6th Grade · 6.SP.B.5

Mean absolute deviation in 6th grade

Mean absolute deviation, or MAD, is the average distance between the numbers in a set of data and their mean. It is part of a bigger 6th grade skill, summarizing data: describing a list of numbers in a few clear sentences. Your child learns to report what was measured, how many values there are, a typical value, how spread out the values are and anything unusual.

What the standard says

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

6.SP.B.5 Summarize numerical data sets in relation to their context, such as by:
6.SP.B.5.a Reporting the number of observations.
6.SP.B.5.b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
6.SP.B.5.c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
6.SP.B.5.d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

What it means, in plain words

A summary tells someone what a list of numbers says without reading every number. A good 6th grade summary says how many values there are and what was measured, in which unit. It gives a typical value, called the center, and a measure of how spread out the values are, called the variability. It also points out any outlier, a value far away from the rest.

The mean absolute deviation measures spread around the mean. Your child finds the mean, then the distance of each value from the mean, whether the value is above or below it. The mean of those distances is the MAD. A small MAD means most values sit close to the mean. A large MAD means they are scattered.

The interquartile range, or IQR, measures spread around the median. It is the upper quartile minus the lower quartile, so it is the width of the middle half of the data. Your child learns to pair the measures: the mean with the mean absolute deviation, the median with the IQR. When the data has an outlier, the median and the IQR are the better pair. One extreme value pulls the mean and the MAD a long way but barely moves the median and the IQR.

Your child should be able to:

Report how many values there are, what was measured and the unit
Find the mean absolute deviation of a small set of data
Find the median and the interquartile range
Choose mean and MAD or median and IQR to fit the data, and explain the choice

Example

Four plants grew 3, 5, 7 and 9 centimeters in one month. What is the mean absolute deviation?

Mean: $3 + 5 + 7 + 9 = 24$ , and $24 \div 4 = 6$ centimeters.
Distance of each value from 6: 3, 1, 1 and 3.
Mean of the distances: $3 + 1 + 1 + 3 = 8$ , and $8 \div 4 = 2$ .
The MAD is 2 centimeters. On average, a plant grew 2 centimeters more or less than the mean of 6 centimeters.

Example

Nine students said how many books they read over the summer: 2, 3, 3, 4, 5, 5, 6, 8 and 19. Write a summary of the data.

There are 9 values, and each one is a number of books.
The values are in order, so the median is the fifth value, 5 books.
The lower half is 2, 3, 3 and 4, so the lower quartile is 3. The upper half is 5, 6, 8 and 19, so the upper quartile is 7. The IQR is $7 - 3 = 4$ books.
The value 19 is far from the rest, so it is an outlier. It pulls the mean up to about 6.1 books, which is higher than 7 of the 9 values.
Summary: 9 students reported how many books they read over the summer. The median is 5 books and the IQR is 4 books. Most read between 2 and 8 books, and one student read 19.

Where children usually get stuck

Your child subtracts the mean from each value, gets $-3$ , $-1$ , $1$ and $3$ , adds them and says the MAD is 0.
The differences below the mean are negative and cancel the ones above it, so they always add up to 0. The MAD uses distances, and a distance is never negative.
What helps. Mark the values and the mean on a number line and count the steps from each value to the mean. Steps are always counted as positive.
Your child's summary says "the median is 5 and the IQR is 4" and stops there.
The numbers feel like the answer. A reader also needs to know what was measured, in what unit, from how many people and whether anything was unusual. Without that, the 5 means nothing.
What helps. Ask "5 what, and from how many people?" until the summary answers both questions.
Your child uses the mean for every set of data, even when one value is far bigger than the rest.
The mean was learned first and feels like the real average. An outlier pulls the mean toward it, so the mean stops being a typical value.
What helps. Take weekly allowances of 5, 5, 6 and 6 dollars and add one of 50 dollars. The mean is 14.40 dollars and the median is 6 dollars. Ask which one describes a typical child.

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
The line plot below shows pencil lengths in centimeters.
Select all statements that are correct for this data set.
A
There are 6 measurements.
B
The range is 7 cm.
C
There are 7 measurements.
D
The mean length is 4 cm.
E
The mean length is 5 cm.
Show the answer and how to solve it
The two correct statements are 'There are 7 measurements' and 'The mean length is 5 cm'.
1. 1Read the dots left to right: 3, 4, 5, 5, 5, 6, 7.
2. 2Count each dot. There are 7 measurements.
3. 3The data is symmetric, so the mean is a good summary. Add the values and divide by 7 to get a mean of 5 cm.
4. 4So the two correct statements are: there are 7 measurements, and the mean length is 5 cm.
Common mistake. A common mistake is to misread the largest value (7) as the range or as the mean rather than computing each statistic from the full data set.
The idea underneath. When data is symmetric with no outliers, the mean is an appropriate summary of the typical value.

2On level

This line plot shows how many minutes it took students to finish a math quiz.

Minutes

Complete the summary table.

Item	Response	Item	Response
Data		N
Best measures		Center	min
Variability	min	Pattern

Show the answer and how to solve it

Use quiz completion times (minutes), N = 10, median and IQR, median 12.5 min, IQR 3 min, and the statement that most quiz completion times are 10 to 15 minutes with one much longer time.

