NORM.S.DIST function

The NORM.S.DIST function in Excel returns the standard normal distribution (a normal distribution with a mean of 0 and a standard deviation of 1) for a given value. This function is used to calculate the cumulative probability for a value in a standard normal distribution.

Syntax:

NORM.S.DIST(z, cumulative)

Arguments:

• z: Required. The value for which you want to calculate the distribution. This is typically a z-score, which represents the number of standard deviations a value is away from the mean.
• cumulative: Required. A logical value that determines the form of the function:
  • TRUE: Returns the cumulative distribution function (CDF), which gives the probability that a value is less than or equal to z in the standard normal distribution.
  • FALSE: Returns the probability density function (PDF), which gives the height of the standard normal distribution curve at z.

Output:

• If cumulative = TRUE, the function returns the cumulative probability that a value from the standard normal distribution is less than or equal to z.
• If cumulative = FALSE, the function returns the probability density at the value of z, which is the height of the standard normal distribution curve at that point.

How It Works:

• The standard normal distribution has a mean of 0 and a standard deviation of 1. The function uses the z-score (how many standard deviations a value is from the mean) to calculate the probability.
• The cumulative distribution function (CDF) calculates the probability that a value from the distribution is less than or equal to z.
• The probability density function (PDF) calculates the likelihood of observing a specific value of z in the distribution.

Example 1: Cumulative Probability

Suppose you want to calculate the cumulative probability for a z-score of 1.5 in the standard normal distribution.

Use the formula:

=NORM.S.DIST(1.5, TRUE)

This will return the cumulative probability that a value is less than or equal to 1.5 standard deviations above the mean in a standard normal distribution.

Example 2: Probability Density

If you want to find the probability density for a z-score of 1.5 in the standard normal distribution, use the formula:

=NORM.S.DIST(1.5, FALSE)

This will return the height of the standard normal distribution curve at the value of 1.5, which represents the likelihood of observing a value exactly at 1.5 standard deviations above the mean.

Example 3: Negative Z-Score

If you want to find the cumulative probability for a z-score of -2 (2 standard deviations below the mean), use the formula:

=NORM.S.DIST(-2, TRUE)

This will return the cumulative probability of a value being less than or equal to -2 standard deviations in the standard normal distribution.

Key Points:

• NORM.S.DIST is used to calculate probabilities for the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
• The cumulative argument determines whether the function returns the cumulative probability or the probability density.
• Z-scores represent the number of standard deviations a value is from the mean, and this function calculates probabilities based on those z-scores.

Use Cases:

• Statistics: Calculate cumulative probabilities or probability densities for z-scores in hypothesis testing or confidence interval estimation.
• Finance: Estimate the probability of financial returns falling within a certain range (e.g., a z-score of 1.96 corresponds to 95% confidence in a standard normal distribution).
• Quality Control: Determine the likelihood of outcomes within specified tolerance limits in a normally distributed process.
• Education: Calculate the probability of test scores falling below a certain level in standardized testing.

Notes:

• NORM.S.DIST is used when you’re working with a standard normal distribution. If your data follows a normal distribution but with a different mean and standard deviation, use NORM.DIST instead.
• Z-scores are often used in hypothesis testing and statistical inference to standardize data points and compare them across different datasets.