1Which of the following is an example of categorical (qualitative) data?
types of data
Easy
A.Temperature in Celsius
B.Height of a person in cm
C.Number of cars sold
D.Blood type of a patient
Correct Answer: Blood type of a patient
Explanation:
Blood type (A, B, AB, O) represents categories with no inherent numeric value, making it categorical data. The others are numeric (quantitative).
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2Data that can only take specific separate values, such as the number of students in a class, is called:
types of data
Easy
A.Ordinal data
B.Continuous data
C.Discrete data
D.Nominal data
Correct Answer: Discrete data
Explanation:
Discrete data consists of countable, separate values (e.g., 20 or 21 students), unlike continuous data which can take any value in a range.
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3A ranking of customer satisfaction as poor, average, good, excellent is an example of which data type?
types of data
Easy
A.Nominal data
B.Interval data
C.Ratio data
D.Ordinal data
Correct Answer: Ordinal data
Explanation:
Ordinal data has categories with a meaningful order or ranking, but the differences between them are not measurable.
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4Which scale of measurement has a true zero point?
types of data
Easy
A.Interval scale
B.Ordinal scale
C.Ratio scale
D.Nominal scale
Correct Answer: Ratio scale
Explanation:
The ratio scale has a true zero (e.g., weight, height), which allows meaningful ratios like "twice as heavy".
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5Replacing a missing numeric value with the average of the available values is known as:
dealing with missing data
Easy
A.Encoding
B.Normalization
C.Mean imputation
D.Deletion
Correct Answer: Mean imputation
Explanation:
Mean imputation fills missing values using the mean of the observed values for that variable, a simple and common technique.
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6Removing all rows that contain any missing values is referred to as:
dealing with missing data
Easy
A.Mean imputation
B.Listwise deletion
C.Standardization
D.Interpolation
Correct Answer: Listwise deletion
Explanation:
Listwise deletion (complete-case analysis) discards entire rows with any missing values, which is simple but may lose useful data.
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7In many datasets, missing values are commonly represented by which of the following?
dealing with missing data
Easy
A.Median
B.Mean
C.NaN
D.Mode
Correct Answer: NaN
Explanation:
NaN (Not a Number) is a standard placeholder used to indicate missing or undefined values in datasets.
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8Which technique estimates a missing value using neighboring known data points, often in a sequence?
dealing with missing data
Easy
A.Deletion
B.Binning
C.Interpolation
D.One-hot encoding
Correct Answer: Interpolation
Explanation:
Interpolation estimates missing values based on surrounding known values, useful for ordered or time-series data.
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9A scatter plot is primarily used to show the relationship between:
scatter plot
Easy
A.A single category
B.Two numeric variables
C.Frequencies of a category
D.One variable over time only
Correct Answer: Two numeric variables
Explanation:
A scatter plot displays points for pairs of values, revealing correlation or patterns between two numeric variables.
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10In a scatter plot, if points rise together from lower-left to upper-right, the relationship is:
scatter plot
Easy
A.Perfectly random
B.No correlation
C.Positive correlation
D.Negative correlation
Correct Answer: Positive correlation
Explanation:
An upward-sloping pattern from lower-left to upper-right indicates a positive correlation, where both variables increase together.
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11Each individual point in a scatter plot represents:
scatter plot
Easy
A.A category label
B.The mean of the dataset
C.One observation with two values
D.A single frequency count
Correct Answer: One observation with two values
Explanation:
Every point corresponds to one observation, plotted using its value and value on the two axes.
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12A histogram is mainly used to display the:
histogram
Easy
A.Relationship between two variables
B.Change of trend over regions
C.Ranking of categories
D.Distribution of a numeric variable
Correct Answer: Distribution of a numeric variable
Explanation:
A histogram groups numeric data into intervals (bins) and shows how frequently values fall into each bin, revealing the distribution.
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13The intervals into which data is grouped in a histogram are called:
histogram
Easy
A.Nodes
B.Points
C.Cells
D.Bins
Correct Answer: Bins
Explanation:
Bins are the consecutive, non-overlapping intervals used to group data values in a histogram.
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14How does a histogram differ from a bar chart?
histogram
Easy
A.Histograms only show categories
B.Histograms cannot show frequencies
C.Histogram bars are for continuous data and touch
D.Histogram bars always have gaps
Correct Answer: Histogram bars are for continuous data and touch
Explanation:
Histograms display continuous numeric data with adjacent (touching) bars, while bar charts show categorical data with gaps between bars.
