## Standard Normal Distribution

The model for all normal distributions.

 The standard normal distribution (SND) is a normal distribution with mean = 0 and standard deviation = 1. The normal distribution is the most fundamental statistical distribution because it can describe most natural world phenomena. According to the Central Limit Theorem, even if a population is not normally distributed, a large number of samples with size > 20 will result in a sampling distribution that is approximately normal. With the SND (also called the Z Score) for the parent population mean to fall within specified ranges can be estimated from the probability distribution table of the normal distribution.

## Normal Distribution:

A normal distribution is known by the characteristic bell shaped curve of its probability distribution function (PDF). Mathematically, the PDF of normal distribution is expressed as Where μ is the mean, and σ2 is the variance of the distribution.

## Standard Normal Distribution

The bell curve of different normal distributions can vary in height (probability of the distribution to assume mean value) and width (range of the distribution). However the graph of a SND, for which mean = 0 and standard deviation = 1, is always the same. ## Properties of the Standard Normal Distribution

• The mean of the SND is 0 and standard deviation is 1.

• The distribution is symmetric about the mean.

• Total area under the curve is 1.

• Areas under the curve at specific values along the x-axis can be found using normal distribution tables.

• 68% of the population lies within one standard deviation from the mean.

• Approximately 99.5% of the population lies within three standard deviations from the mean.

## Z Scores for Standard Normal Distribution:

The mean values of the SND are plotted along the x-axis. In order to compare any normal distribution (any value of mean, and any value of standard deviation) to the SND, the values of z-score are computed as: Where x is the raw value from the given normal distribution to be standardized, is the population mean and is the population standard deviation. Probabilities are then read from probability distribution table, against the values of normalized z-score. Conversely if a z-score is known, it can be used to calculate sample size, calculate standard deviations or calculate population mean.

## Example Using The Standard Normal Distribution:

Consider a scenario where the average age of workers in a plant is 30 with standard deviation of 4. We want to find the probability that a worker picked at random will have age less than 40.

1. Calculate z-score as z = (40 - 30)/4 = 2.5

2. Refer to the SND table below.

3. 4. Area under the curve between 0 (age = 30) and z = 2.5 (age = 40) is 0.4938

5. Area under the curve to the left of 0 is 0.5

6. Since we want probability for age < 40, not only 30 < age < 40, so we add the two areas (to the left of z=0, and area between z=0 and z=2.5) = 0.5 + 0.4938 = 0.9938

7. Probability that a worker picked at random will have age less than 40, P(x<40) = P(z<2.5) = 0.9938 = 99.38%

## T-Score Table for the t-Distribution

When the sample size is small (< 30), we use the t Distribution and not the normal distribution. Similar to z-score tables for SND, there are t-score tables for t-Distributions according to different sample sizes.

## Summary

 The standard normal distribution is a representation of all normal curves (any combination of mean and standard deviation). Tables of values from this distribution can be used to evaluate any normal distribution.

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