1Each dot stands for one student. Count the dots and read the values from left to right.
2The value 20 minutes is far from the rest of the data, with four empty ticks between 15 and 20. Because of that unusual high value, median and IQR are better measures than mean and MAD.
3Find the median by locating the two middle values of the ordered data.
4Find the IQR by splitting the data into lower and upper halves and locating the middle of each half.
5The context describes quiz completion times, and the axis shows minutes. Most students finished in 10 to 15 minutes, with one student taking 20 minutes—much longer than the others.

Common mistake. A common mistake is using the full range, 20 − 10 = 10, as the variability value. Because the 20-minute time is unusually long, the better summary is median and IQR, so the variability value is the IQR, 14 − 11 = 3.

The idea underneath. Choose mean and MAD for roughly symmetric data. MAD (mean absolute deviation) is the average distance from the mean. Choose median and IQR when an unusual value pulls the mean away from most values. IQR (interquartile range) is the difference between the medians of the upper and lower halves of the ordered data.

3On level
The line plot below shows how many minutes students in one class spent on homework one evening.
Minutes
Select the best complete summary of the data. Use a matched pair of center and variability measures that fits the shape.
A
The data set has 11 values for homework time, measured in minutes. The median is 15 minutes and the IQR is 3 minutes. Most values are clustered between 12 and 17 minutes, with one high outlier at 25 minutes.
B
The data set has 12 values for homework time, measured in minutes. The median is 15 minutes and the IQR is 3 minutes. Most values are clustered between 12 and 17 minutes, with one high outlier at 25 minutes.
C
The data set has 11 values for homework time, measured in minutes. The mean is 15.5 minutes and the IQR is 3 minutes. Most values are clustered between 12 and 17 minutes, with one high outlier at 25 minutes.
D
The data set has 11 values for homework time, measured in minutes. The median is 15 minutes and the IQR is 3 minutes. The values are spread fairly evenly from 12 to 25 minutes, with no unusual values.
Show the answer and how to solve it
The best complete summary reports N = 11 homework times in minutes, median 15 minutes, IQR 3 minutes, and the high outlier at 25 minutes.
1. 1Count each dot on the line plot, not just each occupied number. There are 11 data points, so N = 11.
2. 2There is a big gap between 17 and 25, and 25 sits alone far above the cluster. The high value at 25 pulls the mean upward. The median uses the middle position, and the IQR measures the width of the middle half of the data. They are less affected by an extreme value, so median and IQR are the matched pair to use here.
3. 3With 11 values, the median is the 6th value. Q1 is the median of the lower five values, and Q3 is the median of the upper five values.
4. 4The correct summary is the one that reports N = 11, homework time in minutes, median 15, IQR 3, and a high outlier at 25. The summary reporting 12 observations miscounts the repeated values. The question asks for a matched pair suited to the shape. Mean and IQR can each describe data, but they do not form the requested matched pair. With this high outlier, use median and IQR. The summary claiming the values are spread evenly from 12 to 25 with no unusual values ignores the visible gap and the isolated point at 25.
Common mistake. A common mistake is checking only whether the reported values are correct. Also check the requested matched pair: median and IQR.
The idea underneath. A complete summary gives the count, attribute and unit, a matched center and variability pair suited to the shape, and a true pattern statement.

4Stretch

The line plot below shows the lengths, in centimeters, of oak leaves collected in a school park.

Length

Complete the summary organizer. If there is an outlier, choose measures that change little when that outlier moves farther away. Otherwise, choose the average and the average distance from it.

Summary item	Answer
Number of observations
Attribute
Unit
Center measure
Center value
Variability measure
Variability value
Pattern or outlier note

Show the answer and how to solve it

The completed organizer is: 10 observations, attribute leaf length, unit centimeters, center measure median, center value 10, variability measure IQR, variability value 2, and pattern note has a high outlier.

1Read every dot from left to right: 8, 9, 9, 10, 10, 10, 11, 11, 12, 18. Count all dots, including the three stacked at 10, for a total of 10 observations.
2The context says the data are lengths of oak leaves, so the attribute is leaf length and the unit is centimeters.
3Most values are between 8 and 12, but 18 is far above that group. This high outlier pulls the mean up. The median uses the middle position, and IQR uses the middle half. They change less if 18 moves farther away, so choose median and IQR.
4For 10 values, the median is the average of the 5th and 6th values in order. Both are 10, so the median is 10.
5Split into lower half (8, 9, 9, 10, 10) and upper half (10, 11, 11, 12, 18). The middle value of the lower half is Q1 = 9, and the middle value of the upper half is Q3 = 11. So IQR = 11 − 9 = 2.

Common mistake. A common mistake is counting only the 6 distinct positions on the plot instead of all 10 dots. Another is reporting the range 18 − 8 = 10 for variability instead of IQR = 2.

The idea underneath. MAD (mean absolute deviation) is the average distance from the mean. IQR (interquartile range) is the spread of the middle half of the data: Q3 minus Q1. An outlier pulls the mean toward it. The median and quartiles use middle positions, so they are less affected by how far away an outlier lies.