A sample’s median absolute deviation is $6,745. For normally distributed data, what is the scaled MAD, which approximates the standard deviation?
Correct Answer
C) $10,000
The scaled MAD is MAD ÷ 0.6745 = $6,745 ÷ 0.6745 = $10,000, which approximates the standard deviation for normal data.
Why This Is the Correct Answer
Option C is correct because the scaled MAD is defined as MAD ÷ 0.6745. Here $6,745 ÷ 0.6745 = $10,000. For normally distributed data, this scaled value approximates the standard deviation.
Why the Other Options Are Wrong
Option A: $5,000
Option A, $5,000, divides by 1.349, the factor that converts an interquartile range to a standard deviation: $6,745 ÷ 1.349 = $5,000. The MAD is scaled by dividing by 0.6745: $6,745 ÷ 0.6745 = $10,000.
Option B: $6,745
Option B, $6,745, is the unscaled MAD itself. It must be divided by 0.6745 to approximate the standard deviation, giving $10,000.
Option D: $13,490
Option D, $13,490, comes from doubling the MAD: $6,745 × 2 = $13,490. The scaling constant is 0.6745 used as a divisor, so the answer is $6,745 ÷ 0.6745 = $10,000.
MAD is smaller, so divide
For normal data the MAD is smaller than the standard deviation, so you divide by 0.6745 to scale it up.
How to use: Use this to choose division over multiplication when scaling a MAD.
Exam Tip
Since 0.6745 is less than 1, the scaled MAD must be larger than the raw MAD. An answer smaller than the MAD came from the wrong constant or the wrong operation.
Common Mistakes to Avoid
- -Using the interquartile-range factor 1.349 instead of 0.6745
- -Reporting the unscaled MAD as the standard deviation estimate
- -Assuming the scaling holds for data that are far from normal
Concept Deep Dive
Analysis
This question tests the scaled median absolute deviation (MADN). For normally distributed data, dividing the MAD by 0.6745 converts it into an estimate of the standard deviation. The key is using the MAD constant 0.6745, not the interquartile-range factor 1.349, and dividing rather than reporting the raw MAD.
Background Knowledge
The median absolute deviation is a robust measure of spread that resists outliers. For a normal distribution, the MAD is about 0.6745 times the standard deviation, so dividing the MAD by 0.6745 puts it on the same scale as the standard deviation. This lets analysts compare a robust spread estimate with the usual standard deviation.
Real-World Application
An analyst reviewing a sales data set with a few extreme prices computes the MAD instead of the standard deviation. She divides it by 0.6745 to report a robust standard-deviation estimate in the same units the client is used to seeing.
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