# Z - Table

## Z Table

Z variable represents a standard Normal Random Variable with Mean = 0 and Standard Deviation = 1

Z table displays the area under the curve, which totals to 1. Practically the Z tables are calculated for a range of +/- 3.4 values (refer to six sigma concepts)

### Shows the percent of population:

- Less than or equal to Z (option "Up to Z")
- Greater than Z (option "Z onward") is calculated as 1 - (less than or equal to Z)

**Example 1:** Learn how to use the Z table for calculating the probability of a Continuous Random Variable

Assume a continuous random variable students examination scores following normal distribution, with µ = 711, σ = 29

Find out the probability of students securing a score of x ≤ 680?

**Ans:** Standard normal variable, z value calculated for corresponding continuous random variable x (680)

Step 1: Calculate z score corresponding to 680

formula: z = (x - µ) / σ

z = (680-711)/29

z = -1.06

Step 2: Refer to z table:

Calculate the probabilities using z table

Start at the row with z = -1.0, and the second decimal value is read along column 06.

The value in the z table at a z value of -1.06 is 0.1446

This implies the area under the curve (towards the left tail) from -1.06 of z value is around 14.46%.

This can be read as the students' population falling between 0 and 680 scores has a probability of 14.46% for the given normal distributed data.

**Example 2:** Assume a continuous random variable students examination scores following a normal distribution, with µ = 711, σ = 29

Find out the probability of students securing a score of x > 725?

**Ans:** Standard normal variable, z value calculated for corresponding continuous random variable x (725)

Step 1: Calculate z score corresponding to 725

formula: z = (x - µ) / σ

z = (725 - 711)/29

z = 0.48

Step 2: Refer to z table:

Calculate the probabilities using z table

Start at the row with z = 0.4, and the second decimal value is read along column 08.

The value in the z table at the z value of 0.48 is 0.6844

This implies the area under the curve (towards the left tail) from 0.48 of z value is around 68.44%.

This can be read as the students' population falling between 0 to 725 scores has a probability of 68.44%.

The table also called the Standard Normal Table, represents the area under the curve (probability), from a given quartile value (z value) towards the left tail of the curve.

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