Z-score tables, also known as standard normal tables or Z-tables, provide a way to find the probability of a particular Z-score occurring within a normal distribution. There are two types of Z-score tables: left-tailed (negative Z-scores) and right-tailed (positive Z-scores). These tables help in determining how unusual or typical a particular score is within a dataset.

## Reading the Left Z-score Table (Negative)

The left Z-score table typically lists Z-scores on the leftmost column, representing data points below the mean. The body of the table shows the probability of a score falling at or below a given Z-score. For instance, a Z-score of -1.96 corresponds to a probability of approximately 0.025. This means there’s a 2.5% chance of a score falling below this Z-score in a standard normal distribution.

## Reading the Right Z-score Table (Positive)

Conversely, the right Z-score table deals with positive Z-scores, which are scores above the mean. Reading this table involves understanding the probability of a score falling at or above a particular Z-score. For example, a Z-score of 1.96 in a right-tailed table also indicates a 2.5% chance, but this time it’s the likelihood of a score falling above this value.

## Practical Examples and Applications

In a college entrance exam with a mean score of 500 and a standard deviation of 100, what is the probability of a student scoring less than 450?

Solution: To find this, we first calculate the Z-score.

The formula for Z-score is $$Z = \frac { X – mean}{standard_deviation}$$

For our example Z-score = $$\frac {(450−500)}{100} = -0.5$$

A Z-score of -0.5 indicates that the score of 450 is half a standard deviation below the mean. Now, using the left Z-score table, we look up the probability corresponding to a Z-score of -0.5. In the table, this value is 0.3085. This means there’s a 30.85% chance of a student scoring below 450 on this exam.

Note: Since a Z-score of -0.5 is relatively close to the mean it suggests that scoring below 450 is not highly unusual in this distribution.

## Limitations and Considerations

While Z-score tables are incredibly useful, they have limitations. They are only accurate for normally distributed data. In datasets with significant skew or kurtosis, Z-scores might not provide the correct insights. That’s why it’s crucial to understand the context and nature of the data when applying Z-scores.

Below we explain why context matters, along with some “on the contrary” examples:

a) Normal Distribution Assumption: Z-scores are based on the assumption that the data follows a normal distribution. If the data is heavily skewed or has a significant departure from normality, using Z-scores might lead to misleading interpretations.

• Deviant Scenario: In a dataset representing annual incomes in a region with a large income disparity, the distribution might be right-skewed with a few very high incomes. Using Z-scores here could suggest that moderately high incomes are more ‘unusual’ than they actually are, due to the influence of the extremely high incomes.

b) Outliers Impact: Z-scores are sensitive to outliers. In datasets with extreme values, the mean and standard deviation (which are used to calculate Z-scores) can be heavily influenced by these outliers, leading to distorted Z-scores.

• Deviant Scenario: In a small class’s test scores, if one student scores exceptionally high or low, it can skew the average and standard deviation, thus affecting the Z-scores of all other students. A score that would normally be considered average might appear as an outlier when it is actually not.

c) Scale and Units of Measurement: The meaning and implications of a Z-score can vary depending on the scale and units of measurement in the dataset. This necessitates understanding the data’s context to interpret Z-scores correctly.

• Deviant Scenario: In psychological testing, a Z-score might indicate a clinically significant deviation from the norm. However, the same Z-score in a different context, like height measurements in a population, might be perfectly normal and not indicate any unusual characteristic.

d) Misinterpretation of Probabilities: Z-scores correspond to probabilities in a normal distribution. Misinterpreting these probabilities can lead to incorrect conclusions, especially in non-normal distributions.

• Deviant Scenario: In a distribution of a rare event, like major natural disasters, using Z-scores might suggest a higher probability of extreme values than what is realistically the case, as rare events do not typically follow a normal distribution.

## Z-table in HTML Format

Below you can see the HTML versions of the left Z-table and the right Z-table, which allows copy and pasting.

### Right Z-table

Related: Check out our Z-score calculator for easier computations.

## How do I make my own Z-score tables in Google Sheets?

Start by creating the header column: 0.00 to 0.09 (increments of 0.01). The create the vertical ones 0.0 to whatever you choose (increments of 0.1).

Then use the NORMDIST() function and sum up (for the right Z-table) the two cells you need. In our case D4 is “=NORMDIST($B4 + D$2)”. Blocking the B column and the \$2 row in the formula allos for dragging it to the whole table and generate the needed Z-score values.

You do similarly with the left Z-table except that you need to substract instead of summing the two cells.