9.2 Data Displays & the Shape of Distributions

9.2 Data Displays & the Shape of Distributions

Skills 2 and 3 of Competency 4 are the reading-and-choosing half of statistics: given a display, extract information; given a data type, choose a display.

Matching data type to display

+---------------------------------------------------------------------------+
|  CATEGORICAL data (colors, brands, yes/no)                                |
|     Bar chart        bars SEPARATED by gaps; order is arbitrary           |
|     Circle graph     parts of a whole; requires 100% coverage             |
|     Frequency table  counts per category                                  |
|                                                                           |
|  NUMERICAL data (heights, scores, times)                                  |
|     Histogram        bars TOUCH; x-axis is a number line of intervals     |
|     Dot / line plot  each value a dot; good for small data sets           |
|     Box plot         five-number summary; good for comparing groups       |
|     Stem-and-leaf    preserves every original value                       |
+---------------------------------------------------------------------------+

[!IMPORTANT] Bar chart versus histogram is the most-tested distinction. A bar chart shows categories and its bars have gaps because the categories are not adjacent on any scale. A histogram shows numerical intervals and its bars touch because the intervals are continuous. Reordering a bar chart's bars is harmless; reordering a histogram's bars destroys it.

A circle graph is appropriate only when the categories are mutually exclusive parts of a single whole. Data on "which sports do you play," where students may play several, cannot be shown in a circle graph, because the parts would exceed 100%.

A stem-and-leaf plot is the one display that keeps every original value visible while still showing shape, which makes it useful for finding a median directly from the display.

Reading a box plot

A box plot is built from the five-number summary: minimum, Q1, median, Q3, maximum.

+---------------------------------------------------------------------------+
|      |-----------[======|========]-----------------|                      |
|     min          Q1    med      Q3                max                     |
|                                                                           |
|      <- 25% -><- 25% -><- 25% -><--------- 25% --------->                 |
|                                                                           |
|      BOX = interquartile range = middle 50% of the data                   |
|      WHISKERS reach to min and max (or to the outlier fences)             |
+---------------------------------------------------------------------------+

[!WARNING] Each of the four sections contains about 25% of the data, no matter how wide or narrow it looks. A long right whisker does not mean more values are out there; it means the values that are out there are spread far apart. Confusing width with count is the single most common box plot misreading, and it appears as a distractor on nearly every box plot item.

What a box plot does show: median, quartiles, IQR, range, skew direction, and outliers. What it does not show: the mean, the mode, the sample size, or whether the distribution has more than one peak. A histogram of the same data can look bimodal while its box plot looks perfectly ordinary.

Box plots are best used to compare groups, since several can be drawn on a shared axis.

The shape of a distribution

Skill 2 names the vocabulary explicitly: symmetry, gaps, clusters, outliers, mode, and range.

Symmetric. The left and right halves mirror each other. The mean and median are approximately equal. A bell-shaped distribution is the familiar symmetric case.

Right-skewed (positively skewed). A long tail extends to the right, with most data bunched left.

• mean > median, because the tail pulls the mean up
• Typical examples: incomes, house prices, response times

Left-skewed (negatively skewed). A long tail extends to the left, most data bunched right.

• mean < median
• Typical example: scores on an easy test, where most students cluster near the top

The skew is named for the direction of the tail, not where the bulk of the data sits. A pile of data on the left with a tail stretching right is right-skewed. Reversing this is a classic error.

Clusters are groups of values concentrated together. Gaps are intervals with no data. A distribution with two clear peaks is bimodal, which often signals two distinct subgroups combined in one data set — for example, heights of a mixed group of children and adults.

Outliers are values far from the rest, formally identified with the 1.5 × IQR fences from section 9.1. When an item asks whether to remove an outlier, the answer depends on cause: a recording error can be corrected or removed, but a legitimate extreme value should be retained and reported.

Building and reading a histogram

Choose intervals (bins) of equal width that cover the full range without overlapping. Bin boundaries must be unambiguous — an interval scheme of 10–20 and 20–30 leaves the value 20 undefined, so use 10–19 and 20–29, or state that intervals include the left endpoint only.

Test scores: 62, 68, 71, 74, 74, 79, 83, 85, 88, 91, 95, 96 Bins of width 10 give: 60–69 → 2, 70–79 → 4, 80–89 → 3, 90–99 → 3.

Bin width changes the apparent shape: very wide bins hide structure, and very narrow bins make the data look ragged. Items ask what happens to a histogram's appearance when bin width changes, and the answer is that shape detail is lost as bins widen.

From a histogram you can read the modal interval (the tallest bar), estimate the range, and judge shape. You generally cannot read exact individual values, since they have been absorbed into intervals — which is precisely the trade-off against a dot plot or stem-and-leaf display.

Misleading displays

A recurring item asks what makes a graph misleading. The usual culprits:

• A vertical axis that does not start at zero, exaggerating small differences in a bar chart
• Unequal interval widths on a histogram
• Inconsistent scales when comparing two graphs side by side
• Pictographs with scaled images, where doubling both the height and width of an icon quadruples its apparent area for a doubling of value

Each of these preserves technically correct numbers while distorting the visual message, which is what makes them a genuine interpretation skill rather than a computation.