Z Scores (Z Value) & Z Table (2024)

Z scores (Z value) is the number of standard deviations a score or a value (x) is away from the mean. In other words, the Z-score measures the dispersion of data. Technically, a Z-score tells you how many standard deviations value (x) are below or above the population mean (µ). If the Z value is positive, it indicates that the value or score (x) is above the mean. Similarly, if the Z value is negative, it means the value (x) is below the mean.

What is a Standard Normal Distribution?

A Normal Standard Distribution curve is a symmetric distribution where the area under the normal curve is 1 or 100%. The standard normal distribution is a type of special normal distribution with a mean (µ) of 0 and a standard deviation of 1.

A standard normal distribution always has a mean of zero and has intervals that increase by 1. Each number on the horizontal line corresponds to the z-score. Hence, use Z Scores to transform a givenstandard distributioninto something that is easy to calculate probabilities on as it can determine the likelihood of some event happening.

Any normal distribution with any value of mean (µ) and a sigma can be transformed into the standard normal distribution, where the mean of zero and a standard deviation of 1. This is also called standardization.

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A Z-score tells how much standard deviation a value or score is from the mean (µ). For example, if a Z-score is negative 3 means the value (x) is 3 standard deviations left of the mean. Similarly, if the Z-score is positive 2.5 means the value (x) is 2.5 standard deviations to the right of the mean (µ).

This is a common transformation, so there is a reference chart that allows us to look up values. Those values correlate to the value under the normal distribution curve – in other words, what’s the chance of an event happening? We use the Z table to find the percent chance.

How to Interpret the Z-score Table

Most importantly, the Z-score helps to calculate how much area that specific Z-score is associated with. A Z-score table is also known as a standard normal table used to find the exact area. The Z-score table tells the total quantity of area contained on the left side of any score or value (x).

Z Score Table Download

Using the Z score, find the percentage by using the formula: 1-NORMSDIST(Z), where Z is your calculated Z Score.

How to Calculate a Z Score by Hand

There are 2 different situations you need to be aware of when calculating a z score:

• Z score for a sample
• Z score for a population

While the z-score equations look very similar, remember that calculating the standard deviation of a population is different than the way you calculate the standard deviation of a sample.

The formula for transforming a score or observation x from any normal distribution to a standard normal score is :

Calculating a Z Score for a Population

Z Score for a Sample

How many parts in a population will be longer or greater than some number?

Z score examples using standard deviation

Example 1: Longer than / Greater than

Hospital stays for admitted patients at a certain hospital are measured in hours and were found to be normally distributed with an average of 200 hours and a standard deviation of 75 hours. How many of these stays can be expected to last for longer than 300 hours?

• x=300
• x̅ = 200
• s=75

Z= x- x̅/s =(300-200)/75= 100/75= 1.33

Z score from the table for 1.33 = 0.9082

Since, we are looking for longer, solution is P(x>300) = P(Z>1.33) = 1- P(Z<1.33)= 1-0.9082 = 0.0918 = 9.18%

Z Score Positive Template

• x=75
• x̅ = 200
• s=75

Z= x- x̅/s =(75-200)/75= -125/75= -1.667

Z score from the table for -1.667 = 0.0475

Since we are looking for less than, the solution is = 4.75%

Z Score Negative Template

P(X<175)

• x=175
• x̅ = 194
• s=11.2

Z= x- x̅/s =(175-194)/11.2= -1.6964

Z_score from the table for -1.6964= 0.0455

P(X<175) = 4.55%

P(X<225)

• x=225
• x̅ =194
• s=11.2

Z= x- x̅/s =(225-194)/11.2= 2.7678

Z score from the table for 2.7678 = 0.9971=99.71%

Since we are looking for weights between 175 and 225, P(175<x<225) = 99.71%-4.55% = 95.16%

Additional Notes:

• Z Scores and Process Sigma
• Z Score and Process Capability

What do you do when the sample size is less than 20?

Great question! You’d apply student t-scores.

Six Sigma Green Belt Z Score Questions

Question: This formula Z = (X – μ)/σ is used to calculate a Z score that, with the appropriate table, can tell a Belt what ____________________________________.

A) Ratio the area under the curve to the total population

B) Number of Standard Deviations between X and μ

C) The Median of the sample population is

D) Proportion of the data is between X and μ

Answer:

B: 1.33 This is an easy algebra question. Z = (X – μ)/σ = (28-32) / 3 = 4/3 = 1.33. See Z Scores.

Want a List of Free Z Score Questions and Z Table downloads to help study?