# The Sum of the Squares of the Suit Lengths

## A Measure of the "Sickness" of a Hand Pattern

In any hand, the average of the suit lengths is 3.25 = 13/4, obviously, but what is the standard deviation of the suit lengths?

The standard deviation, it turns out, is related to the sum of the squares of the suit lengths. In particular, the higher the sum of the squares, the higher the standard deviation.

So it should come as no surprise that the more balanced hands should have lower standard deviations, and hence lower sums of squares. In particular, when the sum of the squares of the suit lengths is less than or equal to 47, the hand is "balanced" in the traditional sense: No singleton or void, and at most one doubleton.

Since the smallest value is 43, and all values are odd, we can "normalize" this metric by subtracting 43 and dividing by 2.

We get the following table:

```Normalized | Squares sum | Std Dev  | Shapes
============================================
0 |          43 |  0.5000  | 4-3-3-3
1 |          45 |  0.9574  | 4-4-3-2
2 |          47 |  1.2583  | 5-3-3-2
3 |          49 |  1.5000  | 4-4-4-1, 5-4-2-2
4 |          51 |  1.7078  | 5-4-3-1
5 |          53 |  1.8930  | 6-3-2-2
6 |          55 |  2.0616  | 5-5-2-1,6-3-3-1
7 |          57 |  2.2166  | 5-4-4-0,6-4-2-1
8 |          59 |  2.3629  | 5-5-3-0
9 |          61 |  2.5000  | 7-2-2-2,6-4-3-0
10 |          63 |  2.6300  | 7-3-2-1,6-5-1-1
11 |          65 |  2.7538  | 6-5-2-0
12 |          67 |  2.8723  | 7-3-3-0, 7-4-1-1
13 |          69 |  2.9860  | 7-4-2-0
15 |          73 |  3.2016  | 6-6-1-0, 8-2-2-1
16 |          75 |  3.3040  | 7-5-1-0, 8-3-1-1
17 |          77 |  3.4034  | 8-3-2-0
19 |          81 |  3.5940  | 8-4-1-0
21 |          85 |  3.7749  | 7-6-0-0
22 |          87 |  3.8622  | 9-2-1-1
23 |          89 |  3.9476  | 8-5-0-0, 9-2-2-0
24 |          91 |  4.0311  | 9-3-1-0
27 |          97 |  4.2720  | 9-4-0-0
30 |         103 |  4.5000  | 10-1-1-1
31 |         105 |  4.5735  | 10-2-1-0
33 |         109 |  4.7170  | 10-3-0-0
40 |         123 |  5.1881  | 11-1-1-0
41 |         125 |  5.2519  | 11-2-0-0
51 |         145 |  5.8524  | 12-1-0-0
63 |         169 |  6.5000  | 13-0-0-0

```
In addition, one can get the wildness of a deal by adding up the sum of the squares of all 16 suit lengths, or add up the normalized values. Again, this is correlated to the standard deviation of the suit lengths.

We can also measure the wildness of a "fit" by summing the squares of the fits in each suit. So if I'm 5-3-3-2 and partner is 4-3-2-4, our "fit" is 9-6-5-6. One thing interesting about this is that our opponent's fit wildness is the same as our fit pattern wildness. That's because if our pattern is: s-h-d-c, their pattern is (13-s)-(13-h)-(13-d)-(13-c), and the sum of the squares of these values is:

```(13-s)^s+(13-h)^s+(13-d)^2+(13-c)^2=
s^2 + h^2 + d^2 + c^2 - 26*(s+h+d+c) + 4*13^2
```
But (s+h+d+c) is 26, so the last two terms cancel, and we are left with the original value.

Of course, if you think in terms of standard deviation, this makes more sense than the pure calculation - the opponents' pattern has the same relative distribution, just inverted, so the deviation should be the same.

This table can also be normalized, by subtracting 170 and dividing by 2. There are 103 fit patterns, or 65 if we consider our fit pattern and opponent's as the same (e.g., that 8-6-6-6 is the same as 7-7-7-5. If the sum of the longest fit and the shortest fit is 13, then the pattern is self-dual, for example, if our fit pattern is 9-7-6-4, then so is the opponent's.)

```Normalized |  Squares sum | Patterns
============================================
0 |          170 | 7-7-6-6
1 |          172 | 8-6-6-6,7-7-7-5
2 |          174 | 8-7-6-5
4 |          178 | 9-6-6-5,8-8-5-5,8-7-7-4
5 |          180 | 9-7-5-5,8-8-6-4
6 |          182 | 9-7-6-4
8 |          186 | 10-6-5-5,9-8-5-4,8-8-7-3
9 |          188 | 10-6-6-4,9-7-7-3
10 |          190 | 10-7-5-4,9-8-6-3
12 |          194 | 10-7-6-3,9-9-4-4
13 |          196 | 11-5-5-5,10-8-4-4,9-9-5-3,8-8-8-2
14 |          198 | 11-6-5-4,10-8-5-3,9-8-7-2
16 |          202 | 11-7-4-4,11-6-6-3,10-7-7-2,9-9-6-2
17 |          204 | 11-7-5-3,10-8-6-2
18 |          206 | 10-9-4-3
20 |          210 | 12-5-5-4,11-8-4-3,11-7-6-2,10-9-5-2,9-8-8-1
21 |          212 | 12-6-4-4,9-9-7-1
22 |          214 | 12-6-5-3,11-8-5-2,10-8-7-1
24 |          218 | 12-7-4-3,10-10-3-3,10-9-6-1
25 |          220 | 12-6-6-2,11-9-3-3,11-7-7-1,10-10-4-2
26 |          222 | 12-7-5-2,11-9-4-2,11-8-6-1
28 |          226 | 13-5-4-4,12-8-3-3,10-10-5-1,9-9-8-0
29 |          228 | 13-5-5-3,12-8-4-2,11-9-5-1,10-8-8-0
30 |          230 | 13-6-4-3,12-7-6-1,10-9-7-0
32 |          234 | 13-6-5-2,12-8-5-1,11-10-3-2,11-8-7-0
33 |          236 | 13-7-3-3,10-10-6-0
34 |          238 | 13-7-4-2,12-9-3-2,11-10-4-1,11-9-6-0
36 |          242 | 13-6-6-1,12-9-4-1,12-7-7-0
37 |          244 | 13-7-5-1,12-8-6-0
38 |          246 | 13-8-3-2,11-10-5-0
40 |          250 | 13-8-4-1,12-9-5-0,11-11-2-2
41 |          252 | 12-10-2-2,11-11-3-1
42 |          254 | 13-7-6-0,12-10-3-1
44 |          258 | 13-9-2-2,13-8-5-0,11-11-4-0
45 |          260 | 13-9-3-1,12-10-4-0
48 |          266 | 13-9-4-0
50 |          270 | 12-11-2-1
52 |          274 | 13-10-2-1,12-11-3-0
54 |          278 | 13-10-3-0
60 |          290 | 12-12-1-1
61 |          292 | 13-11-1-1,12-12-2-0
62 |          294 | 13-11-2-0
72 |          314 | 13-12-1-0
84 |          338 | 13-13-0-0
```

 Thomas Andrews (deal@thomasoandrews.com) Copyright 1996-2010. Deal is covered by the GNU General Public License. Plane Dealing graphic above created using POV-Ray.