Summarizing Data: Patterns, Center and Variability | Grade 6 Math | LevelKite

A data set can hold many values. A good summary tells its story in a few words and two numbers.

A measure of center is one number that stands for a typical value in the data.
A measure of variability is one number that tells how spread out the values are.

Every summary also names what was measured, called the attribute, and its unit. So each number you report in this lesson carries a unit, such as points or books.

You will work toward three goals: find a measure of center and a measure of variability for a data set, describe its overall pattern and any value that breaks the pattern, and decide which measures fit the data best.

Start with the mean. Picture collecting every value into one total and then sharing that total out equally among all the data values. That equal share is the mean. For quiz scores, the mean is the score each student would have if the points were shared out equally. To find it, add all the values and divide the sum by the number of values.

The mean gives a center. For the variability around it, ask how far each value sits from the mean. The distance between a value and the mean is the larger number minus the smaller one, and it is $0$ when the two are equal. A distance is never negative.

The mean absolute deviation (MAD) is the mean of those distances. A small MAD means the values are, on average, close to the mean. A large MAD means they are, on average, far from it.

Line plot

Quiz scores in points, one mark per student. The stacks above

5

,

6

,

7

,

8

and

9

hold

1

,

2

,

2

,

2

and

1

marks. Positions

0

through

4

and position

10

are empty.

Worked example

Summarize quiz scores with the mean and the MAD

Eight students took a quiz worth $10$ points. The line plot shows their scores in points: $5, 6, 6, 7, 7, 8, 8, 9$ . Summarize the scores with the mean and the mean absolute deviation (MAD).

Add all 8 scores to get the total number of points. Use every mark on the plot, so a score with two marks is added twice.
$5 + 6 + 6 + 7 + 7 + 8 + 8 + 9 = 56$
Divide the total by the number of scores, 8, to share the points equally. The equal share is the mean. Multiplication checks it: 8 equal shares of 7 points make the same total, 56.
$56 \div 8 = 7 \quad \text{and} \quad 8 \times 7 = 56$
Find how far each score is from the mean of 7. Subtract the smaller number from the larger one, so every distance is 0 or more. For the score 5 the distance is 7 - 5 = 2, and for the score 9 it is 9 - 7 = 2. The distances below are in the same order as the scores.
$2, 1, 1, 0, 0, 1, 1, 2$
Add the 8 distances to get the total distance from the mean.
$2 + 1 + 1 + 0 + 0 + 1 + 1 + 2 = 8$
Divide the total distance by the number of scores, 8. This mean of the distances is the MAD.
$8 \div 8 = 1$

Answer

The mean score is $7$ points and the MAD is $1$ point. A typical score is about $7$ points, and the scores are, on average, $1$ point away from the mean.

A second way to find a center uses order instead of a total. The median is the middle of the data. Put the values in order from least to greatest.

With an odd number of values, the median is the one middle value.
With an even number of values, there are two middle values, and the median is their mean.

The median splits the ordered list into a lower half and an upper half. It comes from the middle position of the ordered list, not from the total. So it answers a different question than the mean: which value sits in the middle when the data is lined up in order?

To measure variability around the median, find the middle of each half. The first quartile, $Q_1$ , is the median of the lower half. The third quartile, $Q_3$ , is the median of the upper half. Build the halves the same way every time:

Even number of values: the lower half is the first half of the ordered list, and the upper half is the second half.
Odd number of values: leave the overall median out of both halves.

The interquartile range (IQR) is the distance from $Q_1$ to $Q_3$ , so $\text{IQR} = Q_3 - Q_1$ . It tells how wide the middle part of the data is. It is a difference between two values on the scale, so it has the unit of the data. It is not a count of data values.

$Q_1$ , the median and $Q_3$ cut the ordered list into four parts, each holding roughly one fourth of the values. That is only roughly true: with a small data set or repeated values, a part can hold more or less than one fourth.

Worked example

Summarize books read with the median and the IQR

Eleven students recorded how many books they read over the summer. In order, the counts in books are $1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 14$ . Summarize the data with the median and the interquartile range (IQR).

Check that the values are in order from least to greatest, then count them. There are 11 values, an odd number, so the median is the one middle value. It is the 6th value, with 5 values before it and 5 values after it.
$\text{median} = 3$
Build the two halves. The number of values is odd, so leave the median out of both halves. The lower half is the first 5 values, and the upper half is the last 5 values.
$\text{lower half: } 1, 2, 2, 3, 3 \qquad \text{upper half: } 3, 4, 4, 5, 14$
Find the first quartile, the median of the lower half. That half has 5 values, an odd number, so take its middle value, the 3rd one.
$Q_1 = 2$
Find the third quartile, the median of the upper half. That half also has 5 values, so take its 3rd value.
$Q_3 = 4$
Subtract the first quartile from the third quartile to get the IQR. The result is a distance of 2 books on the scale. It is not a count of students. In fact, 8 of the 11 values lie from 2 books to 4 books. That is more than half, because the data set is small and values repeat.
$\text{IQR} = Q_3 - Q_1 = 4 - 2 = 2$

Answer

The median is $3$ books and the IQR is $2$ books. A typical student read about $3$ books, and the middle part of the data, from $Q_1 = 2$ to $Q_3 = 4$ , is $2$ books wide.

Two numbers do not tell the whole story. A summary also describes what the data looks like along the scale. That arrangement of the values is called the distribution of the data.

