Quadratic and cubic regression in Excel

Question

I have the following information: ``` Height Weight 170 65 167 55 189 85 175 70 166 55 174 55 169 69 170 58 184 84 161 56 170 75 182 68 167 51 187 85 178 62 173 60 172 68 178 55 175 65 176 70 ``` I want to construct quadratic and cubic regression analysis in Excel. I know how to do it by linear regression in Excel, but what about quadratic and cubic? I have searched a lot of resources, but could not find anything helpful.

Accepted Answer

You need to use an undocumented trick with Excel's `LINEST` function:

```
=LINEST(known_y's, [known_x's], [const], [stats])
```

Background

A regular linear regression is calculated (with your data) as:

```
=LINEST(B2:B21,A2:A21)
```

which returns a single value, the linear slope (`m`) according to the formula:

which for your data:

is:

Undocumented trick Number 1

You can also use Excel to calculate a regression with a formula that uses an exponent for `x` different from `1`, e.g. x1.2:

using the formula:

```
=LINEST(B2:B21, A2:A21^1.2)
```

which for you data:

is:

You're not limited to one exponent

Excel's `LINEST` function can also calculate multiple regressions, with different exponents on `x` at the same time, e.g.:

```
=LINEST(B2:B21,A2:A21^{1,2})
```

Note: if locale is set to European (decimal symbol ","), then comma should be replaced by semicolon and backslash, i.e. `=LINEST(B2:B21;A2:A21^{1\2})`

Now Excel will calculate regressions using both x1 and x2 at the same time:

How to actually do it

The impossibly tricky part there's no obvious way to see the other regression values. In order to do that you need to:

select the cell that contains your formula:

extend the selection the left 2 spaces (you need the select to be at least 3 cells wide):

press F2

press Ctrl+Shift+Enter

You will now see your 3 regression constants:

```
  y = -0.01777539x^2 + 6.864151123x + -591.3531443
```

Bonus Chatter

I had a function that I wanted to perform a regression using some exponent:

y = m×xk + b

But I didn't know the exponent. So I changed the `LINEST` function to use a cell reference instead:

```
=LINEST(B2:B21,A2:A21^F3, true, true)
```

With Excel then outputting full stats (the 4th paramter to `LINEST`):

I tell the Solver to maximize R2:

And it can figure out the best exponent. Which for you data:

is:

Quadratic and cubic regression in Excel

Solution

Problem