Normal Distributions
Learning Objectives
Understand the shape, statistical properties and formulas for Normal Distributions.
Properties of Normal Distributions
A normal distribution is:
- The most common type of distribution
- It is continuous and has a “bell” shape.
- It is ‘symmetric’ about the mean (µ) (see more in the ‘SYMMETRIC’ section)
- The total area under the normal curve is 1.
- Ie: the probability of being anywhere on the distribution=1.
It would require a calculate technique called Integration by Parts to calculate the probabilities by hand for the Normal Distribution. For this reason, we will only use Excel’s NORM.DIST() function to calculate probabilities:
- (at most or less than) =NORM.DIST(, µ, σ, TRUE)
- (at least or more than) =1−NORM.DIST(, µ, σ, TRUE)
- Where µ (mu) is the mean of the distribution and σ (sigma) is the standard deviation.
If we are looking to solve for the -value instead of the probability, this is called an ‘inverse‘ problem and we use Excel’s NORM.INV() function:
- = NORM.INV(Area to left of , µ, σ)
- = NORM.INV(1− Area to right of , µ, σ)
A z-score is:
- “A statistical measurement that describes a value’s relationship to the mean of a group of values.”
- “Measured in terms of standard deviations from the mean.”
- “A measure of an instrument’s variability and can be used by traders to help determine volatility.”
It can be calculated using a formula if the -value, µ (mu) and, σ (sigma) are given:
It can be calculated using Excel’s NORM.S.INV function if the area/probability is given:
- It is symmetric (or identical) on either side of the mean.
- The mean and median are equal.
- The data in this distribution is neither skewed left nor skewed right.
The following metrics apply to Normal Distributions:
- population mean = µ
- sample mean = x̄
- population standard deviation = σ
- sample standard deviation = s
- mode = µ or x̄ (depending if population or sample given)
- variance = σ2 or s2 (depending if population or sample given)
- symmetric (not skewed) and the skewness = 0
Additional Resources:
- Click here to download the Powerpoint slides that accompany the video.
- Click here to download the Excel solutions for the Normal Distribution section.
Key Takeaways: Normal Distributions
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