# Z distribution and relationship

### The Standard Normal and The Chi-Square | STAT /

State the mean and standard deviation of the standard normal distribution; Use a Z table; Use the normal calculator; Transform raw data to Z scores. In probability theory, the normal distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often. In probability theory and statistics, there are several relationships among probability The following are self-replicating: Normal distribution, Cauchy distribution, Logistic distribution, Error distribution, Power distribution, Rayleigh distribution.

What is the chance of a false rejection, i. You most likely have already been exposed to the topic of probability in earlier courses. You may have used probability in your genetics class to construct a Punnet square.

## The Standard Normal and The Chi-Square

You may have used probability theory in chemistry to predict the number of isomers for compounds containing asymmetric carbon groups. Probability Probability is a number from 0. A probability of 1. A probability of 0. It's also common to talk about the chance of occurrence, which is commonly described by percentage figures between 0. What is the chance of tossing a coin and having it land heads up H?

## Normal distribution

Now, if we tossed five coins, what would be the most likely outcome? The answer is some combination of heads and tails. Intuitively, we know that the chance of tossing 5 coins and having all of them come up heads is quite small. That chance is the product of each of the single episodes: There is only a probability of about 0. With a 5 coin toss, it's likely to see some combinations of heads and tails based on these possible outcomes: Similarly, all tails would occur 0.

Intuitively, the greatest possibility lies in having some mixture of H and T, possibly 2. The expected frequency distribution can be seen in the top figure, which shows the distribution of possibilities as fractions of 32nds.

The possibilities total to thirty-two 32nds or unity. The area under the curve is greatest between 2 and 4 heads, i. The frequency distribution for a 10 coin toss is shown in the second figure.

Again, intuitively speaking, the greatest possibility is some mixture of H and T. With a 10 coin toss, achieving half heads or 5 is a real number, and the maximum probability is 0.

• Z-Stats / Basic Statistics
• How to do Normal Distributions Calculations
• Relationships among probability distributions

The distribution of a 20 coin toss is seen in the third figure. Again, the greatest area under the curve indicates some combination of heads and tails.

### Introduction to Normal Distributions

When increasing the number of coins or events, the curves begin to look like normal distribution curves. As the number of events exceeds 30, the graph is considered to approximate a normal curve. The importance of this point is that a normal curve can often be used to assess the probability that an event or observed difference occurs by chance or by cause.

In interpreting experimental results, it is common to use a probability of 0. If the probability is greater than 0. If Pr Standard Normal Distribution The standard normal distribution or the unit normal distribution is a special normal curve made up of z-scores. Remember that a z-score is a standard score also called the standard Gaussian variable that is calculated by subtracting the mean from a value and dividing the result by the standard deviation: For example, if the mean were and the value werethen the difference between the mean and the value is If the standard deviation were 5, then the difference of 10 has 2 SD's in it, thereby it is equal to 2z.

As discussed in lessons 1 and 2, this is one of those statistical forms that appears repeatedly in laboratory statistics.

The formula is used to express the number of standard deviations in the difference between a value and the mean. The values for z range from zero to infinity. The figure shows that the most common z-scores are from 0. In statistical language, this distribution can be described as N 0,1which indicates distribution is normal N and has a mean of 0 and a standard deviation of 1. Area under a normal curve.

The total area under the curve is equal to 1. Half of the area, or 0. The area between the mean and The area between These numbers should seem familiar to laboratorians. Z-scores can also be listed as decimal fractions of the 1's, 2's, and 3's we have been using thus far.

For example, you could have 1. Here the decimal fraction is carried out to the hundreths place. Table of areas under a normal curve. It is often convenient to use a table of areas under a standard normal curve to convert an observed z-score into the area or probability represented by that score.

So how do we calculate the probability below a negative z-value as illustrated below? We start by remembering that the standard normal distribution has a total area probability equal to 1 and it is also symmetrical about the mean. Thus, we can do the following to calculate negative z-values: Making this connection is very important because from the standard normal distribution table, we can calculate the probability less than 'a', as 'a' is now a positive value. From the above illustration, and from our knowledge that the area under the standard normal distribution is equal to 1, we can conclude that the two areas add up to 1.

We can, therefore, make the following statements: To understand the reasoning behind this look at the illustration below: To understand this we need to appreciate the symmetry of the standard normal distribution curve. We are trying to find out the area below: But by reflecting the area around the centre line mean we get the following: