6th Grade · 6.SP.A.3

Mean, median and mode in 6th grade

Mean, median and mode are three ways to describe a whole list of numbers with a single number. In 6th grade, your child learns how to find each one and what it says about the data. Your child also learns a second kind of number, one that tells how spread out the values are.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.SP.A.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

What it means, in plain words

Data is a list of numbers someone collected, such as the heights of the plants in a class garden. Mean, median and mode are called measures of center, because each one gives a typical value from the middle of the data.

The mean is the fair share: add all the numbers and divide by how many there are. The median is the middle number once the list is in order. When the list has an even count, the median is halfway between the two middle numbers. The mode is the number that appears most often. A list can have one mode, several or none.

Mean, median and mode in 6th grade come with a new idea: measures of variability. These describe how spread out the values are, again with one number. The simplest is the range, the largest value minus the smallest. Later in the year your child meets two more, the interquartile range and the mean absolute deviation. A center on its own can hide a lot, because two lists can have the same mean and look very different.

Your child should be able to:

Find the mean, median and mode of a short list of numbers
Explain that a measure of center gives one typical value for a whole data set
Explain that a measure of variability, such as the range, tells how spread out the values are
Say whether a question about data asks for a typical value or for a spread

Example

Five friends counted the books on their shelves: 4, 7, 7, 9 and 13. What are the mean, median and mode?

Mean: $4 + 7 + 7 + 9 + 13 = 40$ , and $40 \div 5 = 8$ .
Median: the list is already in order, and the middle (third) number is 7.
Mode: 7 appears twice and every other number once, so the mode is 7.
The mean is 8 books, the median is 7 books and the mode is 7 books.

Example

One player scored 9, 10, 10 and 11 points in four basketball games. Another scored 2, 18, 4 and 16 points. How are the two players alike, and how are they different?

First player: $9 + 10 + 10 + 11 = 40$ , and $40 \div 4 = 10$ .
Second player: $2 + 18 + 4 + 16 = 40$ , and $40 \div 4 = 10$ .
Both means are 10 points, so the centers match.
The ranges are $11 - 9 = 2$ points and $18 - 2 = 16$ points. The first player scores about the same in every game, and the second player's scores are much more spread out.

Where children usually get stuck

For the list 9, 2, 4, 8, 6, your child circles the number in the middle of the list as written and says the median is 4.
The median was learned as "the middle number", and the list already seems to have a middle. The rule only works after the numbers are put in order: 2, 4, 6, 8, 9 has a median of 6.
What helps. Write the numbers on scraps of paper and have your child line them up from smallest to largest before pointing to the middle.
For 3, 4 and 8, your child gets a mean of 5, then decides it must be wrong because 5 is not in the list.
The median and the mode come from the list itself, so your child expects the mean to as well. The mean is a fair share, and a fair share does not have to match any one value.
What helps. Make piles of 3, 4 and 8 coins and even them out into three equal piles. Each pile ends up with 5, even though no pile started with 5.
Asked for the median, your child finds the mode, or mixes up all three words.
The three words sound alike and are learned in the same week. Nothing in the words themselves points to what they mean.
What helps. Give each word a hook. The median is in the middle, like the median strip down the middle of a highway. Mode and most both start with "mo". The mean is the fair share.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Which measures belong to the category Center? Select all that apply.
A
mean
B
IQR
C
median
D
range
Show the answer and how to solve it
Select mean and median. Those are the measures of center; range and IQR are measures of variation.
1. 1Mean is the average, a balance point for the data. Median marks the middle of the ordered data. Both describe center.
2. 2The prompt asks for Center, so select mean and median.
3. 3Range and IQR stay unselected because they describe variation, not center.
Common mistake. A common mistake is selecting IQR as a center measure because it focuses on the middle half of the data. However, IQR measures how spread out that middle half is, making it a measure of variation.
The idea underneath. Center measures tell where a data set is located overall, while variation measures tell how spread out the data are.
2On level
The IQR of the test scores is 8 points.
Select the statement that best interprets this value.
A
The middle half of the test scores span 8 points.
B
A typical test score is 8 points.
C
The highest and lowest test scores differ by 8 points.
D
On average, the test scores are about 8 points from the mean.
Show the answer and how to solve it
The correct answer is "The middle half of the test scores span 8 points." That matches the meaning of IQR.
1. 1The prompt names IQR, which is a measure of spread, not center.
2. 2IQR measures the width of the middle half of the test scores. Here, that width is 8 points.
3. 3So 8 points describes the span of the middle half of the test scores. Choice-b describes a typical score (center, not spread). Choice-c compares only the two extremes, which is the range. Choice-d uses average-distance-from-mean language, which describes MAD.
4. 4The correct statement is the one that says the middle half of the test scores span 8 points.
Common mistake. A common mistake is choosing the sentence about the highest and lowest test scores differing by 8 points. That sentence interprets the range, not the IQR, because IQR only describes the spread of the middle half of the data.
The idea underneath. Measures of spread are not interchangeable: range uses the extremes, IQR uses the middle half, and MAD uses average distance from the mean.
3On level
A librarian tracked how many pages each student in a reading group finished over the weekend. The mean number of pages was 28. Lucas's claim: "That means everyone read about the same number of pages."
Select the answer that best describes Lucas's use of the mean.
A
It treats the mean as if every student read exactly 28 pages.
B
It treats 28 pages as the total number of pages the group read.
C
It uses the mean to describe how spread out the page counts are instead of a typical value.
D
It correctly describes the mean as a typical value for the reading group.
Show the answer and how to solve it
The best answer is the choice saying Lucas used the mean to describe how spread out the page counts are instead of a typical value. The mean tells a typical value, not whether everyone read about the same amount.
1. 1The statistic given is the mean number of pages read over the weekend.
2. 2A mean of 28 pages tells the typical or average number of pages each student in the group read. It describes a center value, not how similar or different the individual counts were.
3. 3Lucas's claim says everyone read about the same number of pages. The phrase "about the same" is a statement about how tightly grouped the page counts are, which is about spread, not about a typical value.
4. 4The choice that best describes Lucas's use of the mean is the one saying he used it to describe how spread out the page counts are instead of a typical value.
Common mistake. A common mistake is thinking a mean also tells whether the data are tightly clustered. In this problem, 28 pages tells the typical page count, but not whether each student's count was close to 28.
The idea underneath. A measure of center tells what a middle or typical value is, while spread tells how much the data vary around that value.
4Stretch
A teacher collects quiz scores from all the students in her class. One question about the data is: "What single score would best represent what students in this class generally earned?"
Select Center or Variation.
A
Center
B
Variation
Show the answer and how to solve it
The answer is Center because the prompt asks for one value that best represents what students in the class generally earned.
1. 1The quoted question asks for a "single score" that would "best represent what students generally earned."
2. 2Asking what single value best represents the group is asking about a typical or representative result — that is Center, not Variation.
3. 3So this prompt should be answered with Center.
Common mistake. A common mistake is choosing Variation because the data come from many different students. Variation is about how spread out or different the scores are, but this prompt asks for one score that best represents the group.
The idea underneath. Center answers what is typical for the group, while Variation answers how much the values differ.