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Same expected month with half the size and twice the trades: SD of 11.1% falls to 7.9%

Two traders have the same payoff: 45% wins at 1.5R, a 0.125R expectancy per trade, before costs. The first takes 80 trades a month at 1% risk. The second takes 160 trades at 0.5%.

The expected month is the same: 80 x 0.125 x 1% = 10%, and 160 x 0.125 x 0.5% = 10%. But the spread of results is not. The standard deviation of one trade is about 1.24R, and for n trades it grows with the square root of n. For the first trader that is 1.24 x the square root of 80 = 11.1R, or 11.1% at 1% risk. For the second it's 1.24 x the square root of 160 = 15.7R, or 7.9% at 0.5%.

A third trader with 320 trades at 0.25% has the same 10% expected month and a standard deviation of 5.6%.

The chance of a losing month shows the same pattern. Using exact binomial odds, it's 15.6% for 80 trades, 8.8% for 160 trades and 3.1% for 320 trades. In every case the average is a gain of 10%. Dividing the average by the standard deviation, you get 0.90 for the first, 1.27 for the second and 1.80 for the third.

The catch is obvious. The second and third traders need twice and four times the number of setups of the same quality, and each one needs its own entry, stop and management. Whether the market gives you that many is a separate question.

What if you only halve the size and keep the 80 trades? Then the expected month is 5% and the standard deviation is 1.24 x the square root of 80 x 0.5% = 5.6%. The chance of a losing month stays at 15.6%, since that depends on the number of trades and the edge, and not on the size. So smaller size alone makes the swings smaller in dollars and leaves the odds of a red month as they were. Only more trades change the odds. That is a useful thing to know before cutting risk in the hope that the red months will go away.

The ratio of the average to the standard deviation rises by the square root of 2 for each doubling of the trade count: 0.90, then 1.27, then 1.80. Going from 80 to 320 trades takes the ratio up by a factor of 2, and cuts the monthly deviation by half at the same expected return. That is the arithmetic behind running many small bets. For a retail scalper the limit is the supply of good setups and the time at the screen, and not the formula, which is why the table is a target and not a plan.

Dilution is the practical problem. Say the extra 80 trades of the second trader are weaker, with an edge of 0.05R and not 0.125R. Then the expected month is 80 x 0.125 x 0.5% plus 80 x 0.05 x 0.5%, which is 5% plus 2%, so 7%. The standard deviation stays near 7.9%. The ratio is 0.89, no better than the single 80 trade month at 0.90. So extra trades help only when they have close to the same edge as the first batch, and checking that is a matter of splitting your journal in two.