Represent one-variable quantitative data with dot plots, histograms, and box plots, and describe the shape of a distribution (A.DSR, Data and Statistical Reasoning).

What this topic is asking

This Data and Statistical Reasoning (A.DSR) standard, part of the statistics connection (about 15 percent of the EOC), asks you to display one-variable quantitative data with the three standard plots, dot plots, histograms, and box plots, and to describe the shape of a distribution. The Georgia Milestones EOC tests reading values from these displays (especially the five-number summary from a box plot), matching a display to a description, and identifying shape (symmetric, skewed, outliers). It is the foundation for comparing distributions, the next topic.

The three displays

Each display suits a different situation and emphasizes a different feature.

• Dot plot. One dot per data value above a number line. Best for small data sets; shows every value and clusters or gaps directly.
• Histogram. Bars over intervals of values, with bar height showing frequency. Best for larger data sets; shows the overall shape but not individual values.
• Box plot (box-and-whisker). A box from $Q_1$ to $Q_3$ with the median marked, and whiskers to the minimum and maximum. Best for showing spread and center and for comparing groups.

The five-number summary and a box plot

A box plot is built from the five-number summary: minimum, first quartile $Q_1$, median, third quartile $Q_3$, and maximum.

Two spread measures come straight from it: the range (maximum minus minimum) and the interquartile range (IQR) $= Q_3 - Q_1$, the spread of the middle 50 percent.

Describing shape

The shape of a distribution is a frequent one-point item.

• Symmetric: the two halves roughly mirror each other (a balanced histogram or dot plot).
• Skewed right (positively skewed): a long tail to the right, most data on the left.
• Skewed left (negatively skewed): a long tail to the left, most data on the right.
• Outliers: values far from the rest, which stretch the range and pull the mean.

The skew is named for the direction the tail points, not where the bulk of the data sits, which is the usual confusion.

How the Milestones examines this topic

• Numeric entry. Find the IQR, range, or a quartile from a box plot or data set.
• Multiple choice. Identify the shape of a distribution, or match a display to a description.
• Drag and drop. Order the five-number summary, or match plots to data sets.

Why the box plot hides individual values but shows spread

Each display trades detail for clarity in a different way, and knowing the trade-off tells you which to read for a given question. A box plot deliberately compresses a whole data set into five numbers, so you cannot see individual data points or how many values there are, but you can see the center (median), the spread (IQR and range), and the shape (where the median sits in the box, how long each whisker is). That makes a box plot ideal for comparing two groups at a glance, which is the next topic, but useless if a question asks for the exact value of one observation. A dot plot is the opposite: every value is visible, so it answers "how many scored exactly 8," but it gets cluttered for large data sets. Matching the display to what the question needs, individual values versus overall spread, is the reading skill the EOC rewards.

Reading skew from center measures

Shape and center interact in a way the EOC likes to probe. In a symmetric distribution the mean and median are about equal. In a distribution skewed right, the long right tail pulls the mean above the median, because the mean is sensitive to large outlying values while the median is not. In a distribution skewed left, the mean is pulled below the median. So if you are told the mean exceeds the median, you can infer right skew without seeing the graph. This connection between shape and the relationship of mean to median is exactly why the next topic prefers the median and IQR for skewed data, since those resist the distorting pull of a long tail.

Try this

Q1. A box plot has $Q_1 = 15$, $Q_3 = 27$. Find the IQR. [1 point]

• Cue. $27 - 15 = 12$.

Q2. A histogram has most bars on the right and a long tail to the left. Describe the shape. [1 point]

• Cue. Skewed left (the tail points left).