Mean, Median and Mode: Measures of Center | Grade 6 Math | LevelKite

How can you describe a whole list of numbers with just one number? This lesson shows you three ways.

A data set here is a group of numbers collected to answer a question. Each number in the set is one observation. This lesson also calls these numbers the values of the data set.

Suppose you ask 5 classmates how many pets they have. Their answers are $1, 2, 2, 3, 2$ . This data set has 5 observations. Only three different numbers appear ( $1$ , $2$ and $3$ ), but each of the 5 answers is its own observation. So the number of observations is the number of entries in the list, repeats included.

Now look at where the answers sit. Three of them are $2$ , and the other two are just $1$ away from $2$ . If you had to describe this group with one number, $2$ would be a good choice: a typical classmate in this group has about $2$ pets. A single number that describes a whole data set in this way is a measure of center.

Key concept

One number that summarizes the whole data set

A measure of center is a single number that summarizes all the values in a data set of numbers. It describes a typical or central value, so you can talk about the whole set at once.

This lesson uses three measures of center:

The mean is the fair-share value: the amount each observation would have if you shared the total equally.
The median is the middle of the list when the values are in order.
The mode is the value that appears most often. Some data sets have more than one mode, or none.

You find each one in a different way, so the three can be different numbers for the same data set.

Start with the mean. Think of it as a fair share.

Four friends have $3$ , $5$ , $6$ and $10$ stickers. They put all the stickers in one pile:

$3 + 5 + 6 + 10 = 8 + 6 + 10 = 14 + 10 = 24$ stickers.

Then they share the pile equally among the 4 friends:

$24 \div 4 = 6$ stickers each.

The mean of $3, 5, 6, 10$ is $6$ . You can reach the same fair share by moving stickers: the friend with $10$ gives $3$ to the friend with $3$ and $1$ to the friend with $5$ . Then all four friends have $6$ .

Sharing does not change the total. Before sharing, the total was $24$ . After sharing, it is $6 + 6 + 6 + 6 = 24$ . So the mean is the one value that could replace every observation and keep the total the same.

The fair-share idea gives you a rule for any data set of numbers: add all the values, then divide the sum by the number of values.

\text{mean} = \dfrac{\text{sum of all the values}}{\text{number of values}}

Two conditions make the rule work:

The sum must include every value. Add a repeated value each time it appears.
The number of values must count every observation. Count a repeated value each time it appears, and count a value of $0$ as one value too.

You can check a mean with multiplication. Multiplying by the number of values undoes dividing by it, so the mean times the number of values must equal the sum of the values. Use the exact mean for this check, not a rounded one.

Worked example

Find the mean

Five students timed how long they took to solve a puzzle. Their times, in minutes, were $7, 9, 9, 12, 13$ . What is the mean time?

Count the values, because you will divide by this number. The list has 5 times. The time 9 appears twice, and each 9 is its own observation, so count both.
$\text{number of values} = 5$
Add all the values, because the mean shares out the whole total. Add both 9s. Work from left to right, one addition at a time.
$7 + 9 + 9 + 12 + 13 = 16 + 9 + 12 + 13 = 25 + 12 + 13 = 37 + 13 = 50$
Divide the sum by the number of values. This shares the total of 50 minutes equally among the 5 observations.
$50 \div 5 = 10$
Check with multiplication. Multiplying by 5 undoes dividing by 5, so the mean times 5 must give back the sum of 50. It does.
$10 \times 5 = 50$

Answer

The mean time is $10$ minutes. No student took exactly $10$ minutes, and that is fine: the mean does not have to be one of the values in the data set.

Next is the median. The median is the middle of a data set after you write the values in order from least to greatest. Keep every repeated value in the ordered list, because each one is an observation.

The median marks the middle of the ordered list: the same number of values come before the middle as after it. How you find the median depends on how many values there are:

If the number of values is odd, one value sits exactly in the middle. That value is the median. The ordered list $2, 4, 9$ has median $4$ , with one value on each side.
If the number of values is even, two values share the middle. The median is the number halfway between them. To find it, take the mean of those two values: add them and divide by $2$ . The ordered list $2, 4, 6, 9$ has middle values $4$ and $6$ , so its median is $(4 + 6) \div 2 = 10 \div 2 = 5$ .

Worked example

Find the median when the list is not in order

Find the median of the data set $12, 5, 9, 20, 7, 9, 15$ .

