Source: Moore, McCabe & Craig, Introduction to the Practice of Statistics, 10th Ed., Ch. 1–2
Tags: descriptive statistics, frequency distributions, histograms, mean, median, standard deviation, IQR, scatterplot, correlation, regression, least-squares, outliers, ECON 2500
Difficulty: Introductory | Prerequisites: Grade 12 Advanced Functions or equivalent
Chapters 1 and 2 of this course lay the groundwork for everything that follows. Before you can test hypotheses or build models, you need to know how to look at data: how a single variable is distributed (Chapter 1) and how two variables relate to each other (Chapter 2). These chapters introduce the vocabulary, the graphical tools, and the numerical summaries that statisticians and economists use daily. If you are behind, start here, because every later topic assumes you can describe a dataset on sight.
Chapter 1 teaches you to describe a single variable with graphs (histograms, stemplots, boxplots) and numbers (mean, median, standard deviation, IQR). Chapter 2 extends this to pairs of variables, focusing on scatterplots, correlation, and least-squares regression lines. Together, they give you the descriptive toolkit you will use throughout the course.
Individual
An object or person described by a set of data. In simple terms, one row in your dataset.
Variable
A characteristic of an individual. It can be categorical (labels or categories) or quantitative (measured numbers). Think of it as a column in a spreadsheet.
Distribution
The pattern of values a variable takes and how often it takes them. In simple terms, it tells you "what values show up, and how common are they?"
Mean (x-bar)
The arithmetic average: sum all values and divide by the count. Sensitive to outliers.
Median
The midpoint of the data when values are arranged in order. Half fall above, half below. More resistant to outliers than the mean.
Standard deviation (s)
A measure of the average distance of observations from the mean. A larger s means the data are more spread out. Think of it as "on average, how far is each data point from the centre?"
Interquartile range (IQR)
The range of the middle 50% of data: IQR = Q3 - Q1. A resistant measure of spread, unaffected by extreme outliers.
Five-number summary
Minimum, Q1, Median, Q3, Maximum. The skeleton of a boxplot.
Outlier
An observation that falls outside the overall pattern. The 1.5 x IQR rule flags suspected outliers: any value below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR).
Symmetric distribution
A distribution where the left and right sides are roughly mirror images. Mean and median are close together.
Skewed distribution
A distribution with a long tail on one side. Right-skewed (positive skew): tail stretches toward larger values, mean > median. Left-skewed (negative skew): tail stretches toward smaller values, mean < median.
Scatterplot
A graph of two quantitative variables, one on each axis, showing how they relate. In simple terms, each dot is one individual plotted by its x-value and y-value.
Correlation (r)
A number between -1 and +1 that measures the strength and direction of a linear relationship between two quantitative variables. r near +1 = strong positive linear association; r near -1 = strong negative; r near 0 = weak or no linear association.
Least-squares regression line
The line that minimises the sum of the squared vertical distances from the data points to the line. Written as y-hat = a + bx. In simple terms, the "best fit" line for predicting y from x.
Residual
The difference between an observed value and the value predicted by the regression line: residual = y - y-hat. Positive means the actual value was above the line; negative means below.
Coefficient of determination (r-squared)
The fraction of the variation in y that is explained by the linear relationship with x. If r = 0.8, then r-squared = 0.64, meaning 64% of the variation in y is accounted for by x.
Lurking variable
A variable not included in the analysis that influences both the explanatory and response variables, potentially creating a misleading association. Also called a confounding variable in some contexts.
Categorical variables place individuals into groups (e.g. major, gender, province). Summarise with counts and proportions; display with bar charts or pie charts.
Quantitative variables take numerical values for which arithmetic makes sense (e.g. income, GDP, test score). Summarise with means, medians, standard deviations; display with histograms, stemplots, boxplots.
Histogram: Breaks the range of a quantitative variable into equal-width intervals (bins) and shows the count or percentage in each. Look for shape, centre, spread, and outliers.
Stemplot (stem-and-leaf): Retains actual data values while showing the shape. Useful for small datasets.
