Question

Out of 5,000 daily observations, you calculate an average of 1%, and a standard deviation of 5%. If your return distribution is normal, how many observations will fall outside the range [-14% , +16%]?

Answer 1

In statistics, the 68–95–99.7 rule is a shorthand used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of two, four and six standard deviations, respectively; more accurately, 68.27%, 95.45% and 99.73% of the values lie within one, two and three standard deviations of the mean, respectively.

So since the mean is 1%, we could estimate that most of the data falls in the interval

[1−3(5),1+3(5)]=[-14,16].

Around 68% of data is between standard deviation -1 to 1. Here it is between 20(1-1x5) to 28(1+1x5).

Around 96% of data is between standard deviation -2 to 2. Here it is between 16(1-2x5) to 32(1+2x5).

Around 100% of data is between standard deviation -3 to 3. Here it is between 12(1-3x5) to 36(1+3x5)

Since 99.7% observation will between the range between -14% to 16% only 0.3% will fall outside the [-14%, _16] i.e 15 Observations