Z-Score Probability Calculator with Explanation
Calculate z-score probabilities with complete explanations. Learn normal distribution, standard scores, and statistical significance. Free calculator.
Quick Answer
Z-score z = (x - μ) / σ measures standard deviations from mean. Positive z = above mean, negative z = below mean. Use standard normal table or calculator to find probabilities. Essential for hypothesis testing and statistical analysis.
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Calculate With Full ToolWhat is Z-Score?
Z-score (standard score) measures how many standard deviations a data point is from the mean. It standardizes different distributions for comparison and helps determine the probability of specific values occurring in a normal distribution.
How Z-Score Works
The z-score transforms any normal distribution to the standard normal distribution (mean = 0, standard deviation = 1). This allows us to use standard normal tables and calculators to find probabilities for any normally distributed data.
Interpreting Z-Scores
Positive z-scores indicate values above the mean, negative scores indicate below mean. A z-score of 0 equals the mean. The magnitude shows how unusual the value is - larger absolute values indicate more extreme observations.
Z-Score Formula
z = (x - μ) / σ
Formula: z = (x - μ) / σ
Where: x = value, μ = mean, σ = standard deviation
Empirical Rule: 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ
Common Z-Scores: ±1.96 (95% confidence), ±2.58 (99% confidence)
Probability: P(Z < z) from standard normal table
Step-by-Step Example
Example: Student scored 85, class mean = 75, SD = 5
Step 1: Identify values: x = 85, μ = 75, σ = 5
Step 2: Calculate difference: x - μ = 85 - 75 = 10
Step 3: Divide by standard deviation: 10 ÷ 5 = 2
Step 4: Z-score = 2 (2 standard deviations above mean)
Step 5: Find probability: P(Z < 2) ≈ 0.9772
Step 6: Interpret: Score is higher than 97.72% of students
Example: Find probability between two z-scores
Step 1: Calculate z-scores for both values
Step 2: Find P(Z < z₂) and P(Z < z₁) from table
Step 3: Calculate difference: P(Z < z₂) - P(Z < z₁)
Step 4: This gives probability between the two values
These examples show how to calculate and interpret z-scores for individual values and find probabilities between values using the standard normal distribution.
Who Should Use This Calculator?
Statisticians
Perform hypothesis testing and analysis
Researchers
Analyze experimental data and significance
Educators
Standardize test scores and grades
Quality Control
Monitor process variations and defects
Frequently Asked Questions
What does a z-score of 1.5 mean?
A z-score of 1.5 means the value is 1.5 standard deviations above the mean. This corresponds to approximately the 93rd percentile, meaning the value is higher than about 93% of all values in the distribution.
How do you find probability from z-score?
Use a standard normal table, calculator, or software to find P(Z < z). For probabilities between two values, subtract the smaller probability from the larger one. Many calculators and statistical software provide this function.
When should z-scores not be used?
Z-scores assume a normal distribution and shouldn't be used for highly skewed distributions, small sample sizes, or non-normal data. In these cases, consider non-parametric methods or data transformation techniques.
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