# regr_sxy

Calculates the sum of products of centered input pairs.

Dividing this sum by the pair count gives population covariance. Dividing by one less than the pair count gives sample covariance.

**Syntax**

```text
regr_sxy(y, x)
```

**Arguments**

* `y` — The dependent numeric variable.
* `x` — The independent numeric variable.

A row contributes only when both arguments are non-`NULL`. The means and the count in the formulas below use these complete pairs, rather than each column separately.

**Calculation**

```text
Sxy = sum((x - mean_x) * (y - mean_y))
```

Here `n` is the number of complete pairs, `mean_x` and `mean_y` are their arithmetic means, `Sxx` and `Syy` are their centered sums of squares, and `Sxy` is their centered cross-product sum.

**Returned value**

The centered cross-product sum as `REAL`. No complete pairs returns `NULL`; one complete pair returns `0.0`.

The result is approximate, including when the inputs are exact integers or decimals. Nonfinite inputs return `NULL`; finite arithmetic overflow raises `NUMERIC_OVERFLOW`.

**Examples**

**Measurements with a linear relationship**

The complete observations below follow `reading = 2 * hour + 1`. The two incomplete observations are excluded.

```sql
CREATE TABLE measurements (hour INTEGER, reading INTEGER);
INSERT INTO measurements VALUES (1, 3), (2, 5), (3, 7), (4, 9), (5, NULL), (NULL, 100);
```

```sql
SELECT round(regr_sxy(reading, hour), 6) AS result
FROM measurements
FORMAT csv;
```

```text
10.0
```

**No complete observations**

```sql
CREATE TABLE incomplete_measurements (hour INTEGER, reading INTEGER);
INSERT INTO incomplete_measurements VALUES (1, NULL), (NULL, 5);
```

```sql
SELECT regr_sxy(reading, hour)
FROM incomplete_measurements
FORMAT csv;
```

```text
\N
```

**Related functions**

* [`regr_count`](https://coordinatordb.com/docs/reference/functions/aggregate-functions/regr-count) counts the pairs used by the calculation.
* [`regr_slope`](https://coordinatordb.com/docs/reference/functions/aggregate-functions/regr-slope) and [`regr_intercept`](https://coordinatordb.com/docs/reference/functions/aggregate-functions/regr-intercept) describe the fitted line.
* [`covar_pop`](https://coordinatordb.com/docs/reference/functions/aggregate-functions/covar-pop), [`covar_samp`](https://coordinatordb.com/docs/reference/functions/aggregate-functions/covar-samp) and [`corr`](https://coordinatordb.com/docs/reference/functions/aggregate-functions/corr) describe the relationship between the variables.