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     6         


        

Answer 1

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. x^1.2:

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×x^k + 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: