Data Displays

📊 Data Displays

Part 1 of 5 — Dot Plots & Frequency Tables

Topics in This Part

🔑 Key Concept: A data display is a picture of numbers. The whole point is to take a messy list of values and arrange it so you can see the story — which values are common, which are rare, and how spread out everything is.

What Counts as Data?

Data is just a collection of facts you gather, usually numbers. When you ask a question like "How many pets does each student have?" and write down the answers, that list of answers is your data.

Suppose you asked 12 classmates how many pets they own and got:

$2,\; 0,\; 1,\; 3,\; 2,\; 1,\; 0,\; 2,\; 4,\; 1,\; 2,\; 0$

That raw list is hard to read. Two questions are tricky to answer just by staring at it:

• How many students own exactly 2 pets?
• What is the most common number of pets?

To answer those quickly, we organize the data. The first tool is a frequency table.

💡 Frequency is a fancy word for how many times something happens. If three students own 2 pets, the frequency of "2 pets" is 3.

Count From the Raw List 🔽

Look back at the raw pet list: $2,\; 0,\; 1,\; 3,\; 2,\; 1,\; 0,\; 2,\; 4,\; 1,\; 2,\; 0$. Count carefully.

Frequency Tables

A frequency table lists each value once and counts how many times it appears. Here is the pet data organized:

Now the data tells its story instantly:

• 4 students own 2 pets — that's the most common value.
• The frequencies add up to $3 + 3 + 4 + 1 + 1 = 12$, which matches our 12 classmates. ✓

🔑 Always check the total. The frequencies must add up to the number of people (or things) you surveyed. If they don't, you miscounted somewhere.

Concept Check 🎯

Dot Plots (Line Plots)

A dot plot (also called a line plot) shows the same information as a frequency table, but with a stack of dots above each value on a number line. One dot = one data point.

Here is the pet data as a dot plot:

            ●
●     ●     ●
●     ●     ●
●     ●     ●     ●     ●
0     1     2     3     4
   Number of pets

Reading it is easy:

• Count the dots in a column to get that value's frequency.
• The tallest stack is the most common value (here, 2 pets with 4 dots).
• A gap (a value with no dots) means nobody had that amount.

💡 Dot plots shine for small data sets of whole numbers. You can see the shape — where data clumps up and where it thins out — at a single glance.

Read the Dot Plot 🧮

A class recorded how many books each student read last month:

            ●
●           ●     ●
●     ●     ●     ●
●     ●     ●     ●     ●
1     2     3     4     5
     Number of books

1) How many students read exactly 3 books? 2) How many students read 5 books? 3) How many students are in the class in total?

Wrapping Up Part 1

You now have two ways to organize a list of numbers:

Both answer the same questions — what's common, what's rare, how many total — they just look different.

In Part 2 we'll handle data with lots of different values using bar graphs and histograms.

📊 Data Displays

Part 2 of 5 — Bar Graphs & Histograms

🔑 The Big Difference: A bar graph compares separate categories (like favorite colors). A histogram groups numbers into ranges called bins (like test scores 70–79, 80–89). They look similar, but one is for categories and the other is for number ranges.

Bar Graphs

A bar graph uses bars to compare categories. The height (or length) of each bar shows the frequency for that category. Because the categories are separate things, the bars have gaps between them.

Students voted for their favorite school lunch:

Votes
 9 |  █
 6 |  █     █
 3 |  █     █           █
 0 |  █     █     █     █
      Pizza Tacos Salad Pasta

To read a bar graph, line the top of each bar up with the scale on the left.

💡 Categories can be put in any order on a bar graph. Pizza first or pasta first — it doesn't change the data, because the categories aren't numbers on a line.

Concept Check 🎯

Histograms

A histogram looks like a bar graph, but it shows numerical data grouped into equal-width ranges called bins (or intervals). Because the bins are next-door ranges on a number line, the bars touch — no gaps.

A coach recorded how many push-ups each athlete did, then grouped the counts:

Freq
 8 |              █
 6 |        █     █
 4 |        █     █
 2 |  █     █     █     █
 0 |__█_____█_____█_____█__
     0-9  10-19 20-29 30-39

⚠️ A histogram tells you the bin, not the exact values. The "20–29" bar has frequency 8, so 8 athletes did somewhere between 20 and 29 push-ups — but you can't tell from the histogram exactly how many each one did.

Read the Histogram 🧮

Use the push-up histogram above (bins 0–9, 10–19, 20–29, 30–39).

1) How many athletes did 20–29 push-ups? 2) How many athletes did fewer than 20 push-ups? (combine the first two bins) 3) How many athletes were measured in total?

Telling Them Apart

Bar graphs and histograms are easy to mix up. This side-by-side comparison is worth memorizing:

🔑 The fastest tell: look at the bars. Gaps → bar graph. Touching → histogram.

Bar Graph or Histogram? 🔽

For each data set, choose the better display.

📊 Data Displays

Part 3 of 5 — Stem-and-Leaf Plots & Line Graphs

🔑 Why these two? A stem-and-leaf plot keeps every exact value while still showing the shape of the data. A line graph shows how one thing changes over time. Each is the right tool for a different job.

Stem-and-Leaf Plots

A stem-and-leaf plot splits each number into a stem (the left digits) and a leaf (the last digit). For two-digit numbers, the tens digit is the stem and the ones digit is the leaf.

