GAUSS

Returns a value in the standard normal cumulative distribution.

Syntax:

GAUSS(x)

where x is a number, returns NORMSDIST(x) - 0.5.

Example:

GAUSS(0)

returns 0.

Application:

Analyzing Student Test Scores

Scenario: A teacher wants to analyze the scores from a recent math test. The test scores are normally distributed with a mean (μ) of 75 and a standard deviation (σ) of 10. The teacher wants to use the GAUSS function to determine the percentage of students who scored below a certain grade.

The GAUSS function calculates the cumulative probability for a standard normal distribution. To use it with a non-standard distribution (like our test scores), we first need to convert the score (x) to a standard score or z-score using the formula:

$z = \frac{x - μ}{σ}$

where:

z is the standard score
x is the raw score
μ is the mean
σ is the standard deviation

The GAUSS function then takes this z-score as its parameter.

The GAUSS function is defined as:

$G A U SS (z) = \int_{- \infty}^{z} \frac{1}{2 π} e^{- t^{2} /2} d t$

This function returns the area under the standard normal curve to the left of z, which represents the cumulative probability.

Calculation and Analysis

Let's use the GAUSS function to answer the following questions:

What percentage of students scored below 85?
What percentage of students scored below 60?
What percentage of students scored between 60 and 85?

Table of Calculations

	Target Score (x)	Z-Score (z = $\frac{x - 75}{10}$ )	GAUSS(z) Result	Interpretation
	A	B	C	D
1	85	1	0.3413	The probability that a student's score is between 75 and 85 is 34.13%.
2	60	-1.5	-0.4332	The probability that a student's score is between 60 and 75 is 43.32%. The negative sign indicates the score is below the mean.
3	Between 60 and 85		0.7745	The probability that a student scored between 60 and 85 is 77.45%.

Explanation of Results:

For the 85 score: The z-score is 1.0. The GAUSS(1.0) function returns approximately 0.34134. This means there is a 34.13% probability that a student scored within one standard deviation above the mean.
For the 60 score: The z-score is -1.5. The GAUSS(-1.5) function returns approximately -0.43319. The negative sign simply tells us the range is below the mean. The probability is the absolute value, 43.32%, that a student scored within 1.5 standard deviations below the mean.
For the range between 60 and 85: To find the total probability for the range, we add the probability from the mean to 85 and the probability from the mean to 60. GAUSS(1.0)+|GAUSS(−1.5)|=0.3413+0.4332=0.7745. This means 77.45% of students scored between 60 and 85.