Data Displays — I can represent numerical data with dot...

SECTION 1

Where Did Data Displays Come From?

People have been collecting numbers for thousands of years. Ancient civilizations counted crops, people, and animals. But a long list of numbers is hard to understand. Imagine staring at 200 numbers and trying to spot a pattern!

Over time, mathematicians invented pictures — called data displays — to help us see what the numbers are telling us. Each type of display was created to solve a different problem. Let's look at how three of our most useful displays were born.

1786

William Playfair Invents Bar Charts

Scottish engineer William Playfair drew some of the first bar charts and line graphs. His work showed that pictures of data are much easier to read than tables of numbers.

1832

Early Histograms Appear

French mathematician André-Michel Guerry used grouped bar charts — early versions of histograms — to study crime data in France.

1884

Dot Plots Gain Popularity

Simple dot plots became a quick way for scientists and teachers to show every single data point on a number line.

1970

John Tukey Introduces the Box Plot

American statistician John Tukey created the box plot (also called a box-and-whisker plot). It became a favorite tool for summarizing data with just five numbers.

Each of these displays answers a different question. A dot plot asks, "Where does each data point fall?" A histogram asks, "How many values land in each group?" A box plot asks, "How is the data spread out?" The big question for you is: which display should I choose for my data?

SECTION 2

Core Principles of Data Displays

Before we build any chart, let's nail down three key ideas. These principles will help you understand why each display works the way it does.

Dot Plot

A dot plot places one dot above a number line for every data value. It shows each individual value and makes it easy to spot clusters (groups of nearby values), gaps, and outliers (values far from the rest).

Histogram

A histogram groups data into equal-width ranges called bins (sometimes called intervals). The height of each bar shows how many values fall in that bin. It's great for larger data sets.

Box Plot

A box plot uses five numbers — minimum, Q1, median, Q3, and maximum — to show how data is spread out. The "box" covers the middle 50% of the data, and the "whiskers" extend to the smallest and largest values.

Choosing the Right Display

The best display depends on your data and your question. Small data sets? Try a dot plot. Large data sets with many values? A histogram. Want to compare groups or show spread? Use a box plot.

✦ KEY TAKEAWAY

Think of data displays like camera lenses. A dot plot is a close-up — you see every single detail. A histogram is a medium shot — you see the overall shape. A box plot is a wide-angle — you see how far everything stretches.

SECTION 3

Seeing All Three Displays Side by Side

Let's use the same data set — quiz scores for 20 students — and build all three displays. The scores are: 60, 65, 68, 70, 72, 72, 75, 75, 75, 78, 80, 80, 82, 85, 85, 88, 90, 92, 95, 98. Notice how each display reveals something different about the same numbers.

This dot plot shows quiz scores for 20 students. Each cyan dot represents one student's score. Where dots stack up, more students got that score. The score of 75 appears three times — the tallest stack.

In the dot plot above, you can see every single score. The three dots stacked at 75 tell you that 75 is the mode (the most common value). You can also spot a gap between 60 and 65 — no one scored 61, 62, 63, or 64.

💡 When to use a dot plot

A dot plot works best when you have a small data set (roughly 30 values or fewer). If you have 200 scores, the number line would be covered in dots and hard to read!

SECTION 4

The Math Behind Each Display

Building data displays involves a few key calculations. Let's walk through the math you need for histograms and box plots. Dot plots don't need much math — you just plot each value!

Histogram: Choosing Bins

To make a histogram, you group your data into bins of equal width. First, find the range. Then decide how many bins you want (usually 4 to 8 for middle school data). Finally, divide.

RANGE

Range = Maximum value − Minimum value

For our quiz data: Range = 98 − 60 = 38

BIN WIDTH

Bin width = Range ÷ Number of bins

If we choose 4 bins: 38 ÷ 4 = 9.5, so we round up to a bin width of 10.

Box Plot: The Five-Number Summary

A box plot is built from five key numbers. These five numbers are called the five-number summary.

