Represent data with plots on the real number line, including dot plots, histograms, and box plots, and read the five-number summary from a box plot (TN A1.S.ID.A.1).

What this topic is asking

Standard A1.S.ID.A.1 asks you to represent single-variable data with three displays on a number line: dot plots, histograms, and box plots. The graded skills are choosing or reading the right display, and pulling the five-number summary (minimum, $Q_1$, median, $Q_3$, maximum) and the shape from it.

The three displays

• Dot plot. Each data value is a dot above its position on a number line; repeated values stack. Best for small data sets and for seeing clusters and gaps.
• Histogram. Data is grouped into equal-width bins, and a bar's height is the frequency in that bin. Best for large sets; reveals overall shape but hides individual values.
• Box plot (box-and-whisker). Built from the five-number summary. The box runs from $Q_1$ to $Q_3$ with the median marked inside; whiskers extend to the minimum and maximum (or to the last non-outlier).

Reading a box plot

The box plot is the most testable display. From it you read:

• Median (the line in the box): the middle value.
• $Q_1$ and $Q_3$ (the box edges): the 25th and 75th percentiles.
• IQR $= Q_3 - Q_1$: the spread of the middle 50 percent (the box width).
• Range $=$ maximum $-$ minimum: the full spread.

Shape and skew

A histogram or dot plot reveals shape. Symmetric distributions balance around the center. Skewed right has a long tail toward high values; skewed left has a long tail toward low values. The skew is named for the tail, not the peak, which is the most common naming error.

How TNReady examines this topic

• Multiple choice. Compute the IQR or range from a box plot, or name the shape of a histogram.
• Drag and drop. Match a data set to its box plot or histogram.
• Multiple select. Choose all true statements about a display.

A clarifying idea is that the display you choose depends on the question: a box plot is best for comparing centers and spreads of two groups (the center and spread topic), while a histogram is best for seeing overall shape.

Why shape decides which center to report

The shape of a distribution is not just descriptive; it tells you which summary statistics are trustworthy, which is the bridge to the next topic. In a roughly symmetric distribution, the mean and median are close, and the mean is a fair center. In a skewed distribution, the long tail pulls the mean toward it, so the mean overstates a right skew and understates a left skew, while the median stays near the bulk of the data. That is why test scores reported on a skewed distribution often use the median. Reading the shape first, then choosing the median for skewed data and either statistic for symmetric data, is the reasoning A1.S.ID rewards. The box plot helps here too: if the median sits off-center in the box or one whisker is much longer, the data is skewed toward the longer side.

Try this

Q1. A box plot has $Q_1 = 15$ and $Q_3 = 35$. What is the IQR? [1 point]

• Cue. $35 - 15 = 20$.

Q2. A histogram has a long tail stretching to the right. Name the shape. [1 point]

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