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15Group plots are most useful for:
group plots
Easy
A.Showing a single value only
B.Reducing dimensions
C.Encoding missing data
D.Comparing data across multiple groups
Correct Answer: Comparing data across multiple groups
Explanation:
Group plots arrange multiple subplots or grouped visuals side by side to compare distributions or trends across categories.
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16A grouped bar chart is typically used to compare:
group plots
Easy
A.A single continuous distribution
B.Correlation between two variables
C.Values across categories and subgroups
D.Missing value counts only
Correct Answer: Values across categories and subgroups
Explanation:
Grouped bar charts place bars for subgroups next to each other within each category, enabling side-by-side comparisons.
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17The line inside the box of a box plot represents the:
box plots
Easy
A.Median
B.Mean
C.Maximum
D.Range
Correct Answer: Median
Explanation:
The central line in a box plot marks the median (the middle value) of the data.
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18In a box plot, points that fall far outside the whiskers are typically identified as:
box plots
Easy
A.Medians
B.Outliers
C.Quartiles
D.Modes
Correct Answer: Outliers
Explanation:
Points beyond the whiskers are shown as outliers, indicating values that are unusually high or low compared to the rest.
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19The height of the box in a box plot represents the:
box plots
Easy
A.Total range
B.Standard deviation
C.Interquartile range (IQR)
D.Mean value
Correct Answer: Interquartile range (IQR)
Explanation:
The box spans from the first quartile () to the third quartile (), representing the interquartile range where the middle 50% of data lies.
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20What is the main goal of dimensionality reduction?
dimensionality reduction
Easy
A.Delete all missing values
B.Reduce the number of features while keeping key information
C.Increase the number of features
D.Sort data alphabetically
Correct Answer: Reduce the number of features while keeping key information
Explanation:
Dimensionality reduction lowers the number of variables (features) while retaining as much important information as possible, simplifying analysis.
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21A dataset records customer satisfaction as Poor, Average, Good, and Excellent. Which data type best describes this variable?
types of data
Medium
A.Continuous numerical
B.Nominal categorical
C.Discrete numerical
D.Ordinal categorical
Correct Answer: Ordinal categorical
Explanation:
The categories have a meaningful order (Poor < Average < Good < Excellent) but no fixed numeric distance between them, which defines an ordinal variable.
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22Temperature measured in degrees Celsius is an example of which measurement scale?
types of data
Medium
A.Ordinal scale
B.Nominal scale
C.Interval scale
D.Ratio scale
Correct Answer: Interval scale
Explanation:
Celsius has equal intervals but no true zero (0°C does not mean 'no temperature'), so ratios are not meaningful. This makes it an interval scale.
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23You encode the number of children per household as . How should this variable be classified?
types of data
Medium
A.Discrete numerical
B.Continuous numerical
C.Nominal categorical
D.Ordinal categorical
Correct Answer: Discrete numerical
Explanation:
The count takes only whole-number values with no intermediate values possible, which is the defining property of discrete numerical data.
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24A numerical column has a strongly right-skewed distribution with several outliers, and about 5% of values are missing. Which imputation choice is most appropriate?
dealing with missing data
Medium
A.Impute with the median
B.Impute with the mean
C.Delete all rows with any missing value
D.Replace missing values with zero
Correct Answer: Impute with the median
Explanation:
The median is robust to skew and outliers, so it represents the central tendency better than the mean, which would be pulled toward extreme values.
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25Values are missing more often for low-income respondents who chose not to report income. This missingness mechanism is best described as:
dealing with missing data
Medium
A.Structurally missing
B.Missing Not At Random (MNAR)
C.Missing At Random (MAR)
D.Missing Completely At Random (MCAR)
Correct Answer: Missing Not At Random (MNAR)
Explanation:
The probability of being missing depends on the unobserved value itself (income level), which characterizes MNAR and cannot be ignored safely.
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26A feature is missing in 78% of records and is weakly related to the target. What is usually the most sensible action?
dealing with missing data
Medium
A.Impute with the column mean
B.Fill all missing entries with the mode
C.Impute using KNN on the few present values
D.Drop the entire column
Correct Answer: Drop the entire column
Explanation:
With such a high missing rate and low relevance, imputation would introduce more noise than signal, so removing the feature is generally the safest choice.
