# Mean absolute deviation (MAD) analogy to 68-95-99 rule

With MAD, 50% of all values fall within one absolute deviation. How many within two, and three?

• $1)$ Welcome to Cross Validated! What 68-95-99.7 rule applies to normal distributions but not to distributions in general. Do you want to assume a normal distribution? // $2)$ What definition of MAD do you want to use, deviation from the mean or deviation from the median?
– Dave
Commented May 31, 2022 at 19:01
• If you're not specifically interested in normality, nor indeed any other specific distribution, then you can pretty much only hope for bounds. The "continuous, unimodal" case might be somewhat interesting in that case. Commented Jun 1, 2022 at 7:51

## 1 Answer

Assuming Normally distributed data (since the $$68-95-99$$ rule comes from the Normal distribution) I estimate that about $$82\%$$ of samples lie within $$\pm 2\text{MAD}$$ of the median and around $$96\%$$ of samples lie within $$\pm 3 \text{MAD}$$ of the median. This is based upon three assumptions

1. Your estimate of central tendency, $$m$$, is the sample median
2. You estimate MAD by $$\text{MAD} = \frac{1}{n} \sum_{i = 1}^n |x_i - m|$$
3. your data are Normally distributed

R code:

set.seed(123)
x <- rnorm(10^6)
m = median(x)
mad = median(abs(x - m))
mean(abs(x - m) < 1*mad)
[1] 0.5
mean(abs(x - m) < 2*mad)
[1] 0.822464
mean(abs(x - m) < 3*mad)
[1] 0.956922

• Thanks, good idea to simulate it. I usually do so when I can’t understand it from the math directly. If I meant that normal distribution should be assumed, well, I don’t know. Is normal distribution derived from standard deviation being assumed, or does it not matter much if MAD is used instead? Mostly I was thinking there might be some obvious pattern where data falls off in a way that is intuitively easy to get. 68-95-99 isn’t super intuitive, it uses some error function of 1/sqrt(2), etc. Commented May 31, 2022 at 19:59
• There's something a little oxymoronic about this approach, because someone employing MAD likely is concerned that the data are not Normal. Under the assumption of Normality they also would more likely just convert the MAD into an estimate of SD by multiplying it by about 5/4 (approximating $\sqrt{\pi/2}$) and then proceed to apply the 68-95-99.7 rule.
– whuber
Commented May 31, 2022 at 21:05
• Wuhber, if you read what I wrote, I suggested I might not be assuming normal distribution. Jcken likely built on Dave’s comment. I appreciate what Jcken contributed so far. Commented May 31, 2022 at 23:08
• @whuber I agree; however the reference to the 68-95-99 rule in the title implies OP was interested in the Normal distribution. @status; you should consider updating your question to be clearer about what types of distributions you want to learn about. For example, Pukelsheim's 3$\sigma$ rule tells us that under quite weak assumptions, at 95% of values lie within $\pm 3 \sigma$ of the mean jstor.org/stable/2684253 - I am unaware of similar results that use MAD but there may be something out there Commented Jun 1, 2022 at 6:48
• Further experimentation suggests the result of 50% laying within $\pm 1 \text{MAD}$ only holds when the underlying distribution is continuous. Using the data generating distribution as poisson(1) gives $\approx 36%$ within $\pm 1 \text{MAD}$ of the median Commented Jun 1, 2022 at 6:58