Z-Score Table

Use this page to find cumulative probabilities for Z-scores and interpret normal distributions. View the Z-tables for both negative and positive Z-scores below, or use the calculators to compute probabilities and Z-scores dynamically.

Understanding the Negative Z-Score Table

The Negative Z-Score Table represents cumulative probabilities for negative Z-scores in a standard normal distribution. These Z-scores indicate how many standard deviations a data point is below the mean, with probabilities corresponding to the area under the curve to the left of the Z-score.

For example, a Z-score of -1.96 represents the point 1.96 standard deviations below the mean, and the cumulative probability from the table shows the proportion of the distribution falling to the left of this point (approximately 2.5%). Negative Z-scores are used in hypothesis testing, confidence intervals, and determining probabilities in the lower tail of the distribution.

The table is organized as follows:

• The rows correspond to the first digit and first decimal place of the Z-score (e.g., -1.2).
• The columns correspond to the second decimal place (e.g., 0.03 for -1.23).
• The value at the intersection gives the cumulative probability \( P(Z < z) \), the area under the curve to the left of the Z-score.

Use this table to quickly find cumulative probabilities for negative Z-scores and interpret data points relative to the mean in a standard normal distribution.

Understanding the Positive Z-Score Table

The Positive Z-Score Table represents cumulative probabilities for positive Z-scores in a standard normal distribution. These Z-scores indicate how many standard deviations a data point is above the mean, with probabilities corresponding to the area under the curve to the left of the Z-score.

For example, a Z-score of 1.96 represents the point 1.96 standard deviations above the mean, and the cumulative probability from the table shows the proportion of the distribution falling to the left of this point (approximately 97.5%). Positive Z-scores are used in hypothesis testing, confidence intervals, and determining probabilities in the upper tail of the distribution.

The table is organized as follows:

• The rows correspond to the first digit and first decimal place of the Z-score (e.g., 1.2).
• The columns correspond to the second decimal place (e.g., 0.03 for 1.23).
• The value at the intersection gives the cumulative probability \( P(Z < z) \), the area under the curve to the left of the Z-score.

Use this table to quickly find cumulative probabilities for positive Z-scores and interpret data points relative to the mean in a standard normal distribution.

How to Use the Z-Score Tables

Z-Score tables, also known as Standard Normal Distribution tables, are used to find the cumulative probability of a Z-score or determine the Z-score corresponding to a given probability. These are essential in hypothesis testing, confidence intervals, and interpreting standard normal distributions. Follow these steps to use the Z-score tables:

1. Understand the Z-Score: The Z-score indicates how many standard deviations a data point is from the mean of a standard normal distribution. Negative Z-scores represent values below the mean, while positive Z-scores represent values above the mean.
2. Use the Z-Score Table:
   • Find the row: Use the first digit and the first decimal place of your Z-score (e.g., for 1.23, look for the row labeled 1.2).
   • Find the column: Use the second decimal place of your Z-score (e.g., for 1.23, look for the column labeled 0.03).
   • The value at the intersection gives the cumulative probability up to that Z-score.
3. Interpret the Result:
   • Cumulative probability: The value from the table represents the area under the curve to the left of the Z-score.
   • For probabilities to the right: Subtract the cumulative probability from 1 (e.g., \( P(Z > z) = 1 – P(Z < z) \)).

Use these probabilities for hypothesis testing, finding confidence intervals, or interpreting data in the context of standard normal distributions.

• If your calculated Z-score is greater than the critical Z-score (for one-tailed tests), reject the null hypothesis.
• For two-tailed tests, check if the Z-score lies outside the critical values on both sides of the distribution.

For quick calculations, use the Z-Score to P-Value Calculator. .

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