Normal (Gaussian) Distribution

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The normal distribution is a probability distribution that is used to describe many natural phenomena. It is also known as the Gaussian distribution, after the mathematician Carl Friedrich Gauss who first described it. The normal distribution is one of the most commonly used distributions in statistics.

Properties of Normal Distribution:

The normal distribution is a continuous probability distribution that is defined by two parameters: the mean and the standard deviation. It has several important properties, including:

  • Symmetry: The normal distribution is symmetrical around the mean. This means that the probability of a value being above the mean is equal to that of a value being below the mean.
  • Unimodality: The normal distribution is unimodal, meaning that it has a single peak. This peak is located at the mean of the distribution.
  • Asymptotic behaviour: The normal distribution approaches but never reaches the x-axis as the values of x become increasingly large. This means that the probability of a value being extremely large or extremely small is always non-zero but becomes increasingly small as the value becomes more extreme.
  • Bell-shaped curve: The normal distribution has a bell-shaped curve. This means that the probability of a value being close to the mean is higher than the probability of a value being far from the mean.
  • 68-95-99.7 rule: The normal distribution has the property that, for a given standard deviation, approximately 68% of the values will fall within one standard deviation of the mean, 95% of the values will fall within two standard deviations of the mean, and 99.7% of the values will fall within three standard deviations of the mean. This is often referred to as the 68-95-99.7 rule.

These are some of the most important properties of normal distribution. It is a versatile distribution that is often used to model normally distributed data, such as height or intelligence.

Standard Normal Distribution:

The standard normal distribution is a special case of the normal distribution. It has a mean of zero and a standard deviation of one. This means that all values in the standard normal distribution are expressed in terms of how many standard deviations they are away from the mean. The standard normal distribution is often used to calculate probabilities for normally distributed data, as it simplifies calculations by eliminating the need to know the mean and standard deviation of the data.

Z-Score

A z-score, also known as a standard normal score, is a measure of how many standard deviations a value is from the mean of a distribution. It is often used to standardize values from a normal distribution, allowing you to compare them to the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1.

$$z = \frac{x - \mu}{\sigma}$$

where:

  • x is the value for which you want to find the standard normal score
  • \(\mu\) is the mean of the distribution
  • \(\sigma\) is the standard deviation of the distribution

The z-score indicates how many standard deviations a value is from the mean of the distribution. Positive z-scores indicate that the value is above the mean, while negative z-scores indicate that the value is below the mean.

For example, if a normal distribution has a mean of 50 and a standard deviation of 10, and you want to find the z-score for a value of 60, you would use the following calculation:

$$z = \frac{60 - 50}{10} = 1$$

This indicates that the value of 60 is 1 standard deviation above the mean of the distribution.

Area Under the Curve:

The total area under a normal distribution is always equal to 1. This is because a probability distribution is a function that describes the relative likelihood of different outcomes. Since the sum of the probabilities of all possible outcomes must be 1, the total area under a probability distribution must also be 1.

To calculate the area under a normal distribution with a different mean and standard deviation, you can find out the z-score and use the standard normal distribution.

Another method for calculating the area under a normal distribution curve is using a standard normal distribution table. A standard normal distribution table lists the area under the curve for different ranges of Z values.

Probability Density Function (PDF):

The probability density function (PDF) of a normal distribution is given by the following formula:

$$\Large{f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}}$$

where:

  • f(x) is the PDF of the normal distribution
  • x is the value for which you want to find the probability
  • \(\mu\) is the mean of the distribution
  • \(\sigma\) is the standard deviation of the distribution
  • \(\pi\) is the mathematical constant approximately equal to 3.14159
  • \(e\) is the mathematical constant approximately equal to 2.71828

This formula describes the shape of the normal distribution and allows you to calculate the probability of a value occurring within the distribution.

Cumulative Distribution Function (CDF):

The cumulative distribution function (CDF) of a normal distribution is the probability that a random variable x is less than or equal to a given value. It can be calculated using the following formula:

$$F(x) = \frac{1}{2} \cdot \left[1 + \text{erf}\left(\frac{x-\mu}{\sigma \cdot \sqrt{2}}\right)\right]$$

where:

  • $\(F(x)\) is the CDF of the normal distribution
  • \(x\) is the value for which you want to find the probability
  • \(\mu\) is the mean of the distribution
  • \(\sigma\) is the standard deviation of the distribution
  • \(\text{erf}\) is the error function, which is defined as: \(\text{erf}(x) = \frac{2}{\sqrt{\pi}} \cdot \int_{0}^{x} e^{-t^2} \mathrm{d}t\)

This formula allows you to calculate the probability that a random variable \(x\) is less than or equal to a given value in a normal distribution.

Using Microsoft Excel

NORM.DIST:

To calculate the area under a normal distribution curve using Microsoft Excel, you can use the NORM.DIST function, which calculates the probability density function (PDF) or cumulative distribution function (CDF) of a normal distribution.

To use the NORM.DIST(x, mean, standard_dev, cumulative) function, you need to provide the following input arguments:

  • X: The value for which you want to calculate the probability density or cumulative probability.
  • Mean: The mean of the normal distribution.
  • Standard deviation: The standard deviation of the normal distribution.
  • Cumulative: A logical value that specifies whether you want to calculate the PDF (FALSE) or CDF (TRUE) of the normal distribution.

For example, to calculate the area under a normal distribution with a mean of 100 and a standard deviation of 15 within one standard deviation of the mean (between 85 and 115), you would use the following formula:

=NORM.DIST(115, 100, 15, TRUE) - NORM.DIST(85, 100, 15, TRUE) = 0.6827

This formula uses NORM.DIST function to calculate the cumulative probability that a value from the given normal distribution is less than or equal to 115 (the upper bound of the range) and then subtracts the cumulative probability that a value is less than or equal to 85 (the lower bound of the range). This gives you the area under the curve within one standard deviation of the mean.

NORM.INV

NORM.INV is a function in Microsoft Excel that can be used to find the z-score (standard normal score) corresponding to a given probability in a normal distribution. The function has the following syntax:

NORM.INV(probability, mean, standard_dev)

where:

  • probability is the probability for which you want to find the corresponding z-score. This should be a value between 0 and 1.
  • mean is the mean of the normal distribution.
  • standard_dev is the standard deviation of the normal distribution.

The function returns the z-score corresponding to the given probability in the normal distribution with the specified mean and standard deviation.

NORM.S.DIST

The NORM.S.DIST function calculates the probability that a standard normal random variable is less than or equal to a given value. The function has the following syntax:

NORM.S.DIST(z, cumulative)

where:

  • z is the value for which you want to find the probability in the standard normal distribution.
  • cumulative is a logical value that specifies whether to return the cumulative probability (TRUE) or the probability density function (FALSE).

The function returns the probability that a standard normal random variable is less than or equal to the given value. If the cumulative argument is set to TRUE, the function returns the cumulative probability, which is the probability that a standard normal random variable is less than or equal to the given value. If the cumulative argument is set to FALSE, the function returns the probability density function, which is the value of the PDF at the given value.

NORM.Z.INV

The NORM.S.INV function calculates the value of a standard normal random variable that corresponds to a given probability. The function has the following syntax:

NORM.S.INV(probability)

where:

  • probability is the probability for which you want to find the corresponding value in the standard normal distribution. This should be a value between 0 and 1.

The function returns the value of a standard normal random variable corresponding to the given probability.

 


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