This Web site is the official and only host of the famed ** Fundamental Formula of Gambling (FFG)**. The

There is a significant number of Internet searches related to: ** fluctuation, variation, or standard deviation** in random phenomena. Fluctuation or variation can be measured by several methods. The most common method measures fluctuation in rapport to the mean average of a data series. The elements of a data series vary from the average by positive or negative quantities. The method leads to the well-known standard deviation.

The Fundamental Formula of Gambling leads to another precise instrument: the ** FFG deviation**. I found it to be significantly more precise and useful than the standard deviation. The standard deviation is viewed as: 1) a statistical parameter of a numerical series; 2) a probability parameter of binomial events.

1) The statistical standard deviation is calculated as the square root of the variance; the variance is the average of the differences from the mean of the series. A data series like 1, 2, 3, 6 has a mean equal to (1+2+3+6)/4=3. The differences from the mean are: -2, -1, 0, +3. The variance is the measurement of such differences. The variance is calculated as: {(-2)^2 + (-1)^2 + 0 + (3^2)}/4=14/4=3.5. Finally, the standard deviation is equal to the square root of the variance: SQR(3.5)=1.87.

One serious problem with the standard deviation as an analytical tool: It is distorted by extreme values (extremely high, or extremely low) in the data series.

2) The binomial standard deviation applies to events with two outcomes: win or lose. For example, betting on heads in coin tossing can lead to win (the appearance of heads) or loss (the appearance of the opposite; tails, in this case). The binomial standard deviation is calculated by the following formula:

*Standard deviation = SQR{(N*p*(1-p)}*

where p is the probability of appearance and N represents the number of trials.

Suppose we toss a coin 100 times (N=100). The probability of heads is p=1/2=0.5. The standard deviation is SQR{100 * 0.5 * 0.5} = SQR(100 * .25) =SQR(25) = 5. The expected number of heads in 100 tosses is 0.5 * 100 = 50. The rule of normal probability proves that in 68.2% of the cases, the number of heads will fall within one standard deviation from the number of expected successes (50). That is, if we repeat 1000 times the event of tossing a coin 100 times, in 682 cases we'll encounter a number of heads between 45 and 55.

This Web site offers great freeware to do a multitude of calculations on the topic of the Fundamental Formula of Gambling, plus theory of probability, and statistics.

Two programs stand out: **FORMULA** and **SuperFormula**. FORMULA is 16-bit software, now superseded by **SuperFormula**, 32-bit probability and statistics software.

This is the definitive and the ultimate probability, gambling and statistical software. Among many functions, the program can take a data series and calculate the sum, mean average, standard deviation, median, minimum, and maximum.

Here is an example of a data series saved in a lotto 5/39 game file (Pennsylvania lottery *Cash 5*):

The Sum of 13,825 numbers in *\LOTTERY\PALOTTO-5* is: 276,423

Mean Average: 19.99

Standard Deviation: 11.29

Median: 20

Minimum: 1

Maximum: 39.

The data file can be created easily in any text editor, including **MDIEditor Lotto WE**. The file can have uneven lines; i.e. variable numbers of items per line. Or, the data file can consist of one huge column; i.e. one number per line. The numbers can be separated by spaces, commas, tabs, or *Enter*. You can also export data from spreadsheets or databases to text files.

Fluctuation (variation) can be measured by another method: ** chi-squared distribution**. In this case, the terms of a data series are accompanied by the frequencies of the respective terms (elements). The frequencies are compared to the expected (theoretical) frequency. For example, in a lotto 6/49 game the expected frequency of any number in 100 drawings is:

(6 / 49) * 100 = 12.24.

Deduct the frequencies of every number from 12.24 to determine the chi-squared independence.

I prefer the ** normal probability rule** to determine the independence of a data series. Let's use the same example of 6/49 lotto game. The degree of certainty is equal to 99.8% that every lotto number will have a frequency between 2 and 22 in any 100 draws. That is, 3 standard deviations from the expected frequency of 12.

Roulette is a totally different game.

In the case of an event of probability p = .02631579 (1/38) in 100 trials:

The expected (theoretical) number of successes is: 3

Based on the Normal Probability Rule:

· 68.2% of the successes will fall within 1 Standard Deviation from 3 - i.e., between 1 - 5

·· 95.4% of the successes will fall within 2 Standard Deviations from 3 - i.e., between -1 - 7

··· 99.7% of the successes will fall within 3 Standard Deviations from 3 - i.e., between -3 - 9.

Real-life roulette spins will show that some numbers do not come out in 100 spins. There are situations when a roulette number is not drawn in over 200 spins!

The * normal probability rule* indicates a very important factor: What is the minimal number of trials to meet a degree of certainty (or a level of confidence)? In the roulette case, 100 spins are not sufficient to meet a 95% degree of certainty. Negative values for the lower bound mean that the level of confidence cannot be satisfied. The maximum satisfied is 88.15%.

In the case of an event of probability p = .02631579 (1/38) in 100 trials, 88.15% of successes will fall within 3 standard deviation(s) from 3; i.e. between 1 and 5; the standard deviation is: 1.60073.

- The Best Introduction to
**Standard Deviation**:.**Theory, Algorithm, Software** - The Latest On:
, Variability, Fluctuation, Volatility, Variation, Dispersion, Median, Mean Average.**Standard Deviation, Variance** **Standard Deviation, Gauss, Normal, Binomial, Distribution**

Calculate: Median, degree of certainty, standard deviation, binomial, hypergeometric, average, sums, probabilities, odds..**New FORMULA, standard deviation, politics**, around the median.**Generate combinations inside the bell (Gauss) curve**.**Upgrade to FORMULA: standard deviation, binomial distribution**.**Probability, odds, standard deviation, binomial software**.**Software, formulae to calculate lotto odds with hypergeometric distribution probability**.**Primer: Probability, Odds, Formulae, Algorithm, Software Calculator****Download Probability, Mathematics, Statistics, Standard Deviation****Software**.