Represent data with dot plots, histograms, and box plots, and interpret the shape of a distribution (NC.M1.S-ID.1, S-ID.3).

What this topic is asking

NC.M1.S-ID.1 asks you to represent data on a single count or measurement variable using dot plots, histograms, and box plots, and interpret them in context. NC.M1.S-ID.3 asks you to interpret the shape of a distribution, symmetry, skew, and outliers. This is about displaying one-variable data and reading what the display says.

The three displays

Each display suits different data.

The five-number summary and the box plot

A box plot is built from five values.

Describing shape

S-ID.3 asks for shape in words.

• Symmetric: the left and right halves roughly mirror each other (a balanced histogram).
• Skewed right: a long tail toward high values; most data is on the low side.
• Skewed left: a long tail toward low values; most data is on the high side.
• Outliers: values far from the rest, which stretch the range and can mislead the mean.

Choosing the right display

The displays are not interchangeable, and a question may ask which one suits a purpose. A dot plot shows every individual value, so it is ideal for small data sets where you want to see each point and spot repeats. A histogram groups data into intervals, so it is ideal for large data sets where individual values would clutter the picture and you care about the overall shape. A box plot hides individual values but cleanly displays the five-number summary and is the best choice for comparing two or more groups side by side, since the boxes line up on one axis. The same data can be shown three ways, and matching the display to the question, count of points, size of the set, or a comparison, is itself an EOC skill.

How the NC Math 1 EOC examines this topic

• Multiple choice. Identify the shape, read a value from a display, or match a data set to a plot.
• Gridded response. Compute the range or IQR from a five-number summary.
• Technology-enhanced. Build a box plot or match displays to summaries.

Reading a display sets up comparing center and spread, where shape decides which measures to use, and contrasts with the two-variable work in scatter plots.

Why shape decides everything else

The shape of a distribution is not just a description; it guides which summary statistics to trust. In a symmetric distribution, the mean and median nearly coincide, so the mean is a fair center. In a skewed distribution, the long tail drags the mean toward it, so the median is the more honest center and the IQR the more honest spread. Outliers exaggerate this effect. That is why S-ID.3 comes before choosing statistics: you read the shape first, then pick measures that the shape will not distort. A box plot's whiskers and a histogram's tail both reveal skew at a glance, which is what makes these displays so useful for one-variable data.

Try this

Q1. A five-number summary is $3, 6, 9, 14, 20$. Find the IQR. [1 point]

• Cue. $Q_3 - Q_1 = 14 - 6 = 8$.

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

• Cue. Skewed left (long tail toward low values).