The overall pattern is what most of the data does. Say where the values gather (a cluster), which value has the tallest stack (a peak), and whether the two sides of the center look alike (roughly symmetric) or one side stretches out farther than the other (skewed).

A striking deviation is a value that does not follow the overall pattern. It sits far from the rest of the data, often with a gap of empty positions in between.

When you find one, go back to the context. Ask what the value means for the thing that was measured, and whether it is a recording mistake or a real but unusual result. A real value stays in the data, and your description should mention it.

The line plot below shows the book counts from the last example. Look for these features in it.

Line plot

Books read, one mark per student. The stacks above

1

,

2

,

3

,

4

and

5

hold

1

,

2

,

4

,

2

and

1

marks, and one mark sits alone above

14

. Position

0

and positions

6

through

13

are empty.

Worked example

Describe the pattern and the striking deviation

The line plot shows the books read over the summer by the same $11$ students: $1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 14$ . The teacher checked the value $14$ . That student really did finish a $14$ -book series. Describe the overall pattern and any striking deviation.

Match the plot to the data. Each mark is one student, so add the stack heights to count the values. The total is 11, the number of students, even though marks sit above only 6 positions.
$1 + 2 + 4 + 2 + 1 + 1 = 11$
Describe where most of the data is. Ten of the 11 marks form one cluster from 1 book to 5 books. The tallest stack, with 4 marks, is the peak at 3 books. The stacks on the two sides of the peak match, so the cluster is symmetric around 3 books.
$1 + 2 + 4 + 2 + 1 = 10$
Look for a value that breaks the pattern. One mark sits alone at 14 books. Positions 6 through 13 are empty, so a gap separates that mark from the cluster. It is 9 books above the next-highest value, which is 5.
$14 - 5 = 9$
Use the context to explain the deviation. The teacher confirmed that this student finished a 14-book series. So 14 is a real count, not a recording mistake. It stays in the data, and the description should mention it.

Answer

Most students read from $1$ to $5$ books, in a cluster that peaks at $3$ books. One student read $14$ books, $9$ more than the next-highest count. That value is a striking deviation, and the context shows it is real.

Key concept

Let the shape and the context choose the measures

A summary pairs a measure of center with a measure of variability: the mean with the MAD, or the median with the IQR.

When the data is roughly symmetric and has no striking deviation, the mean and the MAD describe it well.
When the data is skewed or has a striking deviation, the median and the IQR usually describe a typical value and the spread better. The mean and the MAD add in the size of every value, so a far-away value pulls on both. The median and the quartiles come from the middle of the ordered list and of its halves.

Then check your choice against the context: what was measured, in what unit, and what a typical value should tell you.

Worked example

Choose the measures for each data set

Choose a measure of center and a measure of variability for each data set. Each summary should show what is typical for a student in the group. Explain each choice from the shape and the context. Quiz scores in points: $5, 6, 6, 7, 7, 8, 8, 9$ . Books read: $1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 14$ .

Quiz scores: look at the shape. The stacks hold 1, 2, 2, 2 and 1 marks at 5, 6, 7, 8 and 9 points, so the two sides of 7 match and no score sits far from the rest. Check the center both ways. The 8 scores have two middle values, the 4th and the 5th. Both are 7, so the median is 7, the same as the mean.
$(7 + 7) \div 2 = 7$
Choose for the quiz scores. The shape is symmetric with no striking deviation, so the mean and the MAD describe the scores well, and they use the size of every score. Report the mean of 7 points and the MAD of 1 point from the first example.
Books read: look at the shape. The data has a cluster from 1 to 5 books and one striking deviation at 14 books. The mean, 4 books, is pulled above the peak of the cluster by the 14. The median, 3 books, sits at the peak, so it shows what is typical for a student. Choose the median.
$\text{mean} = 44 \div 11 = 4, \quad \text{median} = 3$
Test the MAD on the book data. Find each value's distance from the mean of 4 and add the 11 distances. The total is 22, so the MAD is 2 books. But the single value 14 supplies 10 of that 22.
$3 + 2 + 2 + 1 + 1 + 1 + 1 + 0 + 0 + 1 + 10 = 22, \quad 22 \div 11 = 2$
Compare with the ten clustered values alone. Their mean is 30 divided by 10, which is 3 books. Their distances from 3 add to 8, so their MAD would be 0.8 books. The single value 14 lifts the MAD from 0.8 books to 2 books.
$2 + 1 + 1 + 0 + 0 + 0 + 0 + 1 + 1 + 2 = 8, \quad 8 \div 10 = 0.8$
Choose the IQR. It comes from the first quartile, 2, and the third quartile, 4, and both lie inside the cluster. Without the 14 there are 10 values, an even number, so the halves are the first 5 and the last 5: 1, 2, 2, 3, 3 and 3, 3, 4, 4, 5. Their middle values are still 2 and 4, so the IQR stays 2 books. The MAD also equals 2 here, but only by chance. The IQR pairs with the median.
$\text{IQR} = 4 - 2 = 2$

Answer

For the quiz scores, report a mean of $7$ points with a MAD of $1$ point. For the books, report a median of $3$ books with an IQR of $2$ books, and mention the one student who read $14$ books.

The takeaway. A strong summary says what was measured and in what unit, gives one number for the center and one for the variability, and describes the overall pattern along with any striking deviation, explained from the context. Let the shape guide the numbers you choose: the mean with the MAD when the data is roughly symmetric with no striking deviation, and usually the median with the IQR when the data is skewed or has a value far from the rest.