Boxplot: Built from the five-number summary. The box spans Q1 to Q3, a line marks the median, and whiskers extend to the smallest and largest non-outlier values. Points beyond 1.5 x IQR from the box are plotted individually as suspected outliers.
When describing any distribution, always address four things:
Shape: Symmetric, left-skewed, or right-skewed? Unimodal, bimodal, or multimodal?
Centre: Mean or median (use median for skewed data).
Spread: Standard deviation or IQR (use IQR for skewed data).
Outliers: Any unusual observations that fall outside the overall pattern.
Mean: Best for roughly symmetric distributions. Pulled toward the tail in skewed data.
Median: Best for skewed distributions or data with outliers. More resistant (robust) to extreme values.
Range: Maximum minus minimum. Simple but highly sensitive to outliers.
IQR: Q3 - Q1. Resistant to outliers. Describes the middle 50% of the data.
Standard deviation (s): The square root of the variance. Uses every observation. Best paired with the mean for symmetric data.
Variance (s-squared): The average of the squared deviations from the mean. Less intuitive than s because its units are squared.
Many real-world variables follow an approximately normal (bell-shaped) distribution. The 68-95-99.7 rule (empirical rule) applies:
About 68% of observations fall within 1 standard deviation of the mean.
About 95% fall within 2 standard deviations.
About 99.7% fall within 3 standard deviations.
This rule is an approximation. It works well only when the distribution is close to normal.
The explanatory variable (x) is the variable you think explains or predicts changes in another variable.
The response variable (y) is the outcome you are trying to explain or predict.
Example: hours studied (x) and exam score (y). Hours studied explains (in part) the exam score.
A scatterplot shows each individual as a point on an x-y plane. When examining a scatterplot, describe:
Direction: Positive association (both go up together) or negative association (one goes up while the other goes down).
Form: Linear or curved? Is there a clear pattern?
Strength: How tightly do the points follow the form? A tight cluster = strong; a loose cloud = weak.
Outliers: Points that deviate from the overall pattern.
Measures the strength and direction of a linear relationship between two quantitative variables.
Always between -1 and +1.
r is unitless: it does not change if you convert units (e.g. centimetres to inches).
r treats x and y symmetrically: the correlation of x with y equals the correlation of y with x.
r is sensitive to outliers. A single extreme point can dramatically raise or lower the value.
The least-squares regression line is:
y-hat = a + bx
where:
b = r(sy / sx), the slope. It tells you the predicted change in y for a one-unit increase in x.
a = y-bar - b(x-bar), the intercept. It is the predicted value of y when x = 0 (which may not be meaningful).
Key properties:
The line always passes through the point (x-bar, y-bar).
r-squared tells you what fraction of the variation in y the regression accounts for.
Residuals = y - y-hat. A residual plot (residuals vs. x) should show no pattern if the linear model is appropriate.
Extrapolation: Do not predict y for x-values far outside the range of your data. The linear pattern may not hold.
Lurking variables: An apparent association between x and y may be driven by a third variable not in the model.
Correlation does not imply causation: Even a strong correlation does not prove that x causes y. Only a well-designed experiment can establish causation.
Influential observations: Points far from the mean of x can have a disproportionate effect on the slope and intercept. Always check what happens when you remove them.
Sample mean: x-bar = (1/n) x sum of all xi
Sample variance: s-squared = (1/(n - 1)) x sum of (xi - x-bar) squared
Sample standard deviation: s = square root of s-squared
Correlation: r = (1/(n - 1)) x sum of [(xi - x-bar)/sx] x [(yi - y-bar)/sy]
where sx and sy are the standard deviations of x and y respectively. Each term in the sum is the product of the z-scores for that observation.
Regression slope: b = r(sy / sx)
Regression intercept: a = y-bar - b(x-bar)
Residual: residual = observed y - predicted y = y - y-hat
1.5 x IQR rule for outliers: An observation is a suspected outlier if it lies below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR).
