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.

Calculating Probabilities

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:

• [latex]P[/latex](at most or less than) =NORM.DIST([latex]x[/latex], µ, σ, TRUE)
• [latex]P[/latex](at least or more than) =1−NORM.DIST([latex]x[/latex], µ, σ, TRUE)
• Where µ (mu) is the mean of the distribution and σ (sigma) is the standard deviation.

Calculating X-Values

If we are looking to solve for the [latex]x[/latex]-value instead of the probability, this is called an ‘inverse‘ problem and we use Excel’s NORM.INV() function:

• [latex]x[/latex] = NORM.INV(Area to left of [latex]x[/latex], µ, σ)
• [latex]x[/latex] = NORM.INV(1− Area to right of [latex]x[/latex], µ, σ)

Z-Scores

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 [latex]x[/latex]-value, µ (mu) and, σ (sigma) are given:
\[z = \frac{x-\mu}{\sigma}\]

It can be calculated using Excel’s NORM.S.INV function if the area/probability is given: \[ z = \text{NORM.S.INV}(\text{Area to left of z}) = \text{NORM.INV}(1-\text{Area to right of z})\]

Symmetric Property

• 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.

Statistical Properties

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

Video & Resources Explaining Normal Distributions

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 (EXERCISE)

Your Own Notes (EXERCISE)

• Are there any notes you want to take from this section? Is there anything you’d like to copy and paste below?
• These notes are for you only (they will not be stored anywhere)
• Make sure to download them at the end to use as a reference