# Which algorithms or methods can be used to detect an outlier from this data set?

Suppose I have a data set : Amount of money (100, 50, 150, 200, 35, 60 ,50, 20, 500). I have Googled the web looking for techniques that can be used to find a possible outlier in this data set but I ended up confused.

My question is: Which algorithms, techniques or methods can be used to detect possible outlier in this data set?

PS:Consider that the data does not follow a normal distribution. Thanks.

• How do you recognize an outlier on this small set? How would you do "by hand" on slightly bigger data? – Laurent Duval Nov 5 '15 at 23:02

You can use BoxPlot for outlier analysis. I would show you how to do that in Python:

Consider your data as an array:

a = [100, 50, 150, 200, 35, 60 ,50, 20, 500]


Now, use seaborn to plot the boxplot:

import seaborn as sn
sn.boxplot(a)


So, you would get a plot which looks somewhat like this: Seems like 500 is the only outlier to me. But, it all depends on the analysis and the tolerance level of the analyst or the statistician and also the problem statement.

You can have a look at one of my answers on the CrossValidated SE for more tests.

And there are several nice questions on outliers and the algorithms and techniques for detecting them.

My personal favourite is the Mahalanobis distance technique.

• Well, thanks, a nice explanation indeed. Can I apply Mahalanobis distance technique in this case or it is works for multivariate data? – Nation Chirara Nov 4 '15 at 14:19
• You can. But, it's over-qualified for univariate data. Just a Boxplot analysis with some set threshold value should do the job for this data. – Dawny33 Nov 4 '15 at 14:39

One way of thinking of outlier detection is that you're creating a predictive model, then you're checking to see if a point falls within the range of predictions. From an information-theoretic point of view, you can see how much each observation increases the entropy of your model.

If you are treating this data as just a collection of numbers, and you don't have some proposed model for how they're generated, you might as well just look at the average. If you're certain the numbers aren't normally distributed, you can't make statements as to how far 'off' a given number is from the average, but you can just look at it in absolute terms.

Applying this, you can take the average of all the numbers, then exclude each number and take the average of the others. Whichever average is most different from the global average is the biggest outlier. Here's some python:

def avg(a):
return sum(a)/len(a)

l = [100, 50, 150, 200, 35, 60 ,50, 20, 500]
m = avg(l)
for idx in range(len(l)):
print("outlier score of {0}: {1}".format(l[idx], abs(m - avg([elem for i, elem in enumerate(l) if i!=idx]))))
>>
outlier score of 100: 4
outlier score of 50: 10
outlier score of 150: 3
outlier score of 200: 9
outlier score of 35: 12
outlier score of 60: 9
outlier score of 50: 10
outlier score of 20: 14
outlier score of 500: 46

• Well, thanks for the response. I am certain that the numbers are not normally distributed, does your method still apply in that context? – Nation Chirara Nov 4 '15 at 14:32
• Yes, there is always a underlying model that implies predictions about the data. If we get a new data point, we can make a guess about what that value is. If we have no other information about the model (it's just a collection of numbers), then the best guess is that it will look similar to the numbers we already have. If you do end up with a specific model for the data, it's important to incorporate that into your prediction. Using mean doesn't imply we're assuming normal distribution, but other models could use something other than the mean. – Tristan Reid Nov 4 '15 at 19:52
• Hello. I usually don't find that's a good idea to use average for non normal distributed data. In fact, for example, many non parametric statistical tests are using median instead of mean. But that's only an opinion... – Michael Hooreman Nov 10 '15 at 8:11
• Fair enough, but I think it's safe to say if you believe your data is distributed symmetrically, mean is a pretty good idea, otherwise median is a better measure of a 'typical' value. – Tristan Reid Nov 10 '15 at 22:06

A simple approach would be using the same thing as box plots does: away than 1.5 (median-q1) or 1.5 (q3-median) = outlier.

I find it useful in lots of cases even it not perfect and maybe too simple.

It has the advantage to not suppose normality.