Z Score Tables are necessary statistical tools, enabling us to know how separate data points relate to the mean of a data set. The tables graphically express the percentage of values above or below a given Z Score, making it simpler to understand data and spot trends. Knowing how to read and use Z Score Tables is not only a useful skill in data analysis but also an essential skill in many U.S. competitive exams, such as SAT, ACT, and GRE, where statistical knowledge is frequently examined. Being proficient in Z Score Tables gives test-takers the skill to answer probability-based questions effectively, providing them with a competitive advantage and improving their overall performance in these tests.

Z Score Table

A Z Score Table, which is also known as the standard normal table, gives the area under a curve to the left of a Z Score. This area represents the probability that the Z values will fall within a region of the standard normal distribution.

A Z Score Table is a table that tells us what percentage of values fall below a certain Z Score in a standard normal distribution. In other words it tells us how many standard deviations away an individual data value falls from the mean.

There are two tables depending on the Z Score. If we get a positive Z Score value then we use the positive Z Score table otherwise the negative Z Score table.

In the above image the positive Z Score Table is shown.

In the above image the negative Z Score Table is shown.

We do not need both tables for solving sums. We use the positive table for both the positive and negative Z Scores. We will see later in the article how we use the positive table for negative Z Scores also.

Formula of Z Score

In order to find the Z Score, the mean is subtracted from the raw score and then that value is divided by the standard deviation.

The formula for Z Score is written as,

\( z=\frac{x-\mu}{\sigma} \)

Where, z = Z Score value,

x = Raw Score,

\( \mu \) = Mean,

\( \sigma \) = Standard Deviation

Reading a Z score table

We have to find the Z Score using the above mentioned formula. Then we look for the appropriate value for the corresponding Z Score from the table.

The Z Score may come out either in whole numbers or as a fraction. In the table the vertical axis represents the Z Score upto 1 decimal place and the horizontal axis represents the Z score from two decimal places.

For example, for certain observations, we have the following data.

Mean(\( \mu \)): 75

Standard deviation(\( \sigma \)): 5

Raw score(x): 83

So first we find the Z Score using the formula as shown below.

\( z=\frac{x-\mu}{\sigma}=\frac{83-75}{5}=1.6 \)

Here as we have the Z Score as a fraction with only 1 decimal place so we check this Z Score from the vertical axis of the table.

As we get the Z Score as a positive value so we use the positive Z Score Table

The conclusion that can be drawn from the table is that 94.52% of the scores are below the raw score i.e., below 83 and the remaining percentage is higher than the raw score i.e, 5.48% is higher than 83.

Area to the Left and Right of a Z Score

The corresponding value which we obtain from the Z Score Table gives the area under the curve.

• Positive Z Score

If we get the Z Score value as 1.09 then the corresponding value from the positive table obtained is 0.8621. Here 86.21% represents the area to the left of this Z Score and 100-86.21%=13.79% represents the area to the right of this Z Score.

• Negative Z Score

If we get the Z Score value as -1.02 then the corresponding value from the negative table obtained is 0.8461. Here 84.61% represents the area to the right of this Z Score and 100-84.61%=15.39% represents the area to the left of this Z Score.

Types of Z Score Table

There are two types of Z Score Table depending on the value of calculated Z Score.

Negative Z Score Table

A negative Z Score Table has a value that is below or to the left of the given mean of the standard normal distribution. A negative Z Score table displays Z values less than zero. If we get the value of the Z Score in negative, then we refer to the Z Score Table but subtract it from 1 for the conclusion.

For example, if we have calculated the Z Score as -1.4, then we will use the Z Score table and subtract from 1.

So from the table value we can conclude that 100-91.92=8.08% of scores are below the raw score whereas 91.92% of scores are higher than the raw score.

Positive Z Score Table

A positive Z Score has a value that is above or to the right of the mean of the standard normal distribution. A positive Z Score table displays Z values greater than zero. If we calculate the Z Score as a positive value, then we use the positive Z Score Table for conclusion.

For example, we calculate the Z Score of certain data as 0.42, then we use the Z Score Table.

This value tells us that 66.28% of scores are lower than the raw score and 33.72% of the values are higher than the raw score.

Altman Z Score Table

The Altman Z Score is a model created by Edward Altman which assesses a companies’ financial stability and predicts how likely a company is to go bankrupt. This model lets us predict the financial distress of a company and how high or low the probability is of the company becoming insolvent in the near future of about 2 years.

The Altman Z-Score, also known as AZ-Score often, is the output of a credit-strength test of any company based on five key financial ratios, which are profitability, leverage, liquidity, solvency and activity.

The formula for Altman Z Score Table is mentioned below.

\( Z=1.2X_1+1.4X_2+3.3X_3+0.6X_4+X_5 \)

Where, \( X_1=\frac{Working\ Earnings}{Total\ Assets}, \)

\( X_2=\frac{Retained\ Earnings}{Total\ Assets}, \)

\( X_3=\frac{Earning\ before\ interest\ \&\ tax}{Total\ Assets}, \)

\( X_4=\frac{Marking\ Capitalisation}{Total\ Libalities}, \)

\( X_5=\frac{Sales}{Total\ Assets} \)

Z Score Table Solved Examples

Problem 1:We have been given a population which has a mean score of 74 and a standard deviation of 4. What percentage of the given population will have scores between 70 and 80?

Solution:

First we will find the z-score of each of the scores of 70 and 80.

So we have, Mean(\( \mu \)): 74

Standard deviation(\( \sigma \)): 4

1st raw score(\( x_1 \)): 70

2nd raw score(\( x_1 \)): 80

Therefore, \( z_1=\frac{x-\mu}{\sigma}=\frac{70-74}{4}=-1 \)

\( z_2=\frac{x-\mu}{\sigma}=\frac{80-74}{4}=1.5 \)

Next we use the Z table to find the percentages.

So from the above table we get area to the left of a score of 80(\( z_2 \)): 0.9332

So from the above table we get the area to the left of a score of 70(\( z_1 \)):1-0.8414= 0.1586

Now to find the area between we subtract the values = 0.9332-0.1586 = 0.77456

Thus 77.46% of scores will fall between 70 and 80.

Conclusion 

Therefore, Z Score Tables are important tools in statistical analysis as they assist in establishing how data points differ from the mean within any dataset. Through the use of such tables, one can derive information related to the likelihood that values will lie above or below a particular Z Score, enabling better understanding of the distribution of data. Regardless of whether a positive or negative Z Score Table is used, the interpretation is the same—allowing for informed decision-making based on the percentage of data within specified ranges. Becoming proficient in the application of Z Score Tables allows analysts to effectively interpret data and draw well-informed conclusions.