Z-Score Cut-Off Calculator
Result:
Z = (X - μ) / σ
Where: Z = standard score, X = raw score, μ = mean, σ = standard deviation
About Z-Score Cut-Offs
A z-score (standard score) represents how many standard deviations an element is from the mean. This calculator helps you find:
- The probability (p-value) associated with a given z-score
- The z-score associated with a given probability
- Probabilities between or outside specific z-scores
Common z-score cut-offs:
- ±1.96 for 95% confidence (two-tailed)
- ±2.576 for 99% confidence (two-tailed)
- -1.645 for left-tailed 5% significance
- 1.645 for right-tailed 5% significance
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Here's a clear description of each calculation type in the Z-Score Cut-Off Calculator, along with explanations of Z-Scores and P-values:
Calculation Types:
Left-Tailed (P(Z ≤ z))
Calculates the probability that a standard normal random variable Z is less than or equal to a given z-score
Example: P(Z ≤ -1.5) = 0.0668 (6.68%)
Used when interested in values below a certain threshold
Visual: Area under the curve to the left of the z-score
Right-Tailed (P(Z ≥ z))
Calculates the probability that Z is greater than or equal to a given z-score
Example: P(Z ≥ 1.96) = 0.025 (2.5%)
Used when interested in values above a certain threshold
Visual: Area under the curve to the right of the z-score
Between Two Z-Scores (P(a ≤ Z ≤ b))
Calculates the probability that Z falls between two specified z-scores
Example: P(-1 ≤ Z ≤ 1) = 0.6826 (68.26%)
Visual: Area under the curve between the two z-scores
Outside Two Z-Scores (P(Z ≤ a or Z ≥ b))
Calculates the probability that Z is either below the first z-score or above the second
Example: P(Z ≤ -2 or Z ≥ 2) = 0.0455 (4.55%)
Visual: Combined areas in both tails outside the specified range
Two-Tailed (P(|Z| ≥ z))
Calculates the probability in both tails beyond a given absolute z-score
Example: P(|Z| ≥ 1.96) = 0.05 (5%)
Commonly used for hypothesis testing with symmetrical rejection regions
Visual: Equal areas in both tails beyond ±z
:Key Concepts
Z-Score:
A measure of how many standard deviations a value is from the mean
Formula: Z = (X - μ)/σ
X = raw score
μ = population mean
σ = population standard deviation
Interpretation:
Z = 0: Exactly at the mean
Z > 0: Above the mean
Z < 0: Below the mean
Probability (P-value):
The area under the standard normal curve corresponding to the specified z-score(s)
Represents the probability of observing a value as extreme as the z-score
Range: 0 to 1 (or 0% to 100%)
Common critical values:
P(Z ≤ -1.96) = 0.025
P(Z ≥ 1.96) = 0.025
P(|Z| ≥ 1.96) = 0.05
Practical Examples:
- Left-Tailed:"What percentage of students scored below 600 if SAT scores have μ=500, σ=100?"Z = (600-500)/100 = 1 → P(Z ≤ 1) = 0.8413 (84.13%)
- Right-Tailed:"What's the probability of getting a value more than 2.5σ above the mean?"P(Z ≥ 2.5) = 0.0062 (0.62%)
- Between Two Values:"What proportion of people have IQ between 85 and 115 (μ=100, σ=15)?"Z-scores: -1 and 1 → P(-1 ≤ Z ≤ 1) = 0.6826
- Two-Tailed Test:"Is this result statistically significant at α=0.05 level?"Critical value: |Z| ≥ 1.96 corresponds to p ≤ 0.05
Z-Score Cut-Off Calculator User Guide
Introduction
Welcome to the Z-Score Cut-Off Calculator! This tool helps you work with standard normal distributions to find probabilities associated with specific z-scores and vice versa.
The calculator supports five different calculation types to meet various statistical needs.
Calculation Types
1. Left-Tailed (P(Z ≤ z))
Calculates the probability that a value is less than or equal to your z-score.
When to use: When you want to know the probability of values below a certain point.
2. Right-Tailed (P(Z ≥ z))
Calculates the probability that a value is greater than or equal to your z-score.
When to use: When you want to know the probability of values above a certain point.
3. Between Two Z-Scores (P(a ≤ Z ≤ b))
Calculates the probability that a value falls between two z-scores.
When to use: When you want to know what percentage of data falls within a specific range.
4. Outside Two Z-Scores (P(Z ≤ a or Z ≥ b))
Calculates the probability that a value is outside a specified range.
When to use: When you want to know the probability of extreme values on both ends.
5. Two-Tailed (P(|Z| ≥ z))
Calculates the combined probability in both tails beyond a z-score.
When to use: For symmetrical hypothesis testing with rejection regions in both tails.
Understanding Z-Scores
A z-score tells you how many standard deviations a value is from the mean.
Formula: Z = (X - μ) / σ
- X = raw score
- μ = population mean
- σ = population standard deviation
Key Interpretation:
- Z = 0: Exactly at the mean
- Z > 0: Above the mean
- Z < 0: Below the mean
Understanding P-Values
The p-value represents the probability of obtaining a result at least as extreme as the observed result.
| P-Value Range | Interpretation |
|---|---|
| p ≤ 0.01 | Highly statistically significant |
| 0.01 < p ≤ 0.05 | Statistically significant |
| p > 0.05 | Not statistically significant |
Important: A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis.
How to Use the Calculator
- Select your calculation type from the dropdown
- Enter your z-score(s) in the input field(s)
- For probability-to-z calculations, enter your p-value
- Click the Calculate button
- Read your result in the output box
Practical Examples
Example 1: Left-Tailed
Scenario: What percentage of students scored below 600 if SAT scores have μ=500, σ=100?
Calculation: Z = (600-500)/100 = 1 → P(Z ≤ 1) = 0.8413 (84.13%)
Example 2: Right-Tailed
Scenario: What's the probability of getting a value more than 2.5σ above the mean?
Calculation: P(Z ≥ 2.5) = 0.0062 (0.62%)
Example 3: Between Two Values
Scenario: What proportion of people have IQ between 85 and 115 (μ=100, σ=15)?
Calculation: Z-scores: -1 and 1 → P(-1 ≤ Z ≤ 1) = 0.6826 (68.26%)
