@Fat Tails, @panjm

Thank You very much for being so generous with your time and attention to detail ... it is much appreciated

Fat Tails 

The standard deviation is a statistical measure for the dispersion of data points around an arithmetic mean. Let us take your example of 5 data points, such as 5 consecutive closes as shown on the chart. I prefer a real example, so let us look at the last five closes of yesterday's regular session for YM 09-13. The close values were 14820, 14812, 14824, 14814, 14830 The arithmetic mean of those data points - identical to a SMA(5) - is obtained by adding up the five values and dividing by 5. arithmetic mean = SMA(5) = (14820 + 14812 + 14824 + 14814 + 14830)/5 = 14820 The arithmetic mean of 14820 does not tell us how far the single data points are away from the mean value. For example, if you had the values 13820, 12812, 15824, 13814, 17830, you would obtain the same mean value, but quite obviously the data points are further away from the mean. The standard deviation is calculated as the square root of the variance. The variance is the sum of the squares of the distances of each data point from the mean. For your example above we calculate variance = [(14820 - 14820)^2 + (14812 - 14820)^2 + (14824 - 14820)^2 + (14814 - 14820)^2 + (14830 - 14820)^2] / 5 = [0 + 64 + 16 + 36 + 100]/5 = 43.2 standard deviation = sqrt (43.2) = 6.57 Now this is the calculation method used for the Standard Deviation and Bollinger Band indicators. The method is not really accurate, because it systematically underestimates the expected standard deviation for a small sample size. Using a modified formula for the population variance would result int values of 54.00 for the variance and 7.35 for the standard deviation. Meaning of 1 Standard Deviation At least for normal (Gaussian, bell-shaped) distributions, you may expect that 68.2% of all data points can be found within the distance of 1 standard deviation from the mean. About 95.4% of all data points can be found with the distance of 2 standard deviations from the mean. Applied to our example this means that we may expect about 3.4 out of 5 data point to lie within the interval [14820 - 7.35, 14820 + 7.35] or [14813, 14827]. In fact 3 data points can be found within this interval, while two data points are located outside the interval. If we take the interval based on two standard deviations [14806, 14834], we would expect 95% of all data points to lie within this interval. For our example, all values can be found within that interval. Although these estimations only apply to normal distributions, they are used for other distributions as well, not because they are correct, but because these estimations are easy to calculate.

@Fat Tails,

If I may ask a couple more questions ...

Is there an indicator that uses the modified formula for the population variance?

Am I correct in assuming that if one standard deviation represents 68.2% of the data points, that by simply using a standard deviation of 1.0263929 to represent 70% of the data points is not accurate because of the Gaussian, bell-shape?

Thank You for your time.

rdtw 

@Fat Tails, If I may ask a couple more questions ... Is there an indicator that uses the modified formula for the population variance? Am I correct in assuming that if one standard deviation represents 68.2% of the data points, that by simply using a standard deviation of 1.0263929 to represent 70% of the data points is not accurate because of the Gaussian, bell-shape? Thank You for your time.

All indicators that I know (including StdDev, Bollinger Bands and the VWAP) use the simple formula.

The rule that one standard deviation represents approximately 68.2% of all data points is only true for normal distributions. It can be entirely false for other distributions. Prices are not distributed normally. Price volume distributions can follow anything from lognormal to power laws. On many days the volume does not even follow a unimodal (single peek) distribution, but you will often observe double or even triple distribution days. Although the standard deviation cannot be used to describe a defined percentage of volume, it is used by many traders, because it is easy to calculate.