FIVE-NUMBER SUMMARY

Minimum, Q₁, Median, Q₃, Maximum

Q₁ (first quartile) is the median of the lower half. Q₃ (third quartile) is the median of the upper half. The IQR (interquartile range) = Q₃ − Q₁. It measures the spread of the middle 50%.

INTERQUARTILE RANGE

IQR = Q₃ − Q₁

A larger IQR means the middle half of the data is more spread out. A smaller IQR means values are bunched together.

SECTION 5

Histogram and Box Plot — A Closer Look

Now let's see the same 20 quiz scores as a histogram and a box plot. Each display highlights different features of the data.

The histogram (left) groups scores into four bins. The tallest bar is 70–79, with 7 students. The box plot (right) shows the five-number summary. The pink line inside the box is the median (79). The box stretches from Q₁ = 72 to Q₃ = 87.

Look at the histogram first. The tallest bar is the 70–79 bin, which means that range holds the most scores. The histogram shows the shape of the data — it's roughly symmetric with a slight lean to the left.

Now look at the box plot. The box covers Q₁ to Q₃ — the middle 50% of scores. The left whisker is longer than the right whisker. This tells us the lower scores are more spread out. The box plot is great for comparing center (median = 79) and spread (IQR = 15).

SECTION 6

Worked Example — Building All Three Displays

Here is a new data set: the number of books 15 students read over the summer. 1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 8, 10, 12, 15. Let's build each display step by step.

Building a Dot Plot, Histogram, and Box Plot

Step 1 — Order the DataThe data is already in order from least to greatest: 1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 8, 10, 12, 15. There are 15 values.

n = 15; Minimum = 1; Maximum = 15

Step 2 — Dot PlotDraw a number line from 1 to 15. Place one dot above the number line for each value. The value 5 gets three dots stacked because it appears three times. The values 3 and 7 each get two dots.

Mode = 5 (three dots). Cluster around 3–7. Gap between 8 and 10.

Step 3 — Histogram (Bins of Width 3)Range = 15 − 1 = 14. Let's use 5 bins of width 3: [1–3], [4–6], [7–9], [10–12], [13–15]. Count values in each bin: [1–3] → 4 values; [4–6] → 4 values; [7–9] → 3 values; [10–12] → 2 values; [13–15] → 2 values. Wait — check: that gives us only 15. We wrote 15 twice, but 13–15 has only the value 15, so that bin has 1 value. Let's recount. [1–3]: 1, 2, 3, 3 = 4. [4–6]: 4, 5, 5, 5 = 4. [7–9]: 6, 7, 7, 8 = 4 — wait, 6 is in [4–6]? No: 6 is in [4–6] if our bins are 1–3, 4–6, 7–9. Let's use 6 in the [4–6] bin. So [4–6]: 4, 5, 5, 5, 6 = 5. [7–9]: 7, 7, 8 = 3. [10–12]: 10, 12 = 2. [13–15]: 15 = 1. Draw bars with those heights.

Bins: [1–3] = 4, [4–6] = 5, [7–9] = 3, [10–12] = 2, [13–15] = 1. Tallest bar is [4–6].

Step 4 — Five-Number Summary for the Box PlotMinimum = 1. Maximum = 15. The median is the middle value. With 15 values, the median is the 8th value: 5. The lower half is the first 7 values: 1, 2, 3, 3, 4, 5, 5. Q₁ is the median of these 7 values, which is the 4th value: 3. The upper half is the last 7 values: 6, 7, 7, 8, 10, 12, 15. Q₃ is the median of these 7 values, which is the 4th value: 8.

Five-number summary: Min = 1, Q₁ = 3, Median = 5, Q₃ = 8, Max = 15. IQR = 8 − 3 = 5.

Step 5 — Choose the Best DisplayWith only 15 values, a dot plot shows every point clearly. The histogram also works but hides individual values. The box plot is handy if you want to compare this data to another class. For this small data set, the dot plot is probably the best first choice.

Best display for 15 data points: dot plot. Use a histogram or box plot when you have more data or need to compare groups.

