Z-table

A Z-table, also known as a standard normal distribution table or a Z-score table, is a reference table used in statistics to find the cumulative probability associated with a specific standard score (Z-score) in a standard normal distribution. The standard normal distribution is a bell-shaped curve with a mean (average) of 0 and a standard deviation of 1. This distribution is widely used in statistical analysis to make inferences about data.

Here’s a brief explanation of the Z-table and its significance:

What Is the Z-Table?

The Z-table provides a way to find the probability that a randomly selected value from a standard normal distribution falls below a particular Z-score, or within a specified range. Z-scores represent how many standard deviations a data point is from the mean. They are useful for comparing and understanding the relative position of data points in a distribution.

How It Works:

1. Find the Z-score for the value you’re interested in using the formula:

   Z = (X – μ) / σ

   • Z is the Z-score.
   • X is the value you want to convert to a Z-score.
   • μ is the mean of the distribution (usually 0 in a standard normal distribution).
   • σ is the standard deviation of the distribution (usually 1 in a standard normal distribution).
2. Look up the Z-score in the Z-table to find the cumulative probability associated with it. This probability represents the area under the standard normal distribution curve to the left of the Z-score.
3. The Z-table provides a value between 0 and 1, and you can interpret it as a probability. For example, if the Z-score is 1.96, the table would show a cumulative probability of approximately 0.9750. This means that 97.50% of the data falls below a Z-score of 1.96 in a standard normal distribution.

Significance:

The Z-table is essential in statistical analysis and hypothesis testing. It allows researchers to make inferences about data by calculating probabilities associated with Z-scores. It helps answer questions like “What is the probability of observing a value as extreme as the one we have in our data under the null hypothesis?”

Interpretation:

• A Z-score greater than 0 indicates that the value is above the mean.
• A Z-score less than 0 indicates that the value is below the mean.
• The magnitude of the Z-score indicates how many standard deviations the value is from the mean. For example, a Z-score of 2.0 indicates a value is two standard deviations above the mean.

Applications:

• Z-tables are widely used in quality control, finance, psychology, and other fields to analyze and make decisions based on standard scores.
• They are also used to calculate confidence intervals, conduct hypothesis tests, and assess the normality of data distributions.

In summary, the Z-table is a valuable tool in statistical analysis for understanding the relative position and probability associated with Z-scores in a standard normal distribution. It provides a convenient way to calculate and interpret the cumulative probability of observing specific values in the context of the standard normal distribution.

Z table (right)

Z Table (Left)