Changing Data Values and Checking Measures of Center | Grade 6 Math | LevelKite

A data set is a group of numbers collected to answer a question. Each number in it is a data value, also called an observation.

A measure of center is one number that summarizes all the values in a data set. You already know two of them:

The mean is the sum of the data values divided by the number of data values.
The median is the middle value when the data values are in order from least to greatest. When the number of values is even, the median is the mean of the two middle values.

This lesson answers two questions. What happens to the mean and the median when one data value changes? And how do you find the mistake when someone's mean or median is wrong?

Start with what each measure means.

Think of the mean as a fair share. You add all the data values to get a total. Then you share that total equally, with one share for each data value. In symbols:

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

In this lesson, a change means that you replace one data value with a different number. The number of data values stays the same.

Look at the fraction above. A change makes the sum on top larger or smaller, but the number on the bottom stays the same. So the fair share must change too.

The median works by position, not by total. Line the values up in order from least to greatest. This lesson calls that the ordered list. The median is whatever sits in the middle of the ordered list. So a change matters to the median only when it changes what sits in the middle.

Worked example

Changing the greatest value

Maya recorded how many minutes she read on each of $5$ days: $10$ , $20$ , $25$ , $30$ , $40$ . Then she corrects one value: the $40$ should be $60$ . How does this change affect the mean and the median?

Find the mean before the change. Add the 5 values to get the total minutes. Then divide by 5, because the mean shares the total equally among the 5 days.
$10+20+25+30+40=125,\qquad 125\div 5=25$
Find the median before the change. The values are already in order from least to greatest. With 5 values, the middle one is the 3rd, because 2 values come before it and 2 come after it.
$10,\ 20,\ \underline{25},\ 30,\ 40\qquad \text{median}=25$
Replace 40 with 60 and find the new mean. There are still 5 values, so you still divide by 5.
$10+20+25+30+60=145,\qquad 145\div 5=29$
Find the new median. The 60 is still the greatest value, so the list is still in order and the 3rd value is still 25.
$10,\ 20,\ \underline{25},\ 30,\ 60\qquad \text{median}=25$
Compare before and after. The mean rose from 25 to 29. That fits the fair-share idea: the total grew by 20 minutes, and sharing 20 extra minutes among 5 days adds 4 minutes to each share.
$29-25=4,\qquad (60-40)\div 5=20\div 5=4$

Answer

The mean increases by $4$ minutes, from $25$ minutes to $29$ minutes. The median stays the same at $25$ minutes.

Key concept

How one changed value affects the mean and the median

This holds when you replace exactly one data value with a different number and the number of data values stays the same.

The mean always changes. The total changes by the amount of the change, and the mean shares that amount equally among all the data values. The mean goes up when the value goes up and down when the value goes down.
The median may stay the same or may change. It changes only when the new ordered list has a different middle than the old ordered list. So put the new values in order and find the middle again.

\text{amount the mean changes}=\dfrac{\text{amount the data value changes}}{\text{number of data values}}

Worked example

A value that moves across the middle

A data set has $6$ values: $5$ , $7$ , $8$ , $10$ , $12$ , $18$ . Its mean is $10$ and its median is $9$ . The value $18$ changes to $6$ . Describe how the mean and the median change.

Find the amount the data value changes. It goes down from 18 to 6. No other value changes, so the total goes down by this same amount.
$18-6=12$
Share that amount among all 6 values to find how much the mean changes. The value went down, so the mean goes down. Subtract the result from the old mean.
$12\div 6=2,\qquad 10-2=8$
Check by adding the new values. The old total was 10 times 6, or 60, because multiplying by 6 undoes the dividing by 6 that gave the mean. The new total is 12 less, or 48. Dividing the whole difference, 60 minus 12, by 6 gives the same result as dividing 60 by 6 and 12 by 6 and then subtracting. So both ways give 8.
$5+7+8+10+12+6=48,\qquad \dfrac{48}{6}=\dfrac{60}{6}-\dfrac{12}{6}=10-2=8$
Put the new values in order from least to greatest. Order did not matter for adding, but it matters for the median. The 6 belongs between the 5 and the 7, so the 7, 8, 10 and 12 each move one place toward the end.
$5,\ 6,\ 7,\ 8,\ 10,\ 12$
Find the new median. With 6 values, the middle values are the 3rd and 4th, because 2 values come before them and 2 come after them. They are now 7 and 8, so find their mean.
$\dfrac{7+8}{2}=\dfrac{15}{2}=7.5$
Compare the medians. Before the change, the middle values were 8 and 10 and the median was 9. You did not replace either of them, but the middle of the ordered list is different now.
$9-7.5=1.5$

Answer

The mean decreases by $2$ , from $10$ to $8$ . The median decreases by $1.5$ , from $9$ to $7.5$ , because the changed value moved from above the middle of the list to below it.

