Z-Score

How it's calculated

The Z-Score (sometimes called the standard score) is a statistical calculation.

Z-Score = (N*(R-.5)-X)/((X*(X-N))/(N-1))^(1/2)

where

• N = The total number of trades in the sequence.
• R = The total number of runs in the sequence.
• X = 2*W*L
• W = The total number of winning trades in the sequence.
• L = The total number of losing trades in the sequence.

What it means

The Z score is the result of the runs test and will tell us if our system has more (or fewer) streaks of consecutive wins and losses than a random distribution. The Z score shows us how many standard deviations we are away from the mean of a distribution.

What we are trying to answer is how many streaks of wins (losses) can we expect from a given system? Are the win (loss) streaks of the system we are testing in line with what we could expect? If not, is there a high enough confidence limit that we can assume dependency exists between trades -i.e., is the outcome of a trade dependent on the outcome of previous trades?

The Z score only takes into account the dependency from the point of view of whether the last trade was a winner or a loser. It does not take into account the size of each winner or loser. For that we use the Serial Correlation.

How can we profit from it?

The Z score tells the trader the tendency for a system to win or lose in streaks. Streaks are defined by the times a system shifts between losing or winning trades. The Z score is calculated by comparing the number of streaks there are in a set of trades with the number of streaks that could be expected randomly. We can then transform this number into another value called the confidence level which gives us a good idea of the relationship between wins and losses.

Use this table to translate the absolute Z Score to the confidence level:

Z-Score	Confidence Limit (%)
3.00	99.73
2.58	99.00
2.33	98.00
2.17	97.00
2.05	96.00
2.00	95.45
1.96	95.00
1.64	90.00

A confidence level of 95% or above is needed to exploit (for extra profit) the apparent non-randomness of streaks in a system so based on this table we're looking for Z scores of above 1.96 and below -1.96.

A negative Z score means that there are fewer streaks in the trading system than would be expected statistically. This means that winning trades tend to follow winning trades and that losing trades tend to follower losers. A positive Z score means that there are more streaks in the trading system than would be expected. This means that winners tend to follow losers and vice versa.

If you discover that the system that you have back tested or the system that you are currently trading has a reasonable Z score then it is possible to exploit the system for extra profit using money management techniques.

Further reading

If this subject interests you then we can recommend that you read more about this in The Mathematics of Money Management: Risk Analysis Techniques for Traders by Ralph Vince. A number of his ideas were used in the back testing system that we developed.