Descriptive Statistics
Unit 3 of Algebra I. 14 questions below, each with the working. Every answer was checked by a second pass before it was published.
Dot plots, histograms and box plots, centre and spread, two-way tables, scatterplots and linear fits, residuals, correlation versus causation.
How this unit is tested
Start by sorting problems into two families: single-variable (dot plots, histograms, box plots, center and spread) and two-variable (two-way tables for categorical data, scatterplots and regression for numerical data). For single-variable questions, always order the data first — nearly every calculation (median, quartiles, IQR) depends on a correctly sorted list, and this is where most arithmetic errors start.
When a question asks you to choose or compare measures of center and spread, look at the shape of the distribution first. Symmetric data with no outliers: use mean and standard deviation. Skewed data or data with outliers: use median and IQR, since these resist distortion by extreme values. A box plot with a long whisker or asymmetric box is your visual cue that the data are skewed.
For two-way tables, decide whether the question wants a marginal frequency (a row or column total divided by the grand total), a joint frequency (one specific cell divided by the grand total), or a conditional relative frequency (one cell divided by its row or column total, not the grand total). Misreading which total to divide by is the single most common error here.
For scatterplot problems, describe direction (positive, negative, none), form (linear or not), and strength (how tightly points cluster) before touching the regression equation. Once you have a line of best fit ŷ = mx + b, use it to predict, compute residuals as observed minus predicted, and read residual plots for randomness. Finally, whenever a strong correlation appears, ask whether a controlled experiment (supports causation) or an observational study (does not) produced the data, and consider lurking variables.
What you have to know
Five-Number Summary
The five-number summary of a data set is (minimum, Q1, median, Q3, maximum). A box plot is drawn from these five values: the box spans Q1 to Q3 with a line at the median, and whiskers extend to the min and max (excluding outliers).
Interquartile Range and Outlier Rule
IQR = Q3 − Q1. A data value is considered an outlier if it is below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.
Resistance to Outliers
The median and IQR are resistant (little affected) by outliers or skew; the mean and standard deviation are not resistant, since every value enters their calculation directly.
Line of Best Fit and Residual
A line of best fit has the form $\hat{y} = mx + b$, chosen to make the residuals as small as possible overall. For any data point, $\text{residual} = y_{observed} - \hat{y}_{predicted}$. A residual plot with points scattered randomly around zero indicates a linear model is appropriate; a curved pattern indicates it is not.
Correlation Coefficient
The correlation coefficient r measures the strength and direction of a linear relationship between two numerical variables, with $-1 \le r \le 1$. Values near ±1 indicate a strong linear relationship, values near 0 indicate a weak or no linear relationship, and the sign of r matches the sign of the slope of the line of best fit.
Correlation vs. Causation
A strong correlation between two variables does not by itself prove that one causes the other; an observed association may be due to a lurking (confounding) variable or coincidence. Causation can only be established through a well-designed controlled experiment.
14 practice questions
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A data set of six values is 5, 6, 7, 8, 9, 50. Which measure of center best describes a typical value, and why?
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Answer. The median (7.5), because the mean (about 14.2) is pulled upward by the outlier 50 and no longer represents a typical value.
Mean = (5+6+7+8+9+50)/6 ≈ 14.17, far above every value except the outlier. Median = (7+8)/2 = 7.5, which sits right in the middle of the ordinary values, so it better represents the typical data point when an outlier is present. -
Find the interquartile range of the data set: 3, 7, 8, 10, 12, 13, 14, 18, 20, 22.
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Answer. IQR = 10
With 10 values, the lower half is 3,7,8,10,12 (Q1 = median = 8) and the upper half is 13,14,18,20,22 (Q3 = median = 18). IQR = Q3 − Q1 = 18 − 8 = 10. -
A data set is 2, 4, 5, 6, 7, 8, 9, 10, 30. Using the 1.5×IQR rule, is 30 an outlier?
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Answer. Yes, 30 is an outlier.
Lower half 2,4,5,6 gives Q1 = 4.5; upper half 8,9,10,30 gives Q3 = 9.5; IQR = 5. Upper fence = 9.5 + 1.5(5) = 17. Since 30 > 17, it is classified as an outlier. -
Two box plots for class test scores have the same median but Class A's box and whiskers are much wider than Class B's. What does this tell you?
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Answer. Class A's scores are more spread out (more variable) than Class B's, even though both classes have the same middle score.
Box plot width reflects spread (IQR and range), not frequency. A wider box/whisker span means the data cover a larger range of values, indicating greater variability in Class A. -
A survey of 120 students gives this two-way table: Males who like sports = 45, males who dislike = 15 (total 60); females who like sports = 25, females who dislike = 35 (total 60). What percent of females like sports?