Now switch roles. Instead of finding a measure of center yourself, you check someone else's work. To identify a mistake means to find the step that went wrong and say why. Redo the work one step at a time and compare. If your steps are right, the first step where the two results differ is where a mistake is.

These questions catch many common mistakes:

For a mean: Does the total include every observation exactly once? Does the work divide the total by the number of observations?
For a median: Are the values in order from least to greatest? Does the middle come from all the observations? Remember that an even number of values has two middle values.

You can also test a mean quickly. Multiply it by the correct number of observations. Multiplying by a number undoes dividing by that same number, so a correct mean gives back the correct total. If you get a different number, the mean is wrong.

Data often comes as a table of counts instead of a list. The table below shows the goals a soccer team scored in each game of its season. It holds the same information as a line plot, where each mark is one observation.

Each row stands for one exact number of goals, not a range of numbers. There is a row for every whole number from $0$ to $5$ , even for $2$ , which no game had.
The Goals in a game column gives that number of goals. The Number of games column counts how many games had exactly that many goals.

Each game gives one observation: its number of goals. So the number of observations is the sum of the counts. It is not the number of rows, and it is not the number of rows that have games.

Goals the team scored in each game of its season

Goals in a game	Number of games
$0$	$2$
$1$	$4$
$2$	$0$
$3$	$1$
$4$	$2$
$5$	$1$

Worked example

Find the mistakes in Kai's work

Kai uses the table above to find the mean and the median number of goals in a game. He adds $0+4+0+3+8+5=20$ goals. He says $5$ rows have games, so the mean is $20\div 5=4$ goals. Then he lists the rows that have games, $0, 1, 3, 4, 5$ , and says the median is $3$ goals. Find Kai's mistakes and correct them.

Find the number of observations first. Each game gives one observation, so add the counts in the Number of games column.
$2+4+0+1+2+1=10$
Check Kai's total. In each row, multiply the goals by the number of games, because that many games each had that many goals. Then add the products. Kai's total of 20 goals is correct.
$0\times 2+1\times 4+2\times 0+3\times 1+4\times 2+5\times 1=0+4+0+3+8+5=20$
Check what Kai divided by. He divided by 5, the number of rows that have games. But the mean shares the total among the observations, and there are 10. This is his first mistake. Divide by 10 instead.
$20\div 10=2$
Test both means. A correct mean times the 10 observations gives back the total of 20 goals, because multiplying by 10 undoes dividing by 10. Kai's mean fails the test, and the corrected mean passes.
$4\times 10=40\neq 20,\qquad 2\times 10=20$
Check Kai's median. He found the middle of the 5 rows that have games. This is his second mistake, because the median is the middle of all 10 observations. With 10 values in order, the middle values are the 5th and 6th: 4 values come before them and 4 come after them.
Count down the rows, which already go from the fewest goals to the most. The first 2 games have 0 goals. The next 4 games, the 3rd through the 6th, have 1 goal each. So the 5th and 6th values are both 1. Find their mean.
$\dfrac{1+1}{2}=1$

Answer

Kai treated each row that has games as one observation, but the table shows $10$ observations, one for each game. The correct mean is $2$ goals, and the correct median is $1$ goal.

The takeaway. The mean follows the total, and the median follows the middle of the ordered list. When you replace one data value with a different number and the number of values stays the same, the mean always moves. It moves by the amount of the change divided by the number of values, in the same direction as the change. The median moves only when the ordered list has a different middle, so reorder and look again.

To find a mistake in someone's mean or median, redo the work step by step. Check the total, the number of observations, the order of the values, and the middle. In a table of counts, the number of observations is the sum of the counts, not the number of rows.