Univariate Outlier Detection Based On Normal Distribution

Detection of Univariate Outlier Based On Normal Distribution
Data involving only one attribute or variable are called univariate data. For simplicity, we often choose to assume that data are generated from a normal distribution. We can then learn the parameters of the normal distribution from the input data, and identify the points with low probability as outliers.
Let’s start with univariate data. We will try to detect outliers by assuming the data follow a normal distribution.
Univariate outlier detection using maximum likelihood: 

Suppose a city’s average temperature values in July in the last 10 years are, in value-ascending order, 24.0°C, 28.9°C, 28.9°C, 29.0°C, 29.1°C, 29.1°C, 29.2°C, 29.2°C, 29.3°C and 29.4°C. Let’s assume that the average temperature follows a normal distribution, which is determined by two parameters: the mean, μ, and the standard deviation, σ.
We can use the maximum likelihood method to estimate the parameter μ and σ. That is, we maximize the log-likelihood function

Where n is the total number of samples, which is 10 in this sample.
Taking derivatives with respect to μ and σ2 and solving the result system of first order conditions leads to the following maximum likelihood estimates:

In this example, we have

Accordingly, we have .

The most dividing value, 24.0ºC, is 4.61ºC away from the estimated mean. We know that the region contains 99.7% data under the assumption of normal distribution. Because the probability that the value 24.0ºC is generated by the normal distribution is less than 0.15%, and thus can be identified as an outlier.