Count the values so you can check your ordered list later. The list has 7 values. The value 9 appears twice, so count it twice.
Write the values in order from least to greatest, because the median is the middle of the ordered list. Keep both 9s. The ordered list still has 7 values, so no value is missing.
$5, 7, 9, 9, 12, 15, 20$
Find the middle. There are 7 values, an odd number, so one value sits exactly in the middle. It is the 4th value: 3 values come before it and 3 come after it, and 3 + 1 + 3 = 7.
$5, 7, 9, \boxed{9}, 12, 15, 20$

Answer

The median of the data set is $9$ .

The third measure is the mode. The mode is the value that appears most often. To find it, count how many times each value appears.

If one value appears more times than every other value, that value is the mode. In $3, 5, 5, 8$ , the mode is $5$ .
If two or more values tie for the most appearances, and each of them appears more than once, each of them is a mode. In $1, 1, 4, 6, 6, 9$ , the modes are $1$ and $6$ .
If every value appears just once, no value appears most often, so the data set has no mode. The set $2, 4, 7$ has no mode.

The mode is the value itself, not the number of times it appears. In $3, 5, 5, 8$ , the value $5$ appears $2$ times. The mode is $5$ , not $2$ .

For a longer list, a count table keeps your counting organized. A soccer team played 8 games. Its goals in those games were $2, 0, 1, 5, 1, 4, 1, 2$ .

In the table below, each row is one exact number of goals, from $0$ to $5$ . The first column gives that value. The second column counts how many of the 8 games had that value.

Goals scored by a soccer team in 8 games

Goals in a game (value)	Number of games with that value (count)
$0$	$1$
$1$	$3$
$2$	$2$
$3$	$0$
$4$	$1$
$5$	$1$

The highest count is $3$ , in the row for $1$ goal. So the mode is $1$ goal, the value in that row, not the count $3$ . The team scored exactly $1$ goal more often than any other number of goals.

The row for $3$ goals has a count of $0$ . No game had exactly $3$ goals, so that row adds no observations.

To find the number of observations, add the counts: $1 + 3 + 2 + 0 + 1 + 1 = 8$ . That matches the 8 games. The number of observations is the sum of the counts. It is not the number of rows ( $6$ ), and it is not the number of rows with a count above zero ( $5$ ).

Worked example

Find the mean, median and mode of one data set

A soccer team's goals in 8 games were $2, 0, 1, 5, 1, 4, 1, 2$ . Find the mean, the median and the mode of the number of goals in a game.

Write the values in order from least to greatest. The median needs the order, and the order also puts equal values side by side for the mode. The game with 0 goals is still an observation, so the ordered list keeps all 8 values.
$0, 1, 1, 1, 2, 2, 4, 5$
Add all 8 values to find the total number of goals. Reordering and grouping the values do not change the sum. Adding 0 does not change the sum either, but the 0 still counts as one of the 8 values.
$(0 + 1 + 1 + 1) + (2 + 2) + (4 + 5) = 3 + 4 + 9 = 7 + 9 = 16$
Divide the sum by the number of values. This shares the 16 goals equally among the 8 games. Check: 2 times 8 is 16, which is the sum.
$\text{mean} = 16 \div 8 = 2$
Find the middle of the ordered list. There are 8 values, an even number, so two values share the middle: the 4th and the 5th. Three values come before them and three come after them. The 4th value is 1 and the 5th value is 2.
$0, 1, 1, \boxed{1}, \boxed{2}, 2, 4, 5$
The median is the number halfway between the two middle values, so take their mean: add them and divide by 2.
$\text{median} = (1 + 2) \div 2 = 3 \div 2 = 1.5$
Count how many times each value appears. In the ordered list, equal values sit side by side: 1 appears 3 times, 2 appears 2 times, and 0, 4 and 5 appear once each. These match the counts in the table above. One value has the highest count, so it is the only mode.
$\text{mode} = 1$

Answer

For the number of goals in a game, the mean is $2$ goals, the median is $1.5$ goals, and the mode is $1$ goal. The three measures differ because each one summarizes the same 8 games in a different way.

Look at what each of those numbers tells you about the team.

The mean, $2$ , is the fair share. If you spread the team's $16$ goals evenly over the 8 games, each game would have $2$ goals.
The median, $1.5$ , marks the middle. Four games had fewer than $1.5$ goals and four games had more.
The mode, $1$ , is the most common result. More games had $1$ goal than any other number of goals.

All three describe the center of the same data, each in its own way.

Takeaway: A measure of center is one number that summarizes a whole data set of numbers, and you now know three of them. The mean is the fair share: the sum of all the values divided by the number of values. The median is the middle of the ordered list, or the number halfway between the two middle values when the number of values is even. The mode is the value that appears most often, and a data set can have one mode, more than one mode or no mode. For all three, count every observation, including repeats and zeros.

Next, you will use the mean to find a missing value, and you will find measures of center from data displays.