Z table

What is a Z-Table?

The z-table is known by several names, including z score table, normal distribution table, standard normal table, and standard normal distribution table.
Z score is the value of the standard normal distribution.
The Z-table contains the probabilities that the random value will be less than the Z score, assuming standard normal distribution.
The Z score is the sum of the left column and the upper row.

What is z-score?

A z-score is a statistical measure that quantifies how many standard deviations a data point is away from the mean of the dataset. It can be used to describe the distance of a value from the mean in terms of standard deviations.

What is a good z-score?

There isn't a definite good or bad z-score. However, extreme z-scores may suggest that the assumed mean is incorrect. For instance, when the significance level is 0.05, the likelihood of obtaining a z-score less than -1.96 or greater than 1.96 is less than 0.05. In such situations, we may reject the assumption that we know the mean.

What is the purpose of z-score?

• Z-score can be used to determine whether to reject the null hypothesis (H₀). If the z-score is extreme, then you may reject the null hypothesis.
• Z-score can be used to scale or standardize variables, allowing for comparisons between variables with different units. For instance, you can determine what is more extreme: a 300-gram orange or a 2.05-meter man? Such a comparison is only possible when the variables have been standardized through scaling.

Normal distribution to standard normal distribution:

When working with the normal distribution, it can be useful to convert to the standard normal distribution, which has a mean of 0 and a standard deviation of 1. The z-table is built based on the standard normal distribution. However, if you need to calculate the probability for a non-standard normal distribution, you can use the following z-score formula.

z =	𝑥 - μ
	σ

Where:
𝑥 - the value.
z - the z-score.
μ - the mean.
σ - the standard deviation.
Example
"Compute the z-score for a value of 18, given that the mean is 12 and the standard deviation is 3."

z =	18 - 12	= 2
	3

Is it recommended to use the z-table?

No! This is the 21st century! For any practical use, instead of tables, you can use the normal distribution calculator Simply select the appropriate value from "Probability (p) or percentile (𝑥)": 𝑥₁ for the Z-table or P(X≤𝑥₁) for the inverse Z-table.
If you are a student you may use z-table as needed.

Example, z-table:
"Calculate the probability of a randomly selected value being less than a Z-score of 0.32."
Row 4 (0.3) and column 3 (0.02). 0.32 = 0.30 + 0.02
P(x ≤ 0.32) = 0.6255158347
P(x ≤ -0.32) = 1- P(x ≤ 0.32) = 0.3744841653
The probability is 0.3744841653

Inverse Z table:

Calculates the Z score based on the less than or greater than probabilities
α - contains the probability.
Z_α - the Z-value where p(x ≤ Z_α) = α, critical value of the left-tailed test.
Z_1-α - the Z-value where p(x ≥ Z_1-α) = α, critical value of the right-tailed test.
Z_α/2 - the Z-value where p(x ≤ Z_α/2) = α/2, left critical value of the two-tailed test.
Z_1-α/2 - the Z-value where p(x ≥ Z_1-α/2) = α/2, right critical value of the two-tailed test.

Example, Inverse z-table:
Row number 6 (excluding the header)
P( x ≤ -1.644854 ) = 0.05,  P( x ≥ 1.644854 ) = 0.05,  P( x ≤ -1.959964 ) = 0.025,  P( x ≥ 1.959964 ) = 0.025