Chapter 13 · Statistics and data displays
Describing Variability: Center, Spread, and Shape
Recognize when data vary, describe a distribution on a line plot by its center, spread, and shape, and explain why a statistical question expects variability.
You already know how to spot and write a statistical question. Now look at what comes back when you ask one.
Suppose you ask each of the 23 students in Room 12, "How many books did you read last month?" Each answer is one observation: one recorded value. All the observations together are your data.
You would not expect 23 identical answers. Some students read a few books. Some read many. That difference has a name: variability. Data have variability when the observations are not all the same value. If every observation is the same value, the data have no variability.
In this lesson you will recognize variability, describe it, and explain why some questions expect it before any data come in.
Worked example
Spot variability in a list
Set A is the number of sides on each of 6 square tiles: . Set B is the number of books each of 6 students read last month: . Which set has variability?
Check Set A. Data have variability when the observations are not all the same value, so compare them. All six observations are 4. No two of them differ, so Set A has no variability.
Check Set B the same way. Its first two observations are 3 and 5, which are different values. Three observations are 5, but repeats do not matter. One pair of different values is enough, so Set B has variability.
See how far each set stretches. Subtract its lowest value from its highest value. Set B stretches across 4 books. Set A stretches across 0 sides, which fits a set with no variability.
Answer
Set B has variability because its observations are not all the same. They run from books to books. Set A has no variability because every observation is sides.
Key concept
A distribution has a center, a spread, and a shape
Data with variability are not all one value, so one number cannot describe them fully. You describe their distribution: which values occur and how many observations have each value.
For data that are numbers, describe a distribution with three features.
- Center: a value in the middle of the data. It is typical for the group.
- Spread: how far apart the values are. Give the lowest value, the highest value, and the distance between them. Spread tells you how much the data vary.
- Shape: the pattern the data make. Look at where the observations pile up and whether the two sides match.
A line plot shows a distribution as marks stacked above a number scale. Each number on the scale is a position, and the marks above it form a stack. Read the line plots in this lesson with three rules.
- Each mark is exactly one observation. A mark above 4 means exactly 4. A stack is never a group of nearby values put together.
- The scale starts at 0 and shows every position in order. A position with no marks stays on the scale as an empty spot.
- The number of observations is the total number of marks. Add the counts of all the stacks. Do not count how many positions have marks.
The plot below shows how Room 12 answered the question about books. The stack above 4 has 5 marks, so 5 students read exactly 4 books. Position 0 has no marks, so no student read 0 books. Marks sit above 7 positions, but there are many more than 7 observations.
You can see the variability at once: the marks sit above more than one position. Data with no variability would make a single stack.
Worked example
Describe the books distribution
The line plot above shows how many books each student in Room 12 read last month. Describe the distribution by its center, spread, and shape.
Count the observations. Each mark is one student, so add the count at every position from 0 to 7. The empty position adds 0. The total of the counts is the number of observations, not the 7 positions that have marks.
Find which mark is in the middle. Count the marks in order from left to right. An odd number of marks has exactly one middle mark. With 23 marks it is the 12th, because 11 marks come before it and 11 come after it.
Find where the 12th mark sits. The stacks at 1, 2, and 3 hold the first 9 marks. The stack at 4 holds the next 5 marks, which are marks 10 through 14. The 12th mark is in that stack, so the center is 4 books.
Describe the spread. Values are read from the scale, so find the lowest and highest positions that have marks. They are 1 and 7. Subtract to find the distance between them, which is 6 books.
Describe the shape. The tallest stack is called the peak, and it is at 4. Moving left from the peak, the counts are 4, 3, 2. Moving right, they are also 4, 3, 2. The two sides match like mirror images, so the shape is symmetric. No position between 1 and 7 is empty.
Answer
The observations have a center of books. They spread from book to books, a distance of books. The shape is symmetric with its peak at .
Not every distribution is symmetric. Room 12 also answered a second question: "How many pets live in your home?" The plot below shows the answers. The counts above 0 through 8 are 5, 5, 7, 2, 2, 1, 0, 0, and 1.
Two more shape words will help you describe it.
- A distribution has a tail when the marks stretch much farther from the peak on one side than on the other.
- A gap is one or more empty positions that have marks on both sides.
Worked example
Describe a distribution with a tail and a gap
The line plot above shows how many pets live in the home of each student in Room 12. Describe the distribution by its center, spread, and shape.
Count the observations. Each mark is one student, so add the count at every position from 0 to 8. Seven positions have marks, but the number of observations is the total of the counts. It is 23, one for each student.
Find the center. The middle of 23 marks is again the 12th mark. The stacks at 0 and 1 hold the first 10 marks. The stack at 2 holds the next 7 marks, which are marks 11 through 17. The 12th mark is at 2, so the center is 2 pets.
Describe the spread. The lowest position with a mark is 0. Zero is a real value here: it means no pets. The highest position with a mark is 8, so the distance between them is 8 pets.
Describe the shape. The peak is at 2. Marks reach 2 positions to its left, down to 0. They reach 6 positions to its right, up to 8. The right side stretches much farther, so there is a tail to the right and the shape is not symmetric. Positions 6 and 7 are empty between marks at 5 and 8. That is a gap.
Answer
The observations have a center of pets. They spread from pets to pets, a distance of pets. The shape is not symmetric: it has a peak at , a tail to the right, and a gap at and .
So far you found variability after the data came in. A statistical question does something earlier. It anticipates variability. That means it expects the observations to differ before anyone collects them. Think back to the books question. Before one student answered, you already expected different numbers.
To explain why a question anticipates variability, give three facts.
- The group. Say who or what is observed. The group needs more than one member.
- The value. Say what is recorded for each member.
- The reason. Say why that value is expected to differ from member to member.
When all three facts hold, the question anticipates variability. Its answer is then a description of a distribution, with a center, a spread, and a shape. One number is not enough.
Worked example
Explain why a question anticipates variability
Explain whether each question anticipates variability. Question 1: "How many pets live in the home of each student in Room 12?" Question 2: "How many sides does each of 6 square tiles have?" Question 3: "How many pets live in the home of the Room 12 teacher?"
Name the group and the value for Question 1. The group is the 23 students in Room 12. For each student you record one number: the pets that live in that student's home.
Give the reason the values are expected to differ. Families choose differently. Some keep no pets and some keep several. You expect different values before you ask anyone, so Question 1 anticipates variability. The pets plot confirms it: the values run from 0 to 8.
Check Question 2. The group is the 6 tiles and the value is the number of sides. Every square has exactly 4 sides, so every observation must be 4. A group with many members is not enough. There is no reason for the values to differ, so Question 2 does not anticipate variability.
Check Question 3. Only one home is observed, so the group has one member and there is one observation. A single value has nothing to differ from. Question 3 does not anticipate variability.
Answer
Question 1 anticipates variability because it records a value for each of students and those values are expected to differ. Questions 2 and 3 do not: every square tile has sides, and one home gives only one value.
Takeaway. Data have variability when the observations are not all the same value. Data that vary form a distribution. You describe it with three features: a center in the middle of the data, a spread from the lowest value to the highest value, and a shape.
On the line plots in this lesson, each mark is exactly one observation, and every position from 0 stays on the scale even when it is empty. Count the observations by adding the stack counts, not by counting the positions that have marks.
A question anticipates variability when it records a value for each member of a group and you expect those values to differ. Its answer describes the whole distribution.