A z-score table is a table that indicates the percentage of observations that fall below a particular z-score in a standard normal distribution. A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Following are the steps to read a z-score table:

## How to read a z-score table?

### Understand the structure of the table

The z-score table is divided into two parts: the first part shows the first digit of the z-score, and the second part shows the second digit of the z-score. For example, the first row of the table may show values from -3.4 to -3.3, while the second row may show values from -3.3 to -3.2.

### Identify the z-score you need to find the percentage for

Find the row and column corresponding to the z-score to find the percentage. For example, if you need to find the percentage of observations that fall below a z-score of 1.50, you would look for the row corresponding to 1.5 in the first part of the table and the column corresponding to 0.00 in the second part of the table.

### Find the percentage

Once you have identified the row and column corresponding to your z-score, look for the corresponding value in the body of the table. This value represents the percentage of observations that fall below the z-score you are interested in. For example, if you find the row corresponding to 1.5 in the first part of the table and the column corresponding to 0.00 in the second part of the table, you will find the value of 0.9332. Approximately 93.32% of the observations fall below a z-score of 1.50.

### Interpret the percentage

Once you have found the percentage of observations that fall below your z-score, you can interpret this percentage in the context of your problem. For example, suppose you are trying to find the probability that a randomly selected person has a height below a certain value. In that case, you could use the z-score table to find the percentage of people with a height below that value.

**What is a Standard Normal Distribution?**

The standard normal distribution is a probability distribution often used in statistics and other quantitative fields to model various phenomena. It is also known as the Gaussian distribution, normal distribution, or bell curve.

In a standard normal distribution, the distribution of values is symmetric around the mean, which is 0. The standard deviation of the distribution is 1. It means that the values in the distribution are spread out around the mean predictably. The standard normal distribution is important because it allows us to standardize data and compare it to a known distribution. It can be useful when we want to compare data from different sources or to see how likely a particular value will occur in a given dataset.

The probability density function (PDF) of a standard normal distribution is given by:

f(x) = (1 / sqrt(2π)) * e^(-x^2/2)

where e is the mathematical constant approximately equal to 2.71828, π is the mathematical constant approximately equal to 3.14159, and x is a random variable.

The cumulative distribution function (CDF) of a standard normal the distribution gives the probability that a random variable from the distribution will be less than or equal to a given value. The CDF of a standard normal distribution is denoted by Φ(z), where z is the standard score (also called the z-score) and is calculated by subtracting the mean from the observed value and dividing by the standard deviation.

Φ(z) = P(Z ≤ z)

## What if the Z-score is off the Chart?

If the z-score is off the chart, the z-score is beyond the range of values included in the standard normal distribution table. In other words, the z-score is so extreme that it falls outside the range of values for which probabilities are listed in the table.

In this situation, you can still estimate the probability associated with the z-score by using statistical software or an online calculator that can calculate probabilities beyond the range of the standard normal distribution table. Alternatively, you can use statistical techniques such as interpolation or extrapolation to estimate the probability associated with the z-score based on the values listed in the table. However, it’s important to note that these estimates may not be as accurate as the probabilities listed in the standard normal distribution table and should be used cautiously.

## FAQS:

### Q1: How do you read a probability Z-table?

A: A standard normal distribution table provides the probabilities associated with different z-scores in a normal distribution. You first need to convert your data into a standard normal distribution with a mean of 0 and a standard deviation of 1. Once you have your data in the form of a standard normal distribution, you can use the z-score to find the corresponding probability in the standard normal distribution table.

### Q2: What is a normal Z-score?

A: Z-scores between -2.0 and 2.0 represent 95% of values in a normal distribution. The mean IQ score is 100 with a standard deviation of 15, meaning most people fall within the 70 to 130 IQ points range. These insights can be useful in a range of contexts.

## Conclusion:

A z-score table provides probabilities associated with different z-scores in a normal distribution. You need to convert your data into a standard normal distribution and locate the corresponding probability for a given z-score. Understanding z-scores and normal distribution can be useful in analyzing and interpreting data in a wide range of fields. By applying these concepts, you can gain insights into the relative rarity of different values in a distribution and make informed decisions based on this information.