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Unit 2 · Standard 6.DS.6c

Describe Data by Mean Absolute Deviation

Lesson 2-9 · Learn It — how the math works

Why we're learning this

The forecast shows the expected high temperatures for a week. The mean high temperature for the week is 79°F. How can you describe the variation in high temperatures over the week?

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What is the mean absolute deviation (MAD)?

The mean absolute deviation, or MAD, is the average distance of each value from the mean. It tells you how spread out the data is: a small MAD means the data is close together, and a large MAD means it is spread out.

Picture itSee what the math looks like before you read about it

The forecast shows the expected high temperature for each day of this week: 75, 77, 78, 79, 80, 82 and 82 degrees Fahrenheit. The mean high for the week is 79 degrees, marked by the dashed line.
Where you see this in real lifeThe forecast shows the expected high temperature for each day of this week: 75, 77, 78, 79, 80, 82 and 82 degrees Fahrenheit. The mean high for the week is 79 degrees, marked by the dashed line.

Watch me solve itI do — read one step at a time

The problem looks like this
Number line with the data 6, 8, 10, 12 and the mean 9 marked. The distances from 9 are 3, 1, 1, 3.345678910111268mean 91012
  1. My data set: 6, 8, 10, 12. First I find the : 6 + 8 + 10 + 12 = 36, then 36 ÷ 4 = 9. The mean is 9.6 + 8 + 10 + 12 = 36
  2. Next I find each (value − ): 6−9 = −3, 8−9 = −1, 10−9 = 1, 12−9 = 3.
  3. Then I take the of each (the distance, never negative (zero is allowed)): 3, 1, 1, 3.
  4. Finally I average those distances: 3 + 1 + 1 + 3 = 8, then 8 ÷ 4 = 2. So the MAD = 2 points.3 + 1 + 1 + 3 = 8

Watch out — mistakes students make here

  • Distances aren't all zero — find each value's gap from 6.
  • 6 is the mean — MAD averages the DISTANCES from the mean.
  • Bigger MAD means times vary more, not faster potential.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. New data set: 9, 10, 11. First, what is the ?
    Check this step
    Add the three numbers and divide by 3.
  2. Now find how far each number is from the . What is the distance for 9?
    Check this step
    For 10? For 11?
  3. Add those distances and divide by 3. Is this MAD small or large?
    Check this step
    What does that tell us about the ?

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Remember: first, then , then average.

Solve: Two runners have the same average time of 60 seconds. Runner A's MAD is 1.2 seconds. Runner B's MAD is 5.8 seconds. Which runner is the coach more likely to pick for a relay race that needs a reliable time?

My answer

Stuck? Open one hint at a time.

Hint 1

'Reliable' is the key word — the coach wants times that barely change from race to race, not just a good average.

Hint 2

Both runners average 60 seconds, so the average can't decide this. MAD tells you how much each runner's times swing around 60.

Hint 3

Read each MAD as an average distance: for Runner A, the recorded times are an average of 1.2 seconds away from the mean of 60 seconds. Compare that with 5.8 seconds for Runner B.

Check my answer
✅ Answer: Runner A — lower MAD means more consistent times
Runner A's MAD of 1.2 seconds means the average distance of the recorded times from the 60-second mean is 1.2 seconds. Runner B has a larger average distance from the same mean, so A was more consistent in these data. MAD does not guarantee how many times fall within 1.2 seconds of the mean.

Turn & TalkSay it out loud with a partner

The forecast shows seven high temperatures and their is 79°F. Which day is farthest from the mean, and what does that tell you about the week?

Try starting with:

  • I think ___ because ___.Pienso que ___ porque ___.
  • I agree / disagree with you because ___.Estoy / no estoy de acuerdo contigo porque ___.
Use these wordsMean Absolute DeviationDeviationAbsolute ValueSpreadData distribution