# regr_slope

Calculates the slope of a least-squares regression line.

For a fitted line `y = slope * x + intercept`, the slope is the change in the predicted `y` for a one-unit increase in `x`.

**Syntax**

```text
regr_slope(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 / Sxx
```

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 slope as `REAL`. Fewer than two complete pairs, or zero variation in `x`, returns `NULL`.

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_slope(reading, hour), 6) AS result
FROM measurements
FORMAT csv;
```

```text
2.0
```

**No complete observations**

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

```sql
SELECT regr_slope(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.