A dot plot puts one mark above each value; nothing is lost. A histogram groups values into intervals and draws a bar for each count, handling large sets. Skew is named for the direction of the tail. An outlier sits far from the rest.
Step 1: Picture It
Start with something you already know.
Sorting buttons into cups
- A jar holds 40 buttons of many sizes.
- Pour them all into one big cup, and you learn only that there are 40.
- Give every size its own tiny cup, and most cups hold one button or none.
- Line up five cups, smallest sizes to largest, and a pattern appears.
- Most buttons are small, and a few big ones trail off at the end.
A histogram is the row of cups: each bar counts the values that landed in one interval, called a bin.
The bin width is the cup size. Too wide hides the shape, and too narrow scatters it into noise.
Where the picture stops: real cups stand apart, but bins share their edges, each one starting where the last ended, which is why histogram bars touch.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Ten neighbors said how many hours they drove last week: 1, 2, 2, 3, 3, 3, 4, 4, 5 and 12. Picture one dot above each value on a number line from 0 to 12. Which sentence describes the picture?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Building a dot plot
Data 2, 3, 3, 4, 4, 4, 5, 5, 9: one dot at 2, two at 3, three at 4, two at 5, one at 9. The pile peaks at 4 and the 9 stands alone on the right.
Bin width matters
Too few bins hides the shape; too many leaves a jagged mess. Fifty test scores in bins of 10 show a clean hump; in bins of 1 they show noise. Try two widths before trusting a histogram.
Shapes
Symmetric: heights. Skewed right, a long tail of high values: incomes, house prices. Skewed left: exam scores on an easy test, piled near 100 with a tail down. Two peaks: heights of a mixed group of adults and children.
Skew names the tail
Skewed right means the tail stretches right, even though most of the data sits on the left. Ask which way the tail points, not where the pile is.
Outliers
A 9 among 2 to 5 may be a typing error, or the one student with a large family. Investigate before deciding. Removing a real value because it is inconvenient is not analysis.
The one-sentence description
"The distribution of pets is skewed right, centered near 2, spread from 1 to 6, with a possible outlier at 6." Shape, center, spread, outlier, in that order, every time.
Step 3: Try It Yourself
Tap and try it out.
Data: 2, 3, 3, 4, 4, 4, 5, 5, 9. Build the dot plot in words and describe the distribution
- 12: one dot; 3: two; 4: three; 5: two; 9: oneone mark per value
Step 0 of 4
123456
Number of pets16 things were measured in all.
The tallest column is at 1, with 6 marks.
Step 4: In Real Life
Commute times
Twenty commute times as dots show every person. Two hundred, in ten-minute bins, are a histogram. The shape, skewed right by a few long drives, is what a city planner reads first.
Step 5: Watch an Example
One step at a time.
Watch Ana Describe a Quiz
Eleven students took an easy quiz scored out of 10. Their scores were 4, 7, 8, 8, 9, 9, 9, 10, 10, 10 and 10. Ana builds the dot plot and describes it in one sentence.
- Step 1
Each score gets one dot above its value. Counting from the list gives one dot at 4, one at 7, two at 8, three at 9 and four at 10. The values 5 and 6 get no dots at all.
45678910Quiz score out of 10The tallest column is at 10, with 4 marks.
Score 4Score 5Score 6Score 7Score 8Score 9Score 10
Watch Leo Read a Histogram
Twenty students timed their trip to school, rounded to the nearest minute. Leo has the histogram, in bins 10 minutes wide. The bars, from the shortest times to the longest, are: 0–9 minutes, 6 students; 10–19 minutes, 8; 20–29 minutes, 3; 30–39 minutes, 2; 40–49 minutes, 1.
- Step 1
Each bar counts the students whose time landed in its bin, and the bars touch because time runs on without gaps. The counts 6 + 8 + 3 + 2 + 1 make 20, one for every student. What the bars do not keep is any single time: a student in the 10–19 bin might have taken 10 minutes or 19.
Step 6: Your Turn
Practice makes it stick.
