Average the sum of rows without a creating new column in Excel

Here's a sample of my matrix:

A   B   C   D   E
1   0   0   1   1
0   0   0   0   0

0   0   1   1   0

You can think of each row as a respondent and each column as an item on a questionnaire.

My goal is to take an average of the sum of each row (i.e. total score for each respondent) without creating a new column AND accounting for the fact that some entries in a given row are empty (e.g., some respondents
didn't complete the questionnaire or missed an item).

The desired solution for this matrix = 1.67, whereby

[1+0+0+1+1 = 3] + [0+0+0+0+0 = 0] + [0+0+1+1+0 = 2]/3 == 5/3 = 1.67

As you can see, we have averaged over three values despite there being four rows because one has missing data.

I am already able to take an average of the sum of rows which are only summed for non-missing entries, e.g.,:

=AVERAGE(IF(AND(A1<>"",B1<>"",C1<>"",D1<>"",E1<>""),SUM(A1:E1)),IF(AND(A2<>"",B2<>"",C2<>"",D2<>"",E2<>""),SUM(A2:E2)),IF(AND(A3<>"",B3<>"",C3<>"",D3<>"",E3<>""),SUM(A3:E3)),IF(AND(A4<>"",B4<>"",C4<>"",D4<>"",E4<>""),SUM(A4:E4)))

However, this results in a value of 1.25 because it treats any row with missing values as = 0 to avoid a #VALUE! error due to averaging a non-number.

It does the following:

[1+0+0+1+1 = 3] + [0+0+0+0+0 = 0] + [0+0+0+0+0 = 0] + [0+0+1+1+0 = 2]/4 == 5/4 = 1.25

Does anyone have any ideas about how to adapt the current code to average over non-missing values or an alternative way of achieving the desired result?