NORM.S.DIST

Calculates the probability density function (PDF) or the cumulative distribution function (CDF) of the standard normal distribution.

Syntax:

NORM.S.DIST(z, cumulative)

z is required, and is the value that you want to use for the standard normal distribution.

cumulative is required, and is a logical value:

TRUE: Returns the cumulative distribution function (the probability that a random variable is less than or equal to z).

FALSE: Returns the probability density function (the height of the standard normal distribution curve at z).

Example:

If z contains 1.5 and cumulative contains TRUE:

NORM.S.DIST(1.5, TRUE)

returns 0.933192799

If z contains 1.5 and cumulative contains FALSE:

NORM.S.DIST(1.5, FALSE)

returns 0.129517596

z:

Cumulative:

Result:

0.933192799

Application:

Analyzing Standardized Test Scores

Imagine a national standardized test where the scores are normalized to a standard normal distribution with a mean of 0 and a standard deviation of 1. The score a student receives is represented by a Z-score. You can use the NORM.S.DIST function to determine the percentage of students who scored below a certain Z-score or to find the probability density at that specific Z-score.

NORM.S.DIST(Z, TRUE) calculates the Cumulative Distribution Function (CDF). This gives you the total probability that a random Z-score will be less than or equal to your specified Z value. This is useful for finding percentiles.
NORM.S.DIST(Z, FALSE) calculates the Probability Density Function (PDF). This gives you the probability density at a specific point Z.

Here is a table demonstrating the use of NORM.S.DIST for various Z-scores.

	Z-Score	CDF (Cumulative Probability)	PDF (Probability Density)
	A	B	C
1	-2	0.0228	0.054
2	-1	0.1587	0.242
3	0	0.5	0.3989
4	1	0.8413	0.242
5	1.5	0.9332	0.1295
6	2	0.9772	0.054

Interpreting the Results

Z-Score of 0.0: A Z-score of 0 is the mean. The CDF value is 0.5000, which means 50% of students scored at or below the mean. The PDF value of 0.3989 indicates the highest point on the bell curve, which is at the mean.
Z-Score of 1.0: A Z-score of 1.0 is one standard deviation above the mean. The CDF value is 0.8413, meaning that 84.13% of students scored at or below this point.
Z-Score of -2.0: A Z-score of −2.0 is two standard deviations below the mean. The CDF value is only 0.0228, indicating that only 2.28% of students scored at or below this point, which is a very low score.
Z-Score of 2.0: A Z-score of 2.0 is two standard deviations above the mean. The CDF value is 0.9772, which means that 97.72% of students scored at or below this point. This is a very high score.