Z-Score Table

Definition of Z-Score Table

A standard normal table that is also known as the z-score table or Z-Score Table, is basically a mathematical table used for the values of ‘ϕ’. These values are of the cumulative distribution function of a usual distribution. However, the Z-Score also called the standard score specifies how many standard deviations an entity is from the mean.

Since the probability tables are not printable for every usual distribution. Moreover, this is because there is an endless variety of usual distribution. In addition, it is a common practice to transform a normal to a standard normal and then apply the z-score table to find out the possible probabilities.

Z-Score Formula

It is a method for comparing the results from a test to a ‘normal’ population. Furthermore, if ‘X’ is a random variable from a usual distribution with a mean ‘μ’ and a standard deviation ‘σ’, therefore, its Z-score might be calculated from ‘X’ by doing the subtraction of ‘μ’ and dividing it with ‘σ’.

Here, ‘x’ = test value

‘μ’ is mean and

‘σ’ is SD i.e. standard value

For the average of a sample from a populace ‘n’ in which we have the mean ‘μ’ and we have the standard deviation ‘σ’.

Interpreting the Z-Score  Table

Here are some ways to interpret the Z-Score:

• Z-score less than ‘0’ signifies an element less than the mean.
• A z-score more than ‘0’ symbolizes an element bigger than the mean.
• Z-score equivalent to ‘0’ denotes an element that is equal to the mean.
• Z-score which is equal to ‘1’ represents an element which is one standard deviation superior to the mean.
• When a z-score is equal to ‘-1’ it represents an element which is one standard deviation less than the mean.
• If the quantity of elements in the set is huge, hence, about 68 percent of all the elements have a z-score between ‘-1’ and ‘1’. Moreover, about 95 per cent will be having a z-score between ‘-2’ and ‘2’, and lastly, about 99 percent of them will be having a z-score between ‘-3’ and ‘3’.

Types of Z-Score Table

There are 2 types of the Z-Score Tables which are as follows:

• Positive Z-Score Table and the Negative Z-Score Table.

Conditions

For the application of the Z-Test, some of these conditions must be fulfilled:

• Firstly, the nuisance parameters must be known. Moreover, or these parameters should be estimated with a high accuracy. Notably, the z-tests concentrate on a single parameter and deal with all the other unknown parameters as fixed. On the other hand, during the practice, due to the Slutsky’s Theorem, ‘plugging in’ consistent estimates of the nuisance parameters is justifiable. However, if the size of the sample is not huge enough for these estimates to be sensibly accurate, the Z-test might not perform fine.
• The test statistic must follow the usual distribution. Therefore, normally, one appeals to the central limit Theorem for justifying the assumption that a test statistic differs usually. Moreover, there is a great deal of statistical research on a question that asks “when a test statistic differs approximately usually”.

Question on Z-Score Table

Question: The test score of some students in a classroom test is having a mean of ‘70’ and having a standard deviation of ‘12’. What will be the probable percentage of the students who have scored more than ‘85’?

Solution: The Z-Score for the data given above is:

Z = 85 – 70/12 = 1.25.

From the Z-Score table, the fraction of the data within this score will be = ‘0.8944’.

This states that ‘89.44’ percent of the students are having the test score within ‘85’. Therefore, the percentage of the students who are above the test score of ‘85’ = (100 – 89.44) percent = ‘10.56’ percent.