Z-Distribution Calculator
Use this calculator to find probabilities associated with a Z-score, or to calculate a Z-score from raw data and then find its probabilities.
Calculate Probabilities from a Z-score
Calculate Z-score and Probabilities from Raw Data
Calculation Results:
Probability P(Z < z):
Probability P(Z > z):
Probability P(-z < Z < z):
Understanding the Z-Distribution and Z-Scores
The Z-distribution, also known as the standard normal distribution, is a special type of normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. It's a fundamental concept in statistics because it allows us to standardize data from any normal distribution, making it easier to compare and analyze.
What is a Z-Score?
A Z-score (also called a standard score) measures how many standard deviations an element is from the mean. A positive Z-score indicates the data point is above the mean, while a negative Z-score indicates it's below the mean. A Z-score of 0 means the data point is exactly at the mean.
The formula to calculate a Z-score from a raw score (x) is:
Z = (x – μ) / σ
- x: The raw score or data point.
- μ (mu): The population mean.
- σ (sigma): The population standard deviation.
Why are Z-Scores Important?
Z-scores are crucial for several reasons:
- Standardization: They allow us to compare observations from different normal distributions. For example, comparing a student's score on a math test with a mean of 70 and standard deviation of 10 to their score on a science test with a mean of 60 and standard deviation of 5.
- Probability Calculation: Once a raw score is converted to a Z-score, we can use the standard normal distribution table (or a calculator like this one) to find the probability of observing a score less than, greater than, or between certain values.
- Outlier Detection: Extremely high or low Z-scores (e.g., beyond ±2 or ±3) can indicate outliers in a dataset.
- Hypothesis Testing: Z-scores are integral to many statistical tests, such as Z-tests, which are used to test hypotheses about population means.
Interpreting Z-Scores and Probabilities
- P(Z < z): This is the cumulative probability, representing the area under the standard normal curve to the left of the given Z-score. It tells you the probability of observing a value less than 'z'.
- P(Z > z): This is the probability of observing a value greater than 'z', representing the area to the right of 'z'. It's calculated as 1 – P(Z < z).
- P(-z < Z < z): This represents the probability of observing a value between -z and +z, indicating the central area under the curve. It's often used for confidence intervals.
Examples of Z-Distribution in Action
Example 1: Calculating Probability from a Given Z-score
Suppose you have a Z-score of 1.5. You want to know the probability of a value being less than this Z-score.
Using the calculator with Z-score = 1.5, you would find:
- P(Z < 1.5) ≈ 0.9332 (or 93.32%)
- P(Z > 1.5) ≈ 0.0668 (or 6.68%)
- P(-1.5 < Z < 1.5) ≈ 0.8664 (or 86.64%)
This means there's a 93.32% chance of a randomly selected value from the standard normal distribution being less than 1.5 standard deviations above the mean.
Example 2: Calculating Z-score and Probability from Raw Data
A class of students took a test. The average score (population mean, μ) was 70, and the standard deviation (σ) was 8. One student scored 82 (raw score, x).
First, calculate the Z-score:
Z = (82 – 70) / 8 = 12 / 8 = 1.5
Now, using the calculator with Z-score = 1.5 (or by entering raw score 82, mean 70, std dev 8), you would get the same probabilities as in Example 1:
- P(Z < 1.5) ≈ 0.9332
- P(Z > 1.5) ≈ 0.0668
- P(-1.5 < Z < 1.5) ≈ 0.8664
This tells us that a student scoring 82 performed better than approximately 93.32% of the students in the class, assuming scores are normally distributed.