SECTION 7

Strengths and Limitations of Each Display

No single display is perfect for every situation. Each one has strengths and trade-offs. The table below will help you pick the right tool for the job.

Comparison of dot plots, histograms, and box plots

Feature	Dot Plot	Histogram	Box Plot
Shows individual values	Yes ✓	No ✗	No ✗
Shows shape of data	Yes ✓	Yes ✓	Partially
Shows median & quartiles	No ✗	No ✗	Yes ✓
Good for large data sets	No — gets crowded	Yes ✓	Yes ✓
Easy to compare groups	Hard	Moderate	Best ✓
Best data set size	Small (< 30)	Medium to large	Any size

✦ KEY TAKEAWAY

Imagine you're packing for a trip. A dot plot is like listing every single item you'll bring — great for a short trip, overwhelming for a month-long vacation. A histogram is like sorting items into categories (shirts, pants, shoes) — you see the big picture. A box plot is like a packing summary that says "suitcase weight ranges from 10 to 50 pounds, middle half is 20–35 pounds."

SECTION 8

Connecting to More Advanced Ideas

The displays you learned here are the foundation for more advanced work in high school and beyond. Here's a peek at what's coming.

From middle school to high school statistics

What You Know Now	What Comes Next
Dot plot shows each value	Scatter plots show how two variables relate to each other
Histogram shows shape of data	Normal distributions (bell curves) describe the shape mathematically
Box plot shows median & IQR	Standard deviation gives a single number for spread; used in AP Statistics
Choosing a display by hand	Software tools (Excel, Desmos, Python) create displays automatically

In high school, you'll also learn about outliers in more detail. A common rule is that any value more than 1.5 × IQR below Q₁ or above Q₃ is considered an outlier. You already have the tools to start spotting them!

🌟 Fun Fact

John Tukey, the inventor of the box plot, also coined the word "software" in 1958! He was a mathematician who loved finding simple ways to understand complex data.

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. Check your answers after each one!

PROBLEM 1 — CONCEPTUAL

A teacher recorded the shoe sizes of 12 students: 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10, 11. Which type of display — dot plot, histogram, or box plot — would best show how many students share the same shoe size? Explain your reasoning.

PROBLEM 2 — BASIC CALCULATION

Use this data set of test scores: 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100. Find the five-number summary (minimum, Q₁, median, Q₃, maximum) that you would need to draw a box plot.

PROBLEM 3 — INTERMEDIATE

A restaurant tracked the number of customers each hour for 20 hours: 12, 15, 18, 20, 22, 22, 25, 25, 28, 30, 30, 32, 35, 38, 40, 42, 45, 50, 55, 60. Create a histogram using bins of width 10, starting at 10. List the bins and the frequency (count) for each bin.

PROBLEM 4 — APPLIED

Two soccer teams tracked the number of goals scored per game over a season. Team A: Min = 0, Q₁ = 1, Median = 2, Q₃ = 3, Max = 5. Team B: Min = 0, Q₁ = 0, Median = 1, Q₃ = 4, Max = 7. Which team is more consistent? Which team has a higher typical score? Use the five-number summaries to justify your answers.

PROBLEM 5 — CRITICAL THINKING

A student says, "A histogram always gives you more information than a box plot, so there's never a reason to use a box plot." Do you agree or disagree? Give at least two reasons to support your argument, and describe a situation where a box plot would be a better choice.

SUMMARY

Lesson Summary

You learned three ways to display numerical data. A dot plot places one dot per value on a number line, showing every data point — best for small data sets where you want to see clusters, gaps, and outliers. A histogram groups values into equal-width bins and uses bar heights to show frequency — ideal for medium to large data sets when you want to see the overall shape. A box plot uses the five-number summary (minimum, Q₁, median, Q₃, maximum) to show center and spread — perfect for comparing groups.

To choose the right display, ask yourself: Do I need to see every value? Use a dot plot. Do I want to see the shape of a large data set? Use a histogram. Do I want to compare the median and IQR across groups? Use a box plot. Remember: the best display depends on your data and your question!