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> Applies Welch's t-test to samples from two populations.

# welchTTest

<h2 id="welchTTest">
  welchTTest
</h2>

Introduced in: v21.1.0

Applies [Welch's t-test](https://en.wikipedia.org/wiki/Welch%27s_t-test) to samples from two populations.

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.
The null hypothesis is that means of populations are equal.
Normal distribution is assumed.
Populations may have unequal variance.

**Syntax**

```sql theme={null}
welchTTest([confidence_level])(sample_data, sample_index)
```

**Parameters**

* `confidence_level` — Optional. Confidence level in order to calculate confidence intervals. [`Float`](/reference/data-types/float)

**Arguments**

* `sample_data` — Sample data. [`Int*`](/reference/data-types/int-uint) or [`UInt*`](/reference/data-types/int-uint) or [`Float*`](/reference/data-types/float) or [`Decimal*`](/reference/data-types/decimal)
* `sample_index` — Sample index. [`Int*`](/reference/data-types/int-uint) or [`UInt*`](/reference/data-types/int-uint)

**Returned value**

Returns a Tuple with two or four elements (if the optional `confidence_level` is specified): calculated t-statistic, calculated p-value, and optionally calculated confidence-interval-low and confidence-interval-high. [`Tuple(Float64, Float64)`](/reference/data-types/tuple) or [`Tuple(Float64, Float64, Float64, Float64)`](/reference/data-types/tuple)

**Examples**

**Basic Welch's t-test**

```sql title=Query theme={null}
CREATE TABLE welch_ttest (sample_data Float64, sample_index UInt8) ENGINE = Memory;
INSERT INTO welch_ttest VALUES (20.3, 0), (22.1, 0), (21.9, 0), (18.9, 1), (20.3, 1), (19, 1);

SELECT welchTTest(sample_data, sample_index) FROM welch_ttest;
```

```response title=Response theme={null}
┌─welchTTest(sample_data, sample_index)──────┐
│ (2.7988719532211235, 0.051807360348581945) │
└────────────────────────────────────────────┘
```

**With confidence level**

```sql title=Query theme={null}
SELECT welchTTest(0.95)(sample_data, sample_index) FROM welch_ttest;
```

```response title=Response theme={null}
┌─welchTTest(0.95)(sample_data, sample_index)─────────────────────────────────────────┐
│ (2.7988719532211235, 0.05180736034858519, -0.026294346671631885, 4.092961013338302) │
└─────────────────────────────────────────────────────────────────────────────────────┘
```

**See Also**

* [Welch's t-test](https://en.wikipedia.org/wiki/Welch%27s_t-test)
* [studentTTest function](/reference/functions/aggregate-functions/studentTTest)