Interpreting Data Displays
Read and interpret histograms, box-and-whisker plots, dot plots, and scatterplots. Describe distributions using shape, center, spread, and notable features. Compare two distributions displayed side by side.
A lesson of Algebra 1, in Data Analysis & Statistics, about 35 minutes.
By the end of this lesson, you will be able to read a dot plot, histogram, box plot and scatterplot, describe a distribution in the four standard terms, compare two distributions, and spot a graph designed to mislead.
Four Displays, Four Jobs
Each display shows something the others hide, so the choice is a real decision rather than a matter of taste.
Dot plot. One dot per value. Shows every individual data point, which makes it ideal for small sets and useless past about fifty values.
Histogram. Bars over intervals, showing how many fall in each. Shows the SHAPE of a large set, at the cost of hiding the individual values inside each bar.
Box plot. Five numbers only: minimum, $Q_1$, median, $Q_3$, maximum. Shows spread and skew compactly, and is the best display for comparing several groups at once. It cannot show gaps or a double peak at all.
Scatterplot. One point per individual, plotted on two variables. The only one of the four that shows a RELATIONSHIP rather than a single distribution.
Histogram bars touch; bar-chart bars do not. That is not decoration: touching bars mean the horizontal axis is a continuous number line, while separated bars mean the categories have no order or spacing.
Sort It: Which Display
The last two are the pair to remember. A box plot shows skew and hides peaks; a histogram shows peaks. A double-peaked distribution and a flat one can produce identical box plots.
Describing a Distribution: Four Things
Every description of a single-variable display should cover all four, in this order.
Shape. Symmetric, skewed right, skewed left, uniform, or bimodal.
Center. Roughly where the middle is.
Spread. How wide the values run.
Unusual features. Outliers, gaps, clusters.
Skew is named for the TAIL, not the bulk. Right-skewed means the long tail points right, so most of the data sits on the LEFT with a few large values stretching out. That naming catches nearly everyone at first.
$$\text{right-skewed} \Rightarrow \text{mean} > \text{median}$$
$$\text{left-skewed} \Rightarrow \text{mean} < \text{median}$$
$$\text{symmetric} \Rightarrow \text{mean} \approx \text{median}$$
The mean chases the tail because it uses the SIZE of the extreme values, exactly as in the previous lesson. So comparing mean and median tells you the skew even without a picture.
Example 1: Describing a Histogram
<!-- SLATE_IF_STRONG: math.problem-solving-data.data-interpretation --> <!-- END_SLATE_IF_STRONG -->Heights of bars, left to right: $2, 5, 12, 18, 14, 6, 2, 1$ over intervals from $0$ to $80$.