📊 Z-Score Calculator

Calculated Result
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Formula

Quick Examples

Notes:

  • Z-score formula: z = (x - μ) / σ (raw to z-score)
  • Raw score formula: x = μ + z × σ (z-score to raw)
  • Z-score represents how many standard deviations from the mean
  • Positive z-score: above mean, Negative z-score: below mean
  • Cumulative probability based on standard normal distribution (μ=0, σ=1)

📊 Z-Score Calculator｜Stats Lovers Rejoice! 2 Modes + Formula Hacks + Pro Tips✨

Still stressing over