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Z-Scores

Main Concept

A standard normal distribution table, also known as the unit normal table or Z table, is used to find the probability that a statistic is observed below, above, or between values in the standard normal distribution, the so-called p-value. More specifically, the table contains values for the cumulative distribution function of the standard normal distribution at a given value, x.  

 

It is common practice to convert any normally distributed data to the standard normal distribution as the standard normal distribution table contains a value for every standardized z-score. Z-scores are calculated by first subtracting the mean of the data set from every observation, then dividing by the standard deviation, such that every standardized observation is a measure of how many standard deviations a given observation is from the sample mean. For more on standardizing data samples, see the Scale command.

z = x μσ

 

Once this z-score is known, its respective probability can be looked up in the standard normal distribution table.

 

The values contained in the standard normal distribution table can also be calculated by hand. For any distribution X, the mean, denoted μ, is the expected value of X. The variance, σ2, is the expected value of the square of the difference between the value of the X and its mean. The square root of the variance, σ, is called the standard deviation.

 

The probability density function of a normal (Gaussian) random variable X is given by:

fx = 1σ2πⅇxμ22σ2

 

The standard normal distribution has a mean μ = 0, and standard deviation σ= 1.

Φ = 12πⅇx22

 

The area under the standard normal distribution curve represents the cumulative probability and as such the total area under the curve is 1.  To find the probability value for a z-score of -1, we need to find the area under the standard normal curve between  and -1.

 

P(X < -1) = &infin;112π&ExponentialE;x22 &DifferentialD;x  =  0.15867

 

Example

Let X be a random variable taken from a standard normal distribution. What is the probability that X is less than 1.5?

 

We can find the probability of a value being less than 1.5 by finding the area of the blue shaded area below.

 

If we refer to the standard normal table it can be observed that for Z = 1.5:

PX &gt; 1.5 &equals; 0.0668

 

Knowing that the area under the standard normal distribution is 1:

PX &gt; 1.5 &equals; 1  PX 1.5

0.0668 1 PX 1.5 

PX 1.5 &equals; 0.9332 or 93.32%.

 

This means that the relative frequency or probability that an event occurs below 1.5 is 0.9332 or 93.32%.

 

 

Adjust to Z to find the corresponding probability.

Z =

 

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