Descriptive statistics and regression are the backbone of applied economics. When economists report that "average household income rose by 3%," they are using the mean. When they note that "the median house price is more informative because the distribution is right-skewed," they are choosing centre measures wisely. Regression lines power everything from predicting GDP growth based on leading indicators to estimating the relationship between education spending and test scores. Every data-driven news article you read rests on the ideas in these two chapters.
Students often think the mean is always the best measure of centre. It is not. For skewed distributions or datasets with outliers, the median is more representative.
Students confuse standard deviation with variance. Standard deviation is the square root of variance and shares the units of the original data. Variance is in squared units.
Students assume that a strong correlation proves causation. It does not. Correlation measures linear association only; causation requires a controlled experiment or careful causal reasoning.
Students forget that r measures only linear relationships. Two variables can have a strong curved relationship and yet r near zero.
Students treat r-squared and r as if they are the same thing. If r = 0.7, then r-squared = 0.49, not 0.7. Nearly half the variation in y is unexplained.
⚠️ You will be asked to describe a distribution. Always mention shape, centre, spread, and outliers. Missing any one of those costs marks.
⚠️ Know when to use mean vs. median and standard deviation vs. IQR. Skewed data or outliers = use median and IQR.
⚠️ Be able to interpret the slope of a regression line in context: "For each additional unit of x, y is predicted to change by b units."
⚠️ Know how to read and interpret a residual plot. A pattern (curve, fan shape) means the linear model is not appropriate.
⚠️ Be ready to compute r-squared from r and explain what it means: the percentage of variation in y explained by the linear model.
True or false: The median is always equal to the mean in a symmetric distribution. Answer: True (approximately, for a perfectly symmetric distribution).
Fill in the blank: r-squared = 0.81 means that ___% of the variation in y is explained by the regression on x. Answer: 81%.
True or false: If the correlation between x and y is r = -0.95, the relationship is weak. Answer: False. r = -0.95 is a strong negative linear relationship.
Fill in the blank: The slope of the regression line equals r multiplied by ___. Answer: sy / sx (the ratio of the standard deviations).
True or false: A residual plot that shows a clear curve suggests the linear model fits well. Answer: False. A curved residual plot indicates the linear model is not appropriate.
Q: A dataset of household incomes in a city is strongly right-skewed. Which measure of centre and which measure of spread should you report, and why?
A: Report the median and the IQR. The mean and standard deviation are pulled upward by the long right tail (a few very high incomes), making them poor summaries of the typical household.
Q: The correlation between hours of TV watched per week and GPA among university students is r = -0.4. A student concludes that watching TV causes lower grades. Is this conclusion justified?
A: No. Correlation does not imply causation. There may be lurking variables (e.g. time management, course load) that influence both TV hours and GPA. A controlled experiment would be needed to establish a causal link.
Q: A regression of exam score (y) on hours studied (x) gives y-hat = 42 + 5.3x with r-squared = 0.64. Interpret the slope and r-squared in context.
A: The slope means that for each additional hour of study, the predicted exam score increases by 5.3 points. r-squared = 0.64 means that 64% of the variation in exam scores among these students is explained by the linear relationship with hours studied.
Q: You compute the regression of y on x and find a large positive residual for one observation. What does that tell you?
A: The observed y-value for that individual was much higher than the regression model predicted. The model underestimated that observation.
Q: If you switch which variable is x and which is y, does the correlation change? Does the regression line change?
A: The correlation r stays exactly the same. The regression line does change: the slope, intercept, and predictions will be different because regression treats x and y asymmetrically (it minimises vertical distances to the line, not horizontal).
These chapters connect directly to Chapter 5 (Sampling Distributions), where you will see that the distribution of sample means is approximately normal regardless of the population shape, provided n is large enough. The regression ideas from Chapter 2 reappear in Chapter 7 when you build confidence intervals and test hypotheses about the slope. Understanding residuals and the distinction between correlation and causation will also matter when you encounter Chapter 3 (Producing Data), which explains how study design determines what conclusions you can draw.
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