Represent and interpret univariate numerical data using dot plots, histograms, and box plots, and describe the shape (symmetric, skewed left, skewed right) of a distribution (MA.912.DP.1.1, MA.912.DP.1.2).

What this topic is asking

MA.912.DP.1 asks you to read and interpret the three standard displays of one-variable numerical data, the dot plot, the histogram, and the box plot, and to describe the shape of the distribution. On the B.E.S.T. Algebra 1 EOC these are core Statistics points, often multiple choice or interpretation, and they are very gettable.

The three displays

• Dot plot. One dot per data value above a number line. Clusters and gaps are easy to see; best for small data sets.
• Histogram. Data grouped into equal intervals (bins); bar height is the frequency in that bin. Bars touch (the scale is continuous), unlike a bar graph of categories. Best for larger data sets.
• Box plot (box-and-whisker). A visual of the five-number summary: a box from Q1 to Q3 (the middle 50 percent), a line at the median, and whiskers extending to the minimum and maximum.

The five-number summary

A box plot is built from five values:

The median splits the data in half; Q1 is the median of the lower half and Q3 the median of the upper half. The box spans Q1 to Q3 (the interquartile range), holding the middle 50 percent of the data.

Describing the shape

Shape is named for where the long tail points:

• Symmetric: balanced, with the two halves mirror images (a roughly centered median).
• Skewed right (positive): a long tail to the right; most data bunched on the left, median pulled left.
• Skewed left (negative): a long tail to the left; most data bunched on the right, median pulled right.

How the B.E.S.T. EOC examines this topic

• Multiple choice. Identify the shape, or read a value (median, quartile) off a display.
• Number entry. Compute a five-number summary or an interquartile range.
• Matching. Pair a data set with its box plot or histogram.

A clarifying idea: each display answers a different question. A dot plot shows every value, a histogram shows the shape of large data through bins, and a box plot compresses the data into five summary numbers, ideal for comparing groups at a glance.

Why skew is named for the tail

Students often mislabel skew by looking at where the data piles up rather than where the tail stretches, so it helps to see why the tail names it. In a right-skewed distribution, a few unusually large values stretch the right side into a long thin tail, while most of the data clusters at the lower (left) end. Those large outliers pull the mean toward them, so the mean ends up greater than the median, and on a box plot the right whisker is long and the median sits low in the box. The mirror image holds for left skew: a few unusually small values create a long left tail, dragging the mean below the median. Because the defining feature is the direction the rare extreme values stretch the distribution, the skew is named for that tail, not for the crowded end, which is exactly the distinction the EOC tests.

Try this

Q1. For $3, 5, 5, 6, 9$, find the median and Q1. [2 points]

• Cue. Median is the 3rd value, $5$; lower half $3, 5$, so Q1 $= 4$.

Q2. A histogram has most bars on the left and a long tail of short bars to the right. What is the shape? [1 point]

• Cue. Skewed right (the long tail points right).