Categorical and Quantitative
A variable is categorical when it sorts things into groups (eye color, favorite sport) and quantitative when it measures an amount (height, test score). The kind of data decides the kind of picture: bar charts and pie charts for categories; dot plots, histograms and box plots for numbers.
The distinction matters because arithmetic only makes sense for quantitative data. The "average zip code" of a class is a number you can compute and it means nothing. Before summarizing anything, ask: would adding two of these values together make sense? If not, the data is categorical, and counts and percentages are the only honest summaries.
A survey records each student's pet type and the number of hours of sleep. Pet type is categorical: the natural picture is a bar chart of counts, one bar per category, in any order. Hours of sleep is quantitative: a dot plot or histogram, whose horizontal axis is a number line, shows the shape of the distribution — where the values cluster and how far they spread.
In a class of , the pets are: dog , cat , fish , none . Relative frequencies divide by the total: , , , , which add to . For a pie chart, each percentage of : , , , . The bar chart shows the same comparison, and it is easier to read — lengths are judged more accurately than angles.
Treating a number that is really a label as quantitative — jersey numbers, ID numbers, zip codes — or treating a rating like "1 to 5 stars" as if the gaps between stars were equal amounts. A "mean jersey number" of is arithmetic without meaning.
Classify each as categorical or quantitative: (a) blood type, (b) number of siblings, (c) commute time, (d) car color.
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(a) categorical, (b) quantitative (a count — adding siblings makes sense), (c) quantitative (a measured amount), (d) categorical.
Dot Plots and Stem-and-Leaf Plots
For a small quantitative data set, a dot plot puts one dot per value above a number line, stacking repeats. A stem-and-leaf plot splits each value into a stem (leading digits) and a leaf (last digit) and lists the leaves beside their stem. Both keep every data value while showing the shape: center, spread, clusters, gaps and outliers.
These displays are histograms that forgot nothing: the height of each stack or row is a count, so the shape is visible, and yet the actual values can still be read off to compute a median or a range. They stop working once there are hundreds of values; then the histogram, which only keeps counts per bin, takes over. The axis must be an honest number line, with equal spacing for equal differences, or the shape is a lie.
Fifteen quiz scores (out of ): . The plot shows a peak at , most scores from to , and one score of far to the left with a gap before it — a possible outlier. The distribution is skewed LEFT: the long tail points toward low scores. The median is the th value, ; the mean is , pulled slightly below the median by the tail.
Resting pulse rates of people: . Stems are the tens digits, leaves the units:
With the median is the average of the th and th values: . The range is , and the stands apart from the rest.
Leaving values off the axis. A dot plot of pulse rates drawn with only the values that occur — equally spaced — puts and side by side and hides the gap between them. The axis must include the empty stretches; likewise a stem plot keeps empty stems (a stem with no leaves stays in the list).
A stem-and-leaf plot has rows , , . Find the median, the mode and the range.
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The values are (). Median ; mode ; range .
Histograms and Shape
A histogram groups values into equal-width bins and draws a bar whose height is the count in each bin. Look for the shape: symmetric, skewed right (a long tail to the right), or skewed left; one peak or two; any gaps or outliers.
Shape is the first thing to report because it decides which summaries are meaningful later. A skewed distribution — incomes, house prices, waiting times — has a mean pulled toward the tail and a median that stays put, so the two disagree, and the direction of the disagreement tells you the direction of the skew. Unlike a bar chart, the bars of a histogram touch: the bins cover a continuous number line.
Commute times (minutes) for workers are binned –: , –: , –: , –: , –: , –: . How many take minutes or more? of , about . Where is the median? It is between the th and th values; the running totals are , so both lie in the – bin. The histogram cannot say more precisely — binning discarded the exact values.
For the commute data, treat each value as its bin midpoint: minutes. It is an estimate, because the true values within a bin are unknown; it lands to the right of the peak, as the tail on the right predicts.
Skewed right means the long tail is on the right (a few large values). For right-skewed data, . For left-skewed data, .
The same values shown with bins of width look jagged; with width , two bars hide everything. Choose bins so that the shape is visible without being noise — usually to bins.
Unequal bins drawn as if equal. If the commute data merged – into one bin of width with count , drawing that bar at height doubles its visual weight. Area, not height, should show the count, so a bin twice as wide needs half the height. Keep bins equal and the problem never arises.
A histogram of household incomes has most bars on the left and a long thin tail to the right. Which is larger, the mean or the median?
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The distribution is skewed right. The few very high incomes in the tail pull the mean above the median, while the median, which depends only on the middle position, barely moves.
Why It Works
The mean is the balance point of the data: it is the unique value for which the deviations sum to zero. A single far-out value contributes a huge deviation, so the balance point must slide toward it to compensate. The median, defined by position alone, does not feel how far away the extreme value is — only that it is on one side. That is the whole reason the two disagree exactly when a distribution is skewed.