Statistics – Z-Scores:

A z-score/ standard score indicates how many standard deviations an element is far from the mean. A z-score can be calculated using the formula:

z = (X – μ) / σ

Where z – z-score, X – value of the element, μ – population mean, and

σ – standard deviation.

Interpretation of z-scores:

If z-score < 0 => an element < mean.

If z-score > 0 => an element > mean.

If z-score = 0 => an element = mean.

If z-score = 1 => an element which is 1σ> mean.

If z-score = 2=> an element which is 2σ> mean.

If z-score = -1=> an element which is 1σ< mean.

If z-score = -2=> an element which is 2σ< mean and so on.

If the number of elements in the set is larger, about

68%  have a z-score between -1 and 1;

95% have a z-score between -2 and 2;

99% have a z-score between -3 and 3.

We can find the z score table online and save it for later calculations.

Example: 

Find the probability for IQ values between 75 and 130, assuming a normal distribution, mean = 100 and standard deviation = 15.
An IQ of 75 corresponds with a z score of -1.67(75-100/15) and an IQ of 130 corresponds with a zscore of 2.00. The value for -1.67 is .4525 from the table. For 2.00 we find .4772. The probability of an IQ between 75 and 130 is the same as .4525+.4772=.9297.

If the mean and standard deviation of a normal distribution are known, the percentile rank of a person obtaining a specific score can be calculated.

Example:

Assume a test in Introductory Psychology is normally distributed with a mean of 80 and a standard deviation of 5. What is the percentile rank of a person who receives a score of 70?

Using the formula, we know that z = 70-80/5=-2. Using the table only 2.3% of the population will be less than or equal to a score 2 below the mean.

The shaded area occupies 2.3% of the total area. The proportion of the area below 70 = the proportion of the scores below 70.

Similarly, a person scoring 75 on the test is 15.9% of the population.

Same way, the percentile rank of a person receiving a score of 90 on the test is 97.7%

Since z = (90 – 80)/5 = 2it can be determined from the table that a z score of 2 is equivalent to the 97.7th percentile: The proportion of people scoring below 90 is thus .977.

Example 3: What score on the Introductory Psychology test would it have taken to be in the 75th percentile? (Mean is 80 and a standard deviation is 5)

We now have to find out z score which can be done by reversing the steps followed in example 2

First, determine  z score from table with value associating to 0.75 by using a z table. The value of z is 0.674.
Second, the standard deviation is 5, sincea little algebra demonstrates that X = μ+ z σ

X = 80 + (.674)(5) = 83.37.