Limited Time Offer: Save 10% on all 2021 and 2022 Premium Study Packages with promo code: BLOG10    Select your Premium Package »

The Standard Normal Distribution

The Standard Normal Distribution

The standard normal distribution refers to a normal distribution that has been standardized such that it has a mean of 0 and a standard deviation of 1. The shorthand notation used is:

$$ N \sim (0, 1) $$

In the context of statistics and mathematics, standardization is the process of converting an observed value for a random variable into a z-value where:

$$ \begin{align*} Z & = \cfrac {(\text{observed value} – \text{population mean})} {\text{standard deviation}} \\ & =\cfrac {(x – \mu)}{\sigma} \\ \end{align*} $$

The z-value, also referred to as the z-score in some books, represents the number of standard deviations a given observed value is from the population mean.

Example: Standard Normal Distribution

The returns on ABC stock are distributed normally. The mean is $0.60 with a standard deviation of $0.20. Calculate the z-scores for a return of $0.10.

Solution

If the return is $0.10, then x = 0.1 (this is our observed value). Therefore,

$$ \begin{align*} z & =\cfrac {(x – \mu)}{\sigma} \\ & =\cfrac {(0.1 – 0.6)}{0.2} \\ & = -2.5 \quad (\text{The return of }$0.1 \text{ is two and a half standard deviations below the mean.}) \end{align*} $$

We could get the z-score for any other observed value following a similar approach. For instance, the z-score for a return of $1 will be:

$$ \begin{align*} z & =\cfrac {(1 – 0.6)}{0.2} \\ & = 2 \quad (\text{The return of } $1 \text{ is two standard deviations above the mean.}) \\ \end{align*} $$

Featured Study with Us
CFA® Exam and FRM® Exam Prep Platform offered by AnalystPrep

Study Platform

Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Online Tutoring
    Our videos feature professional educators presenting in-depth explanations of all topics introduced in the curriculum.

    Video Lessons



    Sergio Torrico
    Sergio Torrico
    2021-07-23
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    diana
    diana
    2021-07-17
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    2021-07-16
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    2021-06-28
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Marwan
    Marwan
    2021-06-22
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    2021-05-10
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.