Z Score Table

Z table or the standard normal distribution table is a mathematical chart used to represent the percentage of values that are less than a particular z score value. There can be two types of z score tables, depending on the values, a positive table and a negative table. Z score tables are useful because they can help us calculate the probability of a data point appearing in a normal distribution.

There are countless normal distributions and it is difficult to print each distribution. Hence, we convert a normal to a standard normal distribution, which is known as the Z score table. Then use the Z-score table to find the probabilities. 
 

Z score table can only be used when we know the value of z. The value represents a number to indicate how many standard deviations the value is below or above the mean. Z score can be used to calculate both sample and population data.

By using the given formulas, z score is calculated:

For population data: z = (x - μ) / 𝞂

Here, x is the raw score 

μ is the population mean

𝞂 is the standard deviation of a population

Next, for sample data: z = (x - x̄) / s

Here,

 x̄ = sample mean

s = sample standard deviation

x = raw score

Take a look at this example,

If the class average on a science test is 60 with a standard deviation of 10, students who scored 80 can. Let’s say a class is averaging 60 on a recently concluded examination, with a standard deviation of 10. Then, the students averaging 80 can calculate their Z score as follows:

Here, the standard deviation (x) = 80

The mean (μ) = 60

the population standard deviation (𝞂) = 10

By using the formula of Z score:  z = (x - μ) / 𝞂

 z = (80 - 60) / 10 = 2 

z = 2 


Thus, the student’s science score is 2 standard deviations above the class average. 
 

The Z score table is divided into two types, negative and positive z score tables. A negative z score means the data point of the random variable is below the mean.

A positive z score means the value is above the mean. We should use the negative z score table to find the values below the mean. Likewise, we use a positive z score table to find the values that fall above the mean or less than the positive z score. The cells in the table describe the area and the rows and columns represent the z score. Here are the negative and positive z score tables, take a look at this: 
 

There are certain steps that should be followed while using a Z score table. The steps are listed below:


Step 1: The first step is to determine the z score for the said data point. Then, we should find the mean and standard deviation. The z score explains how far the data point is from the average or mean. 


Step 2: Analyze the Z score table. The left side column represents the Z scores. The matching percentiles or probabilities are on the table’s body. 


Step 3: Identifying the z score, located on the left-most column of the table, is the next step. If we do not find the exact z score, estimate the probability by interpolating between the nearest values. 

Let us understand this with an example,

 
In a university, the average score of an entrance exam has a mean of 70 and a standard deviation of 10. One of the students wants to find the probability of scoring below 85.


Solution:

First, we need to calculate the Z score. 

z = (x - μ) / 𝞂

z = 85 - 70 / 10 

z = 15 / 10 = 1.5 

Now, from the z score table, we should identify the cumulative probability corresponding to 1.5. The probability for z = 1.5 is 0.9332. 


0.9332 × 100 = 93.32%


The probability of the student scoring below 85 is 93.32%. 
 

In statistics, mathematics, and research Z score is a vital tool to find the probabilities of values. It makes the calculation and analysis of data much easier for students.
 

• Z score table helps students compare their marks and test scores. If students need to compare their marks on different subjects they can use the Z score table to find their performance relative to the class average. 
   
• Kids can use this concept to analyze their scores and evaluate how well they perform compared to other classmates. 
   
• Students who focus on different projects, surveys, and data collection can use the Z score table to understand the current trends and opinions of their subjects. 
   
• Students can understand the concepts of the Z score table and they can prepare for advanced studies such as economics, statistics, mathematics, and engineering.