Here are 11 quiz scores:

$62,\; 75,\; 78,\; 81,\; 83,\; 83,\; 90,\; 71,\; 88,\; 95,\; 79$

Sorted into stems (tens) and leaves (ones):

Stem | Leaf
  6  | 2
  7  | 1 5 8 9
  8  | 1 3 3 8
  9  | 0 5

Key: $8\,|\,3$ means 83.

Read it like this:

• The stem 7 with leaves 1 5 8 9 means the scores 71, 75, 78, 79.
• Count the leaves in a row to get that group's frequency — the 70s row has 4 leaves, so 4 scores were in the 70s.

💡 The best of both worlds: unlike a histogram, a stem-and-leaf plot keeps the exact values. You can see 83 appeared twice (two 3's in the 80s row).

Read the Stem-and-Leaf Plot 🧮

Use the quiz-score plot above (key: $8\,|\,3 = 83$).

1) How many scores were in the 80s? 2) What is the lowest score in the whole data set? 3) How many scores were 80 or higher? (combine the 80s and 90s rows)

Line Graphs

A line graph plots points and connects them with line segments to show how a value changes over time. Time goes along the bottom (the x-axis); the measured value goes up the side (the y-axis).

A plant's height was measured each week:

Height
10 |             •___•
 8 |          •
 6 |       •
 4 |
 2 | •
 0 |__1___2___3___4__  Week

The line tells a story about change:

• A line going up means the value is increasing (the plant grew).
• A steeper segment means a faster change. From week 2 to week 3 the plant grew $9 - 5 = 4$ cm — its fastest week.
• A flat segment would mean no change at all.

⚠️ Use a line graph only for data that changes over time (or another ordered scale). Don't use one for categories like favorite colors — connecting "red" to "blue" with a line is meaningless.

Concept Check 🎯

Two Tools, Two Jobs

This part added two more displays to your toolbox. Keep their jobs straight:

💡 A stem-and-leaf plot answers "what were the actual numbers?" A line graph answers "is it going up or down, and how fast?" Choosing well starts with knowing which question you're asking.

Match the Display 🔽

Pick the best display for each goal.

📊 Data Displays

Part 4 of 5 — Shape, Center & Spread

🔑 Reading the Story: Every display has a shape. By looking at where the data clumps and how it stretches out, you can describe its center (a typical value), its spread (how far apart the values are), and any outliers (values far from the rest).

The Shape of Data

Look at a dot plot or histogram from across the room and notice its shape.

A cluster is where data bunches together. A gap is a range with no data. An outlier is a single value sitting far away from everything else.

●
●  ●
●  ●  ●              ●   ← outlier (far from the cluster)
1  2  3  4  5  6  7  8

💡 Outliers grab attention. In the dot plot above, most values cluster at 1–3, then there's a big gap, then one lonely point at 8. That 8 is an outlier — always ask whether it's a real surprise or just a typo.

Concept Check 🎯

Center: Mean, Median & Mode

The center is a single typical value that summarizes the whole set. Three common measures:

• Mean (average): add all the values, then divide by how many there are.
• Median: the middle value when the data is listed in order.
• Mode: the value that appears most often.

Example. Five quiz scores: $4,\; 8,\; 6,\; 10,\; 7$.

First order them: $4,\; 6,\; 7,\; 8,\; 10$.

$\text{Mean} = \frac{4 + 6 + 7 + 8 + 10}{5} = \frac{35}{5} = 7$

$\text{Median} = 7 \quad (\text{the middle of } 4,\, 6,\, \mathbf{7},\, 8,\, 10)$

There is no mode here, because every value appears exactly once.

⚠️ Median needs ordered data first. A very common mistake is grabbing the middle number of the unsorted list. Always sort, then find the middle.

Find the Center 🧮

A basketball player scored these points in 5 games:

$12,\; 8,\; 15,\; 8,\; 7$

1) What is the mean (average) number of points? 2) What is the median? (remember to sort first) 3) What is the mode?

Spread: The Range

Spread describes how far apart the values are. The simplest measure is the range:

$\text{Range} = \text{maximum value} - \text{minimum value}$

For the basketball points $7,\; 8,\; 8,\; 12,\; 15$:

$\text{Range} = 15 - 7 = 8$

A small range means the values are close together (consistent). A large range means they are spread out (more varied).

💡 Center and spread together tell the whole story. "Typical score about 10, ranging from 7 to 15" describes the data far better than either number alone.

Center & Spread 🔽

The data set is $3,\; 5,\; 5,\; 9$. Fill in each summary value.

📊 Data Displays

Part 5 of 5 — Choosing Displays & Mastery Check

You can now build and read dot plots, frequency tables, bar graphs, histograms, stem-and-leaf plots, and line graphs — and describe a data set's shape, center, and spread. The last skill is choosing the right display and reading displays critically.

Choosing the Right Display

🔑 The key question is always: What kind of data is it, and what do I want to show? Categories vs. numbers, and "over time" vs. "all at once," decide nearly every choice.

Concept Check 🎯

Reading Displays Critically

Displays can mislead if you don't read carefully:

• ⚠️ Check the scale. If a bar graph's vertical axis starts at 50 instead of 0, small differences look gigantic.
• ⚠️ Check the bin width. Two histograms of the same data can look very different if their bins are wide or narrow.
• ⚠️ Watch for missing labels. A graph with no axis titles or no key can't be trusted — you don't know what it's measuring.

💡 A smart reader always asks: Where does the scale start? What is each bar or dot counting? Is anything left out? Good data sense is as much about questioning a display as building one.

Choose the Display 🔽

Pick the best display for each task.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.