Data Science in ClickHouse: How to implement Chebyshev’s Inequality?

Introduction

Chebyshev’s inequality is a statistical theorem that provides a bound on the probability that a random variable deviates from its mean by more than a certain number of standard deviations. Specifically, it states that for any random variable X with finite mean μ and finite variance σ^2, the probability that X deviates from its mean by more than k standard deviations is at most 1/k^2, where k is any positive number greater than 1.

Data scientists can use Chebyshev’s inequality to make statements about the distribution of data, even when they do not know the exact distribution. For example, they can use the inequality to estimate the proportion of data that falls within a certain range of values, or to identify potential outliers in the data.

Chebyshev’s inequality is a fundamental concept in statistics and has several use cases in data science. Some of the common use cases of it in data science are:

1. Outlier Detection: Chebyshev’s inequality provides a bound on the probability of a data point lying outside a certain number of standard deviations from the mean. Data scientists can use this bound to identify potential outliers in the data that fall outside the expected range.
2. Confidence Intervals: Chebyshev’s inequality can be used to estimate the proportion of data that falls within a certain range of values, even when the distribution is not known. This can be useful in estimating confidence intervals for statistical estimates.
3. Sample Size Estimation: Chebyshev’s inequality can also be used to estimate the required sample size to achieve a certain level of precision in statistical estimates.
4. Quality Control: Chebyshev’s inequality can be used to set quality control limits for manufacturing or production processes. The limits can be set based on the expected range of values, and any measurements that fall outside this range can be investigated for potential issues.

Implementing Chebyshev’s Inequality In ClickHouse

In ClickHouse, you can implement Chebyshev’s inequality using the quantiles function. This function calculates the empirical quantiles of a dataset, which can be used to estimate the proportion of data that falls within a certain range of values. To apply Chebyshev’s inequality, you can use the quantiles function to estimate the median and the interquartile range of the data, and then use these estimates to calculate the bounds on the proportion of data that falls within a certain number of standard deviations from the mean.

Here’s an example of how to implement Chebyshev’s inequality in ClickHouse:

-- Calculate the median and interquartile range of a dataset
SELECT quantiles(x, 2) AS median, quantiles(x, 4) - quantiles(x, 1) AS iqr
FROM my_table

-- Calculate the bounds on the proportion of data that falls within k standard deviations from the mean
SELECT count(*) / n() AS proportion
FROM my_table
WHERE x BETWEEN median - k * iqr / 2 AND median + k * iqr / 2

Conclusion

We looked at how data scientists can use Chebyshev’s inequality to make statements about the distribution of data and how the same can be implemented in ClickHouse

To read more about Data Science in ClickHouse, do consider reading the following articles

• Data Science in ClickHouse: Implementation of Black-Scholes Model
• Data Science in ClickHouse: Normal Distributions and t Distributions
• Data Science in ClickHouse: Skewed Q-Q Plots