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119 lines
4.4 KiB
Markdown
119 lines
4.4 KiB
Markdown
---
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slug: /en/sql-reference/aggregate-functions/reference/kolmogorovsmirnovtest
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sidebar_position: 300
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sidebar_label: kolmogorovSmirnovTest
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---
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# kolmogorovSmirnovTest
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Applies Kolmogorov-Smirnov's test to samples from two populations.
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**Syntax**
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``` sql
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kolmogorovSmirnovTest([alternative, computation_method])(sample_data, sample_index)
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```
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Values of both samples are in the `sample_data` column. If `sample_index` equals to 0 then the value in that row belongs to the sample from the first population. Otherwise it belongs to the sample from the second population.
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Samples must belong to continuous, one-dimensional probability distributions.
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**Arguments**
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- `sample_data` — Sample data. [Integer](../../../sql-reference/data-types/int-uint.md), [Float](../../../sql-reference/data-types/float.md) or [Decimal](../../../sql-reference/data-types/decimal.md).
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- `sample_index` — Sample index. [Integer](../../../sql-reference/data-types/int-uint.md).
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**Parameters**
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- `alternative` — alternative hypothesis. (Optional, default: `'two-sided'`.) [String](../../../sql-reference/data-types/string.md).
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Let F(x) and G(x) be the CDFs of the first and second distributions respectively.
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- `'two-sided'`
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The null hypothesis is that samples come from the same distribution, e.g. F(x) = G(x) for all x.
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And the alternative is that the distributions are not identical.
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- `'greater'`
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The null hypothesis is that values in the first sample are *stochastically smaller* than those in the second one,
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e.g. the CDF of first distribution lies above and hence to the left of that for the second one.
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Which in fact means that F(x) >= G(x) for all x. And the alternative in this case is that F(x) < G(x) for at least one x.
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- `'less'`.
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The null hypothesis is that values in the first sample are *stochastically greater* than those in the second one,
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e.g. the CDF of first distribution lies below and hence to the right of that for the second one.
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Which in fact means that F(x) <= G(x) for all x. And the alternative in this case is that F(x) > G(x) for at least one x.
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- `computation_method` — the method used to compute p-value. (Optional, default: `'auto'`.) [String](../../../sql-reference/data-types/string.md).
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- `'exact'` - calculation is performed using precise probability distribution of the test statistics. Compute intensive and wasteful except for small samples.
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- `'asymp'` (`'asymptotic'`) - calculation is performed using an approximation. For large sample sizes, the exact and asymptotic p-values are very similar.
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- `'auto'` - the `'exact'` method is used when a maximum number of samples is less than 10'000.
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**Returned values**
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[Tuple](../../../sql-reference/data-types/tuple.md) with two elements:
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- calculated statistic. [Float64](../../../sql-reference/data-types/float.md).
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- calculated p-value. [Float64](../../../sql-reference/data-types/float.md).
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**Example**
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Query:
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``` sql
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SELECT kolmogorovSmirnovTest('less', 'exact')(value, num)
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FROM
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(
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SELECT
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randNormal(0, 10) AS value,
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0 AS num
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FROM numbers(10000)
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UNION ALL
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SELECT
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randNormal(0, 10) AS value,
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1 AS num
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FROM numbers(10000)
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)
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```
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Result:
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``` text
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┌─kolmogorovSmirnovTest('less', 'exact')(value, num)─┐
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│ (0.009899999999999996,0.37528595205132287) │
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└────────────────────────────────────────────────────┘
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```
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Note:
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P-value is bigger than 0.05 (for confidence level of 95%), so null hypothesis is not rejected.
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Query:
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``` sql
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SELECT kolmogorovSmirnovTest('two-sided', 'exact')(value, num)
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FROM
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(
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SELECT
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randStudentT(10) AS value,
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0 AS num
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FROM numbers(100)
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UNION ALL
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SELECT
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randNormal(0, 10) AS value,
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1 AS num
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FROM numbers(100)
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)
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```
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Result:
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``` text
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┌─kolmogorovSmirnovTest('two-sided', 'exact')(value, num)─┐
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│ (0.4100000000000002,6.61735760482795e-8) │
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└─────────────────────────────────────────────────────────┘
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```
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Note:
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P-value is less than 0.05 (for confidence level of 95%), so null hypothesis is rejected.
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**See Also**
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- [Kolmogorov-Smirnov'test](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test)
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