Z Score from Mean Calculator
Calculate standardized Z-score ($Z = \frac{X - \mu}{\sigma}$), distance from mean ($X - \mu$), percentile rank %, and normal distribution probabilities.
Calculate Z-Score from Mean & Standard Deviation
Enter raw score (X), mean (μ), and standard deviation (σ).
Calculation Results
Calculated using standardized normal distribution equations: Z = \frac{X - \mu}{\sigma}
Quick Summary
The Z Score from Mean Calculator evaluates standardized Z-score ($Z = \frac{X - \mu}{\sigma}$), distance from mean ($X - \mu$), percentile rank %, and tail probabilities.
Formula Explanation
Z = \frac{X - \mu}{\sigma}
\text{Percentile Rank \%} = P(Z < z) \times 100How It Works
A Z-score normalizes raw data scores across different distributions into a standardized bell curve. The Z Score from Mean Calculator subtracts mean $\mu$ from raw score $X$ and divides by standard deviation $\sigma$.
Step-by-Step Worked Example
Practical Problem: An individual scores $X = 120$ on an exam with mean $\mu = 100$ and standard deviation $\sigma = 15$. Calculate Z-score and percentile rank.
- Step 1: Calculate raw deviation from mean ($X - \mu$): $120 - 100 = \mathbf{+20.0000}$.
- Step 2: Divide deviation by standard deviation ($\sigma = 15$): $Z = +20 / 15 = \mathbf{+1.3333}$.
- Step 3: Look up cumulative left-tail probability $P(Z < 1.3333)$: $P(Z < 1.3333) = \mathbf{0.9088}$.
- Step 4: Calculate Percentile Rank: $0.9088 \times 100 = \mathbf{90.88th percentile}$.
- Step 5: Interpretation: The score of 120 is $1.33$ standard deviations above the mean, outscoring 90.88% of participants.
Real-World Calculation Examples
Scenario 1: Exam Score (X = 120, μ = 100, σ = 15)
Parameters: X = 120, mean = 100, SD = 15
Result: Z = +1.3333 (90.88th percentile rank).
Scenario 2: Below Average Score (X = 80, μ = 100, σ = 15)
Parameters: X = 80, mean = 100, SD = 15
Result: Z = -1.3333 (9.12th percentile rank).
Scenario 3: Standardized SAT Test Score (X = 1350, μ = 1050, σ = 200)
Parameters: X = 1350, mean = 1050, SD = 200
Result: Z = +1.5000 (93.32nd percentile).
Scenario 4: Height Comparison (X = 74 in, μ = 70 in, σ = 3 in)
Parameters: X = 74, mean = 70, SD = 3
Result: Z = +1.3333.
Key Benefits of Using This Calculator
Standardized Z-Score Formula
Applies standard $Z = (X - \mu) / \sigma$ normalization for cross-test comparisons.
Raw Deviation Output
Displays exact numerical distance from mean ($X - \mu$).
Percentile Rank & CDF Probability
Calculates cumulative left-tail $P(Z < z)$ and right-tail $P(Z > z)$ normal curve probabilities.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a Z-score?
A Z-score indicates how many standard deviations an individual raw score $X$ lies above or below the mean $\mu$.
How is Z-score calculated from mean and standard deviation?
Subtract mean $\mu$ from raw score $X$ and divide by standard deviation $\sigma$ ($Z = \frac{X - \mu}{\sigma}$).
What does Z = 0 mean?
A Z-score of 0 means the raw score is exactly equal to the mean.
What does a negative Z-score mean?
A negative Z-score means the raw score lies below the mean.
How do I convert Z-score back to raw score X?
Use formula $X = \mu + (Z \times \sigma)$.
How do I calculate Z-score in Excel?
Use formula =STANDARDIZE(x, mean, standard_dev).
What Z-score threshold indicates an outlier?
Z-scores with $|Z| > 2.0$ are unusual and $|Z| > 3.0$ are extreme outliers.
What is standard normal distribution $N(0,1)$?
A normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$.
What is the 68-95-99.7 Empirical Rule?
68% of data falls within $Z = \pm 1$, 95% within $Z = \pm 2$, and 99.7% within $Z = \pm 3$.
Can Z-scores be used to compare two different tests?
Yes — Z-scores standardize scores across tests with different means and standard deviations.