## Normal CDF Transformations

- This tool does multiple Z-to-P and P-to-Z transformations, saving users time and effort.
- Just enter a set of scores, one per line. End each line by pressing the
`Enter`

key so these are recognized as individual scores. - From the selection menu, chose which probability case or convention you want to use.

- In Mathematics and Statistics, the upper-case letter Φ is used as a symbol for the cumulative distribution function (CDF) of the normal distribution. Given a Z score, Φ returns a probability value P.
- Conversely, given a P value, Φ Inverse returns a Z score. Some programmers like to declare these functions by writing
*normal_cdf*and*normal_cdf_inverse*. An inverse CDF is also called the Quantile Function (QF). - Instrumental in these transformations are Z tables.
These typically use at least three different probability cases or conventions (Laerd Statistics, 2016; NIST.Gov, 2003; Wikipedia, 2016):
**Case 1: Cumulative from mean = 0**. This case corresponds to the probability that Z is between 0 and some given value*a*.Example: If

*a*= 0.69, then P(0 ≤ Z ≤*a*) = 0.2549, so there is 25.49% chance that Z lies between 0 and 0.69.**Case 2: Cumulative**. This case corresponds to the probability that Z is less than or equal to*a*.Example: If

*a*= 0.69, then P(Z ≤*a*) = 0.7549, so there is 75.49% chance that Z is less than or equal to 0.69.**Case 3: Complementary cumulative**. This case corresponds to the probability that Z is greater than or equal to*a*.Example: If

*a*= 0.69, then P(Z ≥*a*) = 1 - P(Z ≤*a*) = 1 - 0.7549 = 0.2451, so there is 24.51% chance that Z is greater than or equal to 0.69.

- Data miners, statisticians, or anyone that need to compute Z and P scores without having to consult reference tables.

- A continuous random variable X, with a normal probability distribution, has an arithmetic mean of 14 and a standard deviation of 2.5. What is the probability that X lies between 14 and 17?
- The following exercises were taken from NIST Engineering Statistics Handbook (NIST.Gov, 2003):
- What is the probability that a value X is less than or equal to 1.53?
- What is the probability that a value X is less than or equal to -1.53?
- What is the probability that a value X is between -1 and 0.5?

- John scored 800 on a standardized test. The mean of the test was 700 with a standard deviation of 120. Test scores were normally distributed. What proportion of students had a higher score than John?

- Laerd Statistics (2016). How to do Normal Distributions Calculations.
- NIST.Gov (2003). Cumulative Distribution Function of the Standard Normal Distribution.
- Wikipedia (2016). Standard Normal Table.

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