Z-Score Table

Z-Score Table

A Z-score table, also known as the standard normal table, is a mathematical table used to determine the percentage of values below a given Z-score in a standard normal distribution. Z-scores are used in statistics to describe the position of a value in relation to the mean of a dataset, measured in standard deviations.

What is a Z-Score?

A Z-score represents how many standard deviations a value is from the mean of a dataset. It’s calculated using the formula:

Z=(X−μ)σZ = \frac{(X - \mu)}{\sigma}Z=σ(X−μ)​

Where:

• XXX is the value being evaluated.
• μ\muμ is the mean of the dataset.
• σ\sigmaσ is the standard deviation of the dataset.

A positive Z-score indicates that the value is above the mean, while a negative Z-score indicates that it is below the mean.

How to Use a Z-Score Table

A Z-score table provides the cumulative probability of a value being less than a given Z-score in a standard normal distribution (mean = 0, standard deviation = 1).

• Reading the Z-Score Table: The table is typically organized into rows and columns:
  • The row represents the first two digits of the Z-score (e.g., 1.2).
  • The column represents the second decimal place (e.g., 0.03).
  • The intersection of the row and column gives the cumulative probability (the area under the curve to the left of the Z-score).

For example:

• A Z-score of 1.23 would correspond to a cumulative probability of around 0.8907. This means that approximately 89.07% of the values in the dataset are below this Z-score.

Applications of Z-Scores and Z-Score Tables

1. Statistical Analysis: Z-scores are used to compare scores from different distributions.
2. Hypothesis Testing: Z-scores help in determining how unusual or extreme a particular data point is.
3. Grading: Z-scores are often used in grading systems to standardize scores across different tests or exams.
4. Quality Control: Z-scores are applied in quality control processes to detect anomalies.

Example Calculation

Suppose the mean score of a test is 70 with a standard deviation of 10. If a student scores 85, the Z-score is:

Z=(85−70)10=1.5Z = \frac{(85 - 70)}{10} = 1.5Z=10(85−70)​=1.5

Using a Z-score table, a Z-score of 1.5 corresponds to a cumulative probability of about 0.9332, indicating that the student scored better than approximately 93.32% of the population.

Types of Z-Score Tables

1. Positive Z-Score Table: Used for positive Z-scores (values above the mean).
2. Negative Z-Score Table: Used for negative Z-scores (values below the mean).

Conclusion

Z-score tables are fundamental tools in statistics for calculating the probability associated with a Z-score in a standard normal distribution. They are widely used in various fields like data analysis, psychology, and quality control to make informed decisions based on statistical data.

For a detailed explanation and Z-score table reference, check out the full article: https://www.geeksforgeeks.org/z-score-table/.