The Lunch Orders
Problem 1 of 15
A food truck took 15 lunch orders, in dollars: 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 12, 13 and 31. The owner draws a histogram with bins $5 wide: $5 to $9, $10 to $14, and so on up to $30 to $34. How many of those bins are empty?
bins
The Late Buses
Problem 2 of 15
A dot plot shows how many minutes late 20 buses were: 7 dots at 0, 5 at 1, 4 at 2, 2 at 3, 1 at 4 and 1 at 15. From the lowest value to the highest, how many minutes does the spread cover?
minutes
Read the Plot
Count from the dots or the bars, then say what the count means.
Problem 3 of 15
The dot plot shows how many siblings each of 15 students has. How many of the students have 2 or more siblings?
students
Problem 4 of 15
A histogram’s bars, from the lowest bin to the highest, are 1, 2, 3, 6 and 9. Which way is it skewed?
Problem 5 of 15
Nine phone calls lasted 6, 14, 5, 3, 7, 6, 8, 5 and 6 minutes. Taking the center as the middle value in order, what is it, in minutes?
minutes
Problem 6 of 15
Test scores run from 41 to 98. A histogram uses bins 10 points wide, starting at 40: 40–49, 50–59, and so on. How many bins does it need to hold every score?
bins
Problem 7 of 15
A histogram of the heights of everyone at a family picnic has bars of 4, 7, 2, 1, 3, 8 and 5, from the shortest bin to the tallest. How many peaks does it have?
peaks
Whose Step?
Each page has one line that slipped. Find it, then put it right.
Problem 8 of 15
A histogram has bars of 9, 6, 3, 2 and 1, from the lowest bin to the highest. Dev described its shape. One line is a slip. Which?
Dev’s working: bars 9, 6, 3, 2, 1 from the lowest bin to the highest
- 1counts from low to high: 9, 6, 3, 2, 1
- 2the tallest bars are at the low end
- 3the bars shrink toward the high end
- Answerthe pile is on the left, so skewed left
Problem 9 of 15
Which sentence should Dev’s last line say about the bars 9, 6, 3, 2, 1?
Problem 10 of 15
Ten scores, 58, 63, 66, 70, 71, 75, 78, 84, 89 and 92, go into bins 50–59, 60–69, 70–79, 80–89 and 90–99. Maya counted each bin. One line is a slip. Which?
Maya’s working: ten scores into bins 10 wide
- 150–59: 58, so 1
- 260–69: 63, 66, 70, so 3
- 370–79: 71, 75, 78, so 3
- 480–89: 84, 89, so 2
- Answer90–99: 92, so 1
Problem 11 of 15
With every one of Maya’s ten scores in its proper bin, how many scores does the 70–79 bin hold?
scores
Pick the Move
Which display, or which bin width, shows the data best? No counting needed.
Problem 12 of 15
A shipping center has 2,000 package weights to show. Which display suits them?
Problem 13 of 15
Sixty delivery times run from 10 to 69 minutes. Which bin width is most likely to show the shape?
Same Story?
Judge which descriptions tell the same story, without counting anything new.
Problem 14 of 15
Bars of 8, 5, 3, 2, 1, from the lowest bin to the highest, make a pile on the left and a tail to the right. Select every histogram below that is skewed the same way.
Problem 15 of 15
A dot plot and a histogram are drawn from the same 12 values. Select every statement that is true.
Step 7: Quick Check
Show what you know.
Question 1 of 2
Counts 9, 4, 2, 1. Skewed which way?
Question 2 of 2
Why does bin width matter in a histogram?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Nine whole-number values have a middle value in order of 4, and their spread runs from 1 to 20. What is the smallest the nine values can add up to?
Stretch 2 of 2
Ten values make a symmetric dot plot with a spread from 2 to 8. Five of them are 2, 3, 3, 4 and 5. What do the other five add up to?
What You Learned
- A dot plot keeps every value; a histogram groups them into bins.
- Skew is named for the direction of the tail, not the pile.
- Investigate an outlier before deciding what to do with it.
- Describe shape, center, spread and outliers, in one sentence.
For the grown-up
The misconception this lesson targets is naming the skew after the pile instead of the tail. A student sees most of the data on the left and says "skewed left", because the eye goes to where the data is. The name comes from the thin tail, the few extreme values that stretch the picture, so a pile on the left with a tail to the right is skewed right. Ask "which way does the tail point?" every time, and have the reason said aloud, not only the word. A second, quieter misconception is that a histogram shows the data the way a dot plot does: it shows only counts per bin, and the same data can look quite different under a different bin width.