# Normal distribution (Single variable)

## Where do you meet this distribution?

- Standard score
- Finance, Economics : changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal
- Average of stochastic variables : Central Limit Theorem
- Statistical mechanics : Velocities of the molecules in the ideal gas
- Quantum physics : Probability density function of a ground state in a quantum harmonic oscillator
- Error analysis

## Shape of Distribution

### Basic Properties

- Two parameters are required.
These parameters are Mean and Standard Deviation of the distribution respectively.

- Continuous distribution defined on entire range
- This distribution is always symmetric.

### Probability

- Cumulative distribution function
where is Probability density function of standard normal distribution.

- Probability density function
- How to compute these on Excel.

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A B Data Description 0.5 Value for which you want the distribution 8 Value of parameter M 2 Value of parameter Sigma Formula Description (Result) =NTNORMDIST((A2-A3)/A4,TRUE) Cumulative distribution function for the terms above =NTNORMDIST((A2-A3)/A4,FALSE) Probability density function for the terms above - Function reference : NTNORMDIST
- NtRand Function NTNORMDIST is same as excel function NORMSDIST when 2nd. argument=TRUE.

### Quantile

- Inverse function of cumulative distribution function
where is cumulative distribution function of standard normal distribution.

- NORMSINV is an excel function
- How to compute this on Excel.

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A B Data Description 0.7 Probability associated with the distribution 1.7 Value of parameter M 0.9 Value of parameter Sigma Formula Description (Result) =A4*NORMSINV(A2)+A3 Inverse of the cumulative distribution function for the terms above

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

- Mean of the distribution is given as .

### Standard Deviation – How wide does the distribution spread? (Definition)

- Standard deviation of the distribution is given as .

### Skewness – Which side is the distribution distorted into? (Definition)

- Skewness of the distribution is given as .

### Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

- Kurtosis of the distribution is given as .

## Random Numbers

- How to generate random numbers on Excel.

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A B Data Description 0.5 Value of parameter M 0.5 Value of parameter Sigma Formula Description (Result) =A3*NTRANDNORM(100)+A2 100 Normal deviates based on Mersenne-Twister algorithm for which the parameters above Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

## NtRand Functions

- Generating random numbers based on Mersenne Twister algorithm: NTRANDNORM

## Reference

- Wolfram Mathworld – Normal Distribution
- Wikipedia – Normal distribution
- Statistics Online Computational Resource