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Answer. About 41.7%
This is a conditional relative frequency: restrict to the female row (total 60) and divide the 'likes sports' count by that row total: 25/60 ≈ 0.417 = 41.7%. -
Using the same 120-student survey table (males: 45 like/15 dislike; females: 25 like/35 dislike), what percent of ALL students are males who dislike sports?
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Answer. 12.5%
This is a joint relative frequency, so divide the single cell by the grand total: 15/120 = 0.125 = 12.5%, not by the male row total. -
A scatterplot shows that as hours spent studying increase, test scores also tend to increase, with points clustered fairly tightly around an upward line. Describe the correlation.
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Answer. A strong positive linear correlation.
The pattern is positive because both variables increase together, and it is described as strong because the points lie close to a line rather than being widely scattered. -
A line of best fit for hours studied (x) and test score (y) is ŷ = 3.2x + 65. Predict the test score for a student who studies 10 hours.
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Answer. 97
Substitute x = 10 into the equation: ŷ = 3.2(10) + 65 = 32 + 65 = 97. -
Using the line ŷ = 3.2x + 65, a student who studied 10 hours actually scored 92. Find the residual for this data point.
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Answer. −5
Residual = observed − predicted = 92 − 97 = −5. The negative sign means the actual score was 5 points below what the line predicted. -
A residual plot for a fitted line shows points forming a clear U-shaped curve rather than a random scatter around zero. What does this indicate?
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Answer. A linear model is not appropriate for the data; the relationship is likely curved, not linear.
Random scatter around zero in a residual plot supports a good linear fit. A systematic pattern, like a U-shape, means the line is missing a curved trend in the data, so a linear model underfits it. -
A study reports a correlation coefficient of r = −0.85 between a car's age and its resale value. Interpret this value.
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Answer. There is a strong negative linear relationship: as a car's age increases, its resale value tends to decrease, and the points cluster fairly closely around a downward line.
The sign (negative) tells the direction of the relationship, and the magnitude (0.85, close to 1) tells the strength: values near ±1 indicate a strong linear association. -
Data show that towns with more fire trucks dispatched to a fire also tend to have more property damage, with a high positive correlation. Does this mean fire trucks cause more damage? Explain.
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Answer. No — this is a classic correlation-without-causation example. The lurking variable is the size/severity of the fire: bigger fires cause both more trucks to be sent and more damage.
Correlation only shows association, not cause. Here a third variable (fire size) drives both the number of trucks dispatched and the amount of damage, creating the appearance of a direct relationship between trucks and damage. -
Two data sets have the same mean of 10. Set A: 10, 10, 10, 10, 10. Set B: 5, 8, 10, 12, 15. Which set has the larger standard deviation, and why?
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Answer. Set B has the larger standard deviation.
Standard deviation measures how far values typically lie from the mean. Set A has zero spread (every value equals the mean), while Set B's values are spread out around 10, so its standard deviation is greater than zero. -
A histogram of employee ages shows most bars concentrated at 22–35 years with a long, low tail of bars stretching out to 60–65 years. Describe the shape of this distribution.
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Answer. The distribution is skewed right (positively skewed).
A distribution is skewed in the direction of its longer tail. Here the tail stretches toward the higher ages, so the shape is skewed right, meaning the mean will typically be pulled higher than the median.
What people get wrong
- Using the mean and standard deviation to describe a skewed data set or one with an outlier. Instead, check the shape first (a box plot or dot plot showing asymmetry or an extreme value) and switch to median and IQR when the data are not roughly symmetric.
- Including the overall median in both halves when finding Q1 and Q3 for an odd-sized data set. Instead, split the data into a strict lower half and upper half, excluding the middle value itself, before taking each half's median.
- Computing a residual as predicted minus observed. Instead, always use residual = observed y − predicted y (ŷ), so a positive residual means the actual point lies above the line.
- Reading a two-way table by dividing by the grand total when the question asks for a conditional relative frequency. Instead, identify whether the question restricts attention to one row or one column, and divide only by that row's or column's total.
- Concluding that a high correlation coefficient proves causation. Instead, state only that the variables are strongly associated, and consider whether a lurking variable or reverse causation could explain the pattern.
- Treating a wide box or long whisker on a box plot as automatically meaning 'more data' in that region. Instead, remember a box plot shows spread of values, not frequency — a wide box can contain few or many data points, which a box plot alone cannot show (a histogram would be needed for that).
Drill this unit until it sticks
These questions come back on a schedule built from what you get wrong, alongside the rest of Algebra I. Free, and no account needed to start.