D. Data · Data Literacy and Probability

Shape and Distribution: How Changes in Data Change the Picture

D1.5 D1.6About 50 minutes

Learning goals

I can describe the shape of a data set: its centre, spread, clusters, gaps, peaks and outliers.
I can tell whether a distribution is symmetric or skewed (has a long tail on one side).
I can describe how adding or removing data changes the shape and the measures of central tendency.

Words to know

distribution
How the data values are spread out along the number line.

cluster
A group of data values that are close together.

gap
An empty space in the data where there are no values.

peak
The highest stack (or bar) in a dot plot, line plot or histogram. It shows the mode or the most common interval.

symmetric
A distribution whose two sides are mirror images around the centre.

skewed
Lopsided: most of the data is on one side, with a long tail of a few values stretching out to the other side.

range
Greatest value − least value. It measures spread.

Remember

A line plot (dot plot) puts one dot or ✕ above the number line for each value. In Grade 6 you described histograms by their shape. In the last lesson you saw how adding or removing a value changes the mean, median and mode. Today you'll see how it changes the whole picture.

The shape tells a story

Sixteen Grade 7 students recorded how many books they read over the summer. Look at the dot plot below. Each dot is one student.

Sixteen students. One read 2 books, two read 3, three read 4, four read 5, three read 6, two read 7, and one read 8.

Describe a distribution with these questions:

Centre: Where is the middle? Here the mean, median and mode are all 5.
Spread: How spread out is it? The range is 8 − 2 = 6.
Peak: Where is the tallest stack? At 5.
Shape: Is it symmetric (the same on both sides) or skewed (with a long tail on one side)? This one is symmetric. The left side (2, 3, 4) mirrors the right side (6, 7, 8).
Clusters, gaps and outliers: Are values bunched together? Are there empty spaces? Any values far from the rest? Here there is one cluster from 2 to 8, no gaps and no outliers.

In a symmetric distribution, the mean and median are close together.

Worked example

Example 1: Add an outlier

One more student joins the survey. She read 22 books! Describe how the shape and the measures change.

New total: 80 + 22 = 102 books for 17 students. Mean = 102 ÷ 17 = 6. It went up from 5.
Median: the 9th of 17 values. The 9th value is still 5.
Mode: still 5. Range: 22 − 2 = 20. It more than tripled.
Shape: now there is a gap from 9 to 21 and an outlier at 22. The distribution has a long tail to the right, so it is skewed right.
The outlier pulled the mean toward the tail. Now the mean (6) is greater than the median (5).

The distribution changed from symmetric to skewed right. The mean and range grew, but the median and mode stayed at 5.

The same 16 dots plus one added dot at 22. A gap from 9 to 21 separates it from the rest.

Worked example

Example 2: Add a cluster

Start again with the 16 students. This time, 5 members of a reading club join, and each read 12 books. Describe the changes.

21 students. Total: 80 + 5 × 12 = 140. Mean = 140 ÷ 21 ≈ 6.7.
Median: the 11th of 21 values. The first 10 values are 2 to 5; the 11th is 6.
Mode: 12 now appears 5 times, more than 5 (4 times). The mode jumps to 12!
Shape: now there are two clusters (2 to 8, and 12) with a gap between them, and two peaks.
Is the mode a good "centre" now? Not really. 12 is at the edge of the data. With two clusters, it is better to describe each group.

Mean about 6.7, median 6, mode 12. The shape now has two clusters and two peaks. Adding a group of similar values can create a new cluster and even move the mode far away.

The 16 original dots plus a stack of 5 added dots at 12. There is a gap at 9, 10 and 11.

Removing data changes the picture too

Removing values can close a gap, remove an outlier, or shrink the spread. If you remove the reader who read 22 books, the data becomes symmetric again and the mean drops back to 5.

Be careful: don't remove an outlier just because it is inconvenient. Ask why it is there. Was it a mistake (someone typed 22 instead of 2)? Or is it real (a student who really loves reading)? Real outliers are part of the story.

Shapes to know

Symmetric: mirror image; mean ≈ median.
Skewed right: long tail to the right; the mean is pulled above the median.
Skewed left: long tail to the left; the mean is pulled below the median.
Clusters with gaps: two or more groups. Describe each group separately.

Just for fun

Why does the outlier always feel a bit lonely?

It lives way out at the end of the dot plot, far from the cluster!

Hands-on activity

Dice sum dot plot

You need: two dice, paper and a pencil.

Draw a number line from 2 to 12.
Roll both dice and add. Put a dot above the sum. Do this 30 times.
Describe the shape: where is the peak? Is it roughly symmetric? (Sums near 7 usually come up most often, because more pairs of numbers add to 7.)
Find the mean, median and mode.
What if? Pretend you add 6 more rolls that were all 12. Draw them in a different way (for example, as ✕ instead of dots). How does the shape change? Which measures change?

Math talk: "What happens to the mean when you add values on the tail side?"

Use this dot plot for the Your turn questions. Twelve students: 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 12 and 25 minutes.

Your turn

Look at Minutes to Walk to School. Which value is an outlier?

Describe the shape. Is it symmetric or skewed? Which way?

Find the mean, median and mode.

Why is the mean greater than the median here?

Remove the 25. Find the new mean, median and range.

Which measure best describes a "typical" walk for these 12 students, the mean or the median? Why?

More practice: warm-up

What does a gap in a dot plot look like?

What is a cluster?

A dot plot is the same on both sides of the middle. What is this shape called?

A dot plot has a long tail to the right. What is it called?

Data: 3, 4, 5, 6, 7. Find the range.

More practice: on your own

Data: 10, 11, 11, 12, 12, 12, 13, 13, 14. Describe the shape, and give the mean and median.

Add 30 to the data above. Describe the new shape and the mean and median.

A histogram of the ages of people at a Grade 7 hockey game has one peak at 12 to 13 years and a second peak at 40 to 49 years. What might explain the two clusters?

In a data set, the mean is 50 and the median is 35. Is it more likely skewed right or skewed left? Why?

Test marks: 72, 75, 75, 78, 80, 81, 12. The 12 was entered by mistake; the real mark was 72. How does fixing it change the shape and the mean?

Challenge

Create a data set of 7 values that is symmetric, with a mean of 10 and a range of 6.

Change one value in your symmetric data set to make it skewed left, with the median staying 10. What happens to the mean?

A class of 20 has a symmetric distribution of heights with mean 152 cm. Five Grade 2 students (heights around 120 cm) are added to the data. Describe how the shape, mean and median will change.

Check your understanding

Quiz

A dot plot has most values between 2 and 6, and one value at 30. How would you describe it?

In a symmetric distribution, how do the mean and median compare?

You add an outlier far to the right. What usually happens?

A dot plot shows two groups of dots with an empty space between them. What are the groups and the space called?

The mean of a data set is 8 and the median is 12. What shape is most likely?

Why shouldn't you always delete an outlier?

For parents and teachers

Big idea: The second part of D1.5 asks students to describe how adding or removing data alters the shape and distribution of the data. Students describe centre, spread, clusters, gaps, peaks and outliers, and whether a distribution is symmetric or skewed. The words skewed right/left are a helpful extra; the key idea is the long tail and its pull on the mean. Analysing what the shape means also supports D1.6.

Common difficulties:

Calling the side with most data the skew direction. Skew is named for the tail: a long tail to the right is skewed right.
Assuming the mode is always a good centre. With two clusters it can be at the edge.
Deleting outliers without a reason.

How to help: Roll two dice together and build the dot plot (activity). Ask your child to tell the "story" of the shape in one or two sentences. Real data like family members' heights or ages also make interesting clusters.