The second number

Plinko variance measured: the spread that a return figure never shows

Two paytables can hand back almost the same share of stakes and still produce completely different sessions. The difference is the spread of results, and it can be computed as easily as the return.

Plinko variance is the measure of how far results wander from their average, and it is the number most people never look at. A return figure tells you what a board hands back over an extremely long run. It says nothing about how bumpy the road is, and for anyone playing a few hundred drops, the road is most of what they will experience.

Two tables, one board

To keep the arithmetic honest, take one 8-row board and two invented paytables, read from the edge bin towards the centre:

Made-up table A: 6×, 2×, 1.2×, 0.9×, 0.5×.
Made-up table B: 26×, 4×, 1.5×, 0.3×, 0.2×.

The route counts are the same for both, because the pegs are the same: 1, 8, 28, 56 and 70 of 256 for each position from the edge inwards. Weighting each value by its chance gives a long-run return of 96.48% for table A and 96.72% for table B in this example. Judged by that single figure, they are almost twins. The page on risk settings shows where each table's total comes from.

The spread of a single drop

Variance is the average squared distance of a result from the mean. Its square root, the standard deviation, is easier to read because it comes back in money.

For a US$1 drop, table A has a standard deviation of about 58 cents. Table B has about US$2.43. The average cost of a drop is roughly 3.5 cents for A and 3.3 cents for B, so the cost barely differs while the spread is more than four times wider on B.

The reason is visible on the board. Table A pays between 50 cents and US$6, with most drops landing between 50 cents and US$1.20. Table B pays 20 or 30 cents on 71.1% of drops and saves its return for bins that are reached far less often.

Working out the spread of table B by hand

The standard deviation above is not a mystery figure; it takes five lines. Square each multiplier of the made-up table B, weight it by its share of routes, and add:

Edge pair, 26× squared is 676, times 2 of 256: 5.281.
Next pair, 4× squared is 16, times 16 of 256: 1.000.
Third pair, 1.5× squared is 2.25, times 56 of 256: 0.492.
Fourth pair, 0.3× squared is 0.09, times 112 of 256: 0.039.
Centre, 0.2× squared is 0.04, times 70 of 256: 0.011.

The total is 6.824. Subtract the square of the return, 0.9672 squared, which is 0.935, and 5.888 is left. Its square root, 2.43, is the spread of a one-dollar drop in this example. Notice that the edge pair supplies more than three quarters of the total: rare, large values are where the spread lives.

How plinko variance changes with the number of drops

Here is the rule that matters most. Over a run of N drops, the expected cost grows in step with N, while the spread grows only with the square root of N. Working through the two example tables at US$1 per drop:

100 drops

Table A: expected cost about US$3.52, spread about US$5.81.
Table B: expected cost about US$3.28, spread about US$24.27.

Counting the outcomes exactly, table A finishes ahead about 25.1% of the time and table B about 39.7%. Over a short run the wider table produces more winning sessions and more heavy losses.

1,000 drops

Table A: expected cost about US$35, spread about US$18.
Table B: expected cost about US$33, spread about US$77.

Now table A finishes ahead only about 3.2% of the time, while table B still does about 32.1% of the time. Yet table B is also down US$20 or more in about 58.4% of these runs.

10,000 drops

Table A: expected cost about US$352, spread about US$58.
Table B: expected cost about US$328, spread about US$243.

Using a normal approximation, table A is practically never ahead, and table B is ahead roughly 8.8% of the time. As the run lengthens, the average cost grows faster than the spread and pulls both tables towards a loss.

Reading plinko variance in practice

Plinko variance does not change the cost of playing; the paytable's return does that. What variance changes is the range of sessions you should be ready for. A narrow table makes a session behave much like its average, so the outcome is predictable and dull. A wide table makes individual sessions scatter, and a budget can vanish or grow for a while before the average asserts itself.

Height adds a second layer. On 16 rows each edge bin is reached once in 65,536 drops, so any large value there contributes spread that almost no session will actually meet. The row count page works through those edge chances. The multiplier tables page shows how little a rare top value adds to the return.

No setting, timing or sequence of stakes removes the cost; they only reshape the spread around it. Gambling is for adults aged 18 and over. Pick a budget that could absorb the wide end of the spread, not just the average.

Questions people type about the board

Is a table with higher variance worse?

Not in the long run, if its return is the same. In the two made-up tables here, the wider one even returns slightly more. Higher variance means sessions scatter further in both directions, so a small budget can run out quickly or grow for a while. The cost per drop comes from the return figure, which is why both numbers are worth reading together.

How can I estimate the spread of a real board?

Take the paytable for your rows and risk level and the route counts for that height. For each bin, square the multiplier and weight it by the bin's chance, add these up, then subtract the square of the return. The square root of the result is the standard deviation of a one-unit drop. Multiply by the stake and by the square root of the number of drops.