The Z-score represents the distance between a raw score (x) and the population mean in units of the standard deviation (s): z is negative when the raw score is below the mean, positive when above.

The Z-Score is used by developing the idea of the Standard Deviation a little further and to assess the probability of any occurence over the stated range of values. It is therefore used to predict the probability of an event (see example 1) or of a range of events (see example 2), based on actual observations.

If you have two sets of data which are at the ordinal or interval level AND both sets are normally distributed AND there is a linear correlation, you can also use a Linear Regression analysis to predict values outside of the sample.

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Example 1: what is the probability of a completely dry month of February in San Francisco?

Rainfall (in cm) in the past 10 years in February in SF was as follows:

4.3	1.5	12.1	10.1	8.7
8.4	6.3	8.1	9.0	7.2

• Stated value (= value to have a "completely dry month") is x = 0. The corresponding Z-score will be referred to as Z(0)
• The mean of the sample above is 7.57 and the standard deviation is 2.84 (use calculator or Excel, or go to Basic Statistical Tools to learn how to calculate mean and standard deviation). The Z-score for the value 0 is therefore Z(0) = (0 - 7.57) / 2.84 = -2.67, which means that "the stated value of 0 lies 2.67 standard deviations left (negative value) of the mean"
• Based on the typical bell-curve shown on the diagram above, only a few % of the sample would reside so far left of the bell curve, 2.67 standard deviations or more to the left of the mean: this means that the possibility that the next month be dry is very unlikely to occur
• In order to quantify this likelihood more precisely and interpret this Z-score, you must then use a Z-Table: Z(0) = -2.67, which means it's -2.6 with 0.07 decimals after that: in the Z-Table, find the cell located at the intersection of the line "-2.6" and the column "0.07". The corresponding value found in the Z-table is: 0.0038. This means that there is a 0.0038 x 100 = 0.38% chance that the next month of February is completely dry, based on the data collected in the past ten years. This confirms our graphical estimate from the bell-curve

Example 2: what is the probability to have a range between 6 and 8 cm of rain in the next month of February?

Stated values (based on the values collected in Example 1):

• for x = 6, Z(6) = (6 - 7.57) / 2.84 = -0.55. According to the Z-Table, this means the probability to have 6 cm of rain is 0.226 = 22.6%
• for x = 8, Z(8) = (8 - 7.57) / 2.84 = +0.15. According to the Z-Table, this means the probability to have 8 cm of rain is 0.079 = 7.9%

Add both probabilities: 22.6% + 7.9% = 30.5%

The probability that the next month of February has between 6 and 8 cm of rain is 30.5%, based on the data collected in the past ten years.