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27A scatter plot of two variables shows points tightly clustered along a line that falls from top-left to bottom-right. What does this indicate?
scatter plot
Medium
A.No correlation
B.Strong negative correlation
C.Strong positive correlation
D.A perfect nonlinear relationship
Correct Answer: Strong negative correlation
Explanation:
A downward-sloping, tightly clustered pattern means one variable decreases as the other increases, indicating a strong negative linear correlation.
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28You want to visualize the relationship between two numeric variables while also encoding a third numeric variable. Which scatter plot enhancement is most appropriate?
scatter plot
Medium
A.Increase the marker transparency uniformly
B.Vary the marker size (bubble chart)
C.Add horizontal grid lines
D.Sort the points along the x-axis
Correct Answer: Vary the marker size (bubble chart)
Explanation:
Mapping the third variable to marker size (a bubble chart) lets you show three numeric dimensions on a 2D scatter plot at once.
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29In a large scatter plot, many points overlap and hide the true density of data. Which technique best addresses this overplotting?
scatter plot
Medium
A.Reduce the axis range
B.Change the marker color to red
C.Remove the axis labels
D.Use transparency (alpha blending)
Correct Answer: Use transparency (alpha blending)
Explanation:
Alpha blending makes dense regions appear darker as overlapping points accumulate, revealing the underlying density that solid markers would hide.
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30A histogram built with too many narrow bins appears jagged and noisy. What is the primary effect of increasing the bin width?
histogram
Medium
A.It changes the total count of observations
B.It converts the histogram into a box plot
C.It smooths the distribution but may hide fine detail
D.It always reveals more detail in the data
Correct Answer: It smooths the distribution but may hide fine detail
Explanation:
Wider bins aggregate more values per bar, smoothing the shape, but excessive width can mask meaningful structure like small peaks.
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31A histogram shows most values bunched on the left with a long tail extending to the right. How is this distribution described?
histogram
Medium
A.Left-skewed (negatively skewed)
B.Right-skewed (positively skewed)
C.Uniform
D.Symmetric
Correct Answer: Right-skewed (positively skewed)
Explanation:
A long tail toward higher values on the right indicates positive (right) skew, where the mean is typically greater than the median.
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32Which statement correctly distinguishes a histogram from a bar chart?
histogram
Medium
A.A histogram requires sorted data, while a bar chart does not
B.A histogram displays continuous data in bins, while a bar chart displays categorical counts
C.A histogram can only show two variables, while a bar chart shows one
D.A histogram always has gaps between bars, unlike a bar chart
Correct Answer: A histogram displays continuous data in bins, while a bar chart displays categorical counts
Explanation:
Histograms group a continuous variable into adjacent bins, whereas bar charts compare counts across distinct categories.
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33You want to compare the distribution of test scores across five different schools on a single figure. Which grouped visualization is most suitable?
group plots
Medium
A.Side-by-side box plots grouped by school
B.A single scatter plot of all scores
C.A pie chart of average scores
D.A single histogram of all scores combined
Correct Answer: Side-by-side box plots grouped by school
Explanation:
Grouped box plots place one box per school side by side, making it easy to compare medians, spread, and outliers across groups.
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34A faceted (small-multiples) plot arranges several subplots that share the same axes. What is its main advantage?
group plots
Medium
A.It combines all groups into one aggregated curve
B.It reduces the total number of data points shown
C.It automatically removes outliers from each panel
D.It allows direct visual comparison across subgroups on identical scales
Correct Answer: It allows direct visual comparison across subgroups on identical scales
Explanation:
Shared axes across facets ensure each subgroup is plotted on the same scale, enabling fair, direct comparison between panels.
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35When using a grouped bar chart to compare two categories across several months, what does color (hue) typically encode?
group plots
Medium
A.The order of the months on the x-axis
B.The total sum across all months
C.The confidence interval of each bar
D.The category being compared within each month
Correct Answer: The category being compared within each month
Explanation:
In grouped bar charts, hue distinguishes the subcategories within each x-axis group, so color represents the compared category.
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36In a standard box plot, the length of the box itself represents which quantity?
box plots
Medium
A.The full range (max min)
B.The mean absolute deviation
C.The interquartile range ()
D.The standard deviation
Correct Answer: The interquartile range ()
Explanation:
The box spans from the first quartile to the third quartile , so its length equals the interquartile range (IQR).
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37Using the common rule, a point is flagged as an outlier if it lies:
box plots
Medium
A.More than two bin widths from the mode
B.Below or above
C.Below the median by more than one standard deviation
D.Outside the range to
Correct Answer: Below or above
Explanation:
The Tukey rule marks values beyond past either quartile as outliers, which is how box plot whiskers and points are determined.
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38A box plot has , median , and . What does the position of the median inside the box suggest?
box plots
Medium
A.The data has no variability
B.The data is left-skewed
C.The data is perfectly symmetric
D.The data is right-skewed
Correct Answer: The data is right-skewed
Explanation:
The median is closer to than to (5 vs 15 units), meaning the upper half is more spread out, which indicates right skew.
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39In Principal Component Analysis (PCA), the first principal component is chosen so that it:
dimensionality reduction
Medium
A.Minimizes the number of features in the dataset
B.Aligns exactly with the original first feature axis
C.Maximizes the correlation between original features
D.Captures the direction of maximum variance in the data
Correct Answer: Captures the direction of maximum variance in the data
Explanation:
PCA finds orthogonal components ordered by variance; the first component is the direction along which the data varies the most.
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40Why is it generally recommended to standardize features before applying PCA?
dimensionality reduction
Medium
A.To increase the number of principal components
B.To prevent features with larger scales from dominating the components
C.To guarantee the data becomes perfectly normal
D.To remove all correlations before analysis
Correct Answer: To prevent features with larger scales from dominating the components
Explanation:
PCA is variance-based, so unscaled features with large ranges would dominate. Standardizing puts features on comparable scales for a fair analysis.
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41A researcher encodes customer satisfaction as where the gaps between adjacent levels are not guaranteed to be equal. A colleague computes the arithmetic mean and standard deviation of this variable. What is the most defensible critique?
types of data
Hard
A.The variable is ratio, so a geometric mean should have been used instead
B.The variable is ordinal, so the mean and SD assume interval spacing that may not hold
C.The variable is interval, so only the mode is a valid summary statistic
D.The variable is nominal, so even the median is undefined for it
Correct Answer: The variable is ordinal, so the mean and SD assume interval spacing that may not hold
Explanation:
Satisfaction ratings are ordinal: order is meaningful but distances between levels are not necessarily equal. Means and SDs presuppose interval-level equal spacing, so they can be misleading. The median or mode are safer summaries.
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42Temperature in Celsius and temperature in Kelvin describe the same physical quantity. Which statement correctly distinguishes their measurement scales?
types of data
Hard
A.Both are ratio scales, so ratios are meaningful in either scale
B.Both are interval scales, so ratios are meaningless in either scale
C.Celsius is ratio and Kelvin is interval, so ratios are meaningful only in Celsius
D.Celsius is interval and Kelvin is ratio, so ratios are meaningful only in Kelvin
Correct Answer: Celsius is interval and Kelvin is ratio, so ratios are meaningful only in Kelvin
Explanation:
Kelvin has a true zero (absence of thermal energy), making it a ratio scale where is twice as hot as . Celsius has an arbitrary zero, making it interval, so is not twice .
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43In a dataset, income is missing more often for high earners because wealthy respondents decline to answer. The missingness depends on the unobserved income value itself. This mechanism is best classified as:
dealing with missing data
Hard
A.MNAR (Missing Not At Random)
B.MAR (Missing At Random)
C.MCAR (Missing Completely At Random)
D.Ignorable missingness under listwise deletion
Correct Answer: MNAR (Missing Not At Random)
Explanation:
When the probability of missingness depends on the missing value itself (income), the mechanism is MNAR. This is non-ignorable and cannot be fixed by standard imputation without modeling the missingness process.
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44A data scientist replaces all missing values in a feature with the column mean before computing the sample variance. What is the primary distortion introduced?
dealing with missing data
Hard
A.The correlation with other variables increases artificially
B.The variance is systematically overestimated because imputed values inflate the range
C.The variance is systematically underestimated because imputed values add no spread
D.The mean shifts toward the imputed values, biasing central tendency
Correct Answer: The variance is systematically underestimated because imputed values add no spread
Explanation:
Mean imputation places all missing points exactly at the mean, contributing zero to the sum of squared deviations. This shrinks the estimated variance and standard error, understating true variability.
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45Compared to single imputation, why does multiple imputation with imputed datasets produce more valid standard errors?
dealing with missing data
Hard
A.It always reduces total variance by averaging out random noise across datasets
B.It replaces missing values with the most frequent observed value times
C.It guarantees unbiased estimates even under MNAR mechanisms
D.It incorporates between-imputation variance, reflecting uncertainty about the missing values
Correct Answer: It incorporates between-imputation variance, reflecting uncertainty about the missing values
Explanation:
Multiple imputation combines within-imputation and between-imputation variance (Rubin's rules). The between-imputation term captures the uncertainty inherent in not knowing the true missing values, which single imputation ignores.
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46Two variables have Pearson correlation , yet a scatter plot shows a clear parabolic pattern. What is the correct interpretation?
scatter plot
Hard
A.A log transform of both axes would restore a high linear
B.A strong nonlinear relationship exists that Pearson's fails to capture
C.The variables are statistically independent since
D.The scatter plot must contain a data entry error
Correct Answer: A strong nonlinear relationship exists that Pearson's fails to capture
Explanation:
Pearson's measures only linear association. A symmetric parabola can yield while showing a strong deterministic nonlinear relationship. Zero correlation does not imply independence.
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47When plotting 500,000 points where many overlap, a scatter plot becomes a solid ink blob (overplotting). Which technique best preserves density information?
scatter plot
Hard
A.Use a 2D density/hexbin plot mapping counts to color intensity
B.Connect the points with lines sorted by x-value
C.Plot only the first 100 rows to reduce clutter
D.Increase the marker size so points are easier to see individually
Correct Answer: Use a 2D density/hexbin plot mapping counts to color intensity
Explanation:
With massive overplotting, individual markers convey no density. Hexbin or 2D kernel-density plots aggregate points into bins whose color encodes count, revealing the underlying distribution that a saturated scatter hides.
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48A histogram of the same data looks unimodal with 10 bins but bimodal with 40 bins. What does this most likely indicate?
histogram
Hard
A.The 10-bin version is always correct because fewer bins reduce error
B.Bin width strongly affects the visual, and the extra modes may be noise or genuine structure needing further checks
C.Histograms are unreliable and a pie chart should be used instead
D.The data are definitively bimodal since more bins reveal more truth
Correct Answer: Bin width strongly affects the visual, and the extra modes may be noise or genuine structure needing further checks
Explanation:
Histogram shape is sensitive to bin width. Too few bins over-smooth and hide structure; too many introduce sampling noise. The apparent modes must be validated (e.g., with a KDE or different bandwidths) before concluding bimodality.
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49Using the Freedman–Diaconis rule, bin width is . For a sample with and , what is the approximate bin width?
histogram
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
, so . The Freedman–Diaconis rule uses the IQR to be robust to outliers.
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50A right-skewed variable is plotted as a histogram, then re-plotted after a transform. What effect is most expected in the transformed histogram?
histogram
Hard
A.The left tail is stretched, increasing the skew further
B.The number of modes always increases after the transform
C.The distribution becomes exactly uniform across all bins
D.The right tail is compressed, making the distribution appear more symmetric
Correct Answer: The right tail is compressed, making the distribution appear more symmetric
Explanation:
A log transform compresses large values more than small ones, pulling in a long right tail. For positively skewed positive data this often yields a more symmetric, near-normal shape.
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51You facet (small-multiple) a scatter plot by a categorical grouping variable across 6 panels. What is the key advantage of a shared common axis range across all panels?
group plots
Hard
A.It removes the need to label the axes on individual panels
B.It forces every group to display the same number of points
C.It guarantees each panel uses the maximum possible resolution
D.It allows direct visual comparison of magnitudes and positions across groups
Correct Answer: It allows direct visual comparison of magnitudes and positions across groups
Explanation:
Shared (fixed) axes make positions and scales comparable across panels, so differences between groups are read directly. Free/independent axes maximize per-panel detail but make cross-group comparison misleading.
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52A grouped (clustered) bar chart shows category means for three groups but omits any uncertainty indicators. What is the most serious analytical limitation?
group plots
Hard
A.Bar charts cannot represent means, only counts, so the plot is invalid
B.Clustered bars always exaggerate the largest group's value
C.Differences between group means cannot be judged for significance without variability information
D.The chart implies a time ordering that does not exist
Correct Answer: Differences between group means cannot be judged for significance without variability information
Explanation:
Means alone hide spread and sample size. Without error bars or confidence intervals, viewers cannot assess whether apparent group differences are meaningful or within sampling noise.
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53Using the standard rule with and , at what upper value does a point begin to be flagged as an outlier?
box plots
Hard
A.Above
B.Above
C.Above
D.Above
Correct Answer: Above
Explanation:
. Upper fence . Points strictly above 70 are flagged as outliers.
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54Two box plots have identical medians, quartiles, and whisker lengths, yet their underlying distributions differ dramatically (one bimodal, one unimodal). What limitation does this reveal?
box plots
Hard
A.The two box plots must actually be identical if all summaries match
B.Box plots summarize quantiles and cannot reveal multimodality or internal shape
C.Box plots always misrepresent the median for bimodal data
D.Box plots overstate variability when data are bimodal
Correct Answer: Box plots summarize quantiles and cannot reveal multimodality or internal shape
Explanation:
Box plots encode only five-number summaries. Distributions with equal quantiles but different internal shapes (e.g., bimodal vs. unimodal) look identical. A violin plot or histogram is needed to reveal modality.
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55A notched box plot is used to compare two groups. If the notches of the two boxes do not overlap, the most appropriate conclusion is:
box plots
Hard
A.The two distributions must both be perfectly symmetric
B.There is roughly 95% confidence evidence that the true medians differ
C.The two groups have identical medians with certainty
D.The variances of the two groups are proven to be equal
Correct Answer: There is roughly 95% confidence evidence that the true medians differ
Explanation:
Notches approximate a 95% confidence interval for the median. Non-overlapping notches provide informal evidence that the medians differ significantly; they say nothing about variance or symmetry.
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56Before applying PCA to features measured in wildly different units (e.g., meters and kilograms), why is standardization typically required?
dimensionality reduction
Hard
A.PCA maximizes variance, so unstandardized high-variance features dominate the components
B.Standardization makes the covariance matrix singular, simplifying the math
C.PCA requires all features to be strictly positive to compute eigenvalues
D.Without it, PCA cannot handle more than two features at once
Correct Answer: PCA maximizes variance, so unstandardized high-variance features dominate the components
Explanation:
PCA finds directions of maximum variance. Features with large numeric scales have large variance purely due to units and will dominate the principal components. Standardizing (or using the correlation matrix) puts features on comparable footing.
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57The eigenvalues of a covariance matrix are . What proportion of total variance is explained by the first two principal components?
dimensionality reduction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Total variance . First two components explain . Proportion .
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58A key difference between PCA and t-SNE for visualization is that:
C.t-SNE preserves local neighborhood structure but distorts global distances and cluster sizes
D.PCA preserves local structure while t-SNE preserves global linear structure
Correct Answer: t-SNE preserves local neighborhood structure but distorts global distances and cluster sizes
Explanation:
t-SNE is nonlinear and emphasizes preserving local neighborhoods, so between-cluster distances and cluster sizes in a t-SNE plot are not reliable. PCA is linear and preserves global variance directions.
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59When choosing the number of components via a scree plot, the 'elbow' heuristic suggests keeping components:
dimensionality reduction
Hard
A.Up to the point where the eigenvalue drop levels off sharply
B.All components until cumulative variance reaches 100%
C.Only those with eigenvalues exactly equal to zero
D.The single component with the smallest eigenvalue
Correct Answer: Up to the point where the eigenvalue drop levels off sharply
Explanation:
The scree plot shows eigenvalues in descending order. The 'elbow' is where the curve flattens, indicating that additional components add little variance. Components before the elbow are retained.
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60In a scatter plot of study hours vs. exam score, adding a third variable (course difficulty) reverses the apparent positive trend within each course subgroup. This is an example of:
scatter plot
Hard
A.Overplotting due to too many data points
B.Heteroscedasticity in the residual variance
C.A ceiling effect on the exam score variable
D.Simpson's paradox, where aggregated and subgroup trends conflict
Correct Answer: Simpson's paradox, where aggregated and subgroup trends conflict
Explanation:
Simpson's paradox occurs when a trend present in aggregated data reverses or disappears within subgroups defined by a confounding variable. Coloring or faceting by course difficulty reveals the true within-group relationship.
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