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 Chebyshev’s inequality in data science are:

- 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.
- 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.
- 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.
- 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.

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