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Norm s dist

NORM.S.DIST Function

The NORM.S.DIST function in Excel is used to calculate the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. This function returns either the cumulative probability or the probability density value for a given z-score.

Key Features of NORM.S.DIST:

  • Works with the Z-distribution (standard normal distribution).
  • Can calculate:
    • Cumulative probability: The probability that a random variable is less than or equal to a specified value.
    • Density function: The value of the probability density function for the input z.

Syntax:

NORM.S.DIST(z, cumulative)
  • z: Required. The z-value for which you want to calculate the distribution.
  • cumulative: Required (TRUE or FALSE). This determines the type of result:
    • TRUE: Returns the cumulative distribution function (CDF).
    • FALSE: Returns the probability density function (PDF).

How It Works:

  1. The cumulative distribution function gives the probability that a value is less than or equal to the specified z. It represents the area under the curve to the left of z.
  2. The probability density function provides the height of the curve at a given z, which is useful for examining the relative likelihood of z.

The formula for the standard normal distribution's PDF is:

f(z) = (1 / √(2π)) * e^(-z² / 2)

Where z is the standard score.

The CDF cannot be expressed in closed form, but Excel uses numerical methods to compute it.

Examples:

  1. Cumulative Distribution Example: Find the cumulative probability for a z-value of 1.5: 
    =NORM.S.DIST(1.5, TRUE)
    Result: 0.933193.

This means there’s a 93.32% chance a random variable from the standard normal distribution will fall below 1.5.

  1. Probability Density Example: Find the height of the standard normal curve at z = 0: 
    =NORM.S.DIST(0, FALSE)
    Result: 0.398942.

This represents the peak value of the curve (since the mean of the standard normal distribution is 0).

  1. Tail Area Example: Calculate the probability of a z-value greater than 2.33. Subtract the cumulative probability from 1: 
    =1 - NORM.S.DIST(2.33, TRUE)
    Result: 0.009902.

This indicates a very small chance (0.99%) of obtaining a value greater than 2.33.

Notes:

  • Interpretation of z:
    • A negative z represents a value below the mean.
    • A positive z represents a value above the mean.
  • Input Validation:
    • If z is non-numeric, the function returns #VALUE!.
    • If cumulative is not a valid boolean (TRUE or FALSE), the function returns an error.

Applications:

  • Statistics and Testing: Commonly used in hypothesis testing and confidence interval calculations.
  • Financial Modeling: Used for risk analysis and stock price modeling under normality assumptions.
  • Quality Control: Helps in standardizing variables and identifying outliers in a dataset.

Tip: For normal distributions with other means and standard deviations, use the NORM.DIST function, which allows specifying these parameters.