 8 more than the actual value." Correct this interpretation.
Looking Ahead — Bridge Note
From Algebra I to Algebra II
You have completed Algebra I. The tools you have built here — expressions, equations, inequalities, functions, graphs, linear models, systems, exponents, polynomials, factoring, quadratics, radicals, and data analysis — are the foundation of everything that follows in mathematics.
In Algebra II, you will return to almost every topic in this book, but at greater depth and with greater generality. Functions will be studied as a unified family — polynomial, rational, radical, exponential, logarithmic, and trigonometric. Quadratics will be revisited in vertex form, and complex numbers will be introduced as solutions. The data and statistics work from this chapter will grow into formal regression, probability distributions, and the normal curve. Exponential functions will be paired with logarithms — their inverses — so you can solve equations like 2ˣ = 100 precisely. Sequences and series will formalize the difference and ratio patterns from Section 14.11. Conic sections will describe circles, parabolas, ellipses, and hyperbolas algebraically. Trigonometry will begin with right-triangle ratios and extend to the unit circle and periodic functions.
The work you have done in Algebra I is real mathematics. What comes next builds directly on it. The path forward is clear.
MGU Library Connection
- The next textbook in this series: Algebra II — MGU Mathematics Series
- Geometry — uses coordinate geometry, slope, distance formula, and algebraic proofs throughout
- Statistics and Data Analysis — extends residuals, regression, normal distributions, and inference
- Personal Finance — compound interest, amortization, investment growth models
- Physics — motion, projectile, and wave models built on linear and quadratic functions
- Computer Science — functions, conditionals, and algebraic expressions as core programming structures
- Appendix O of this textbook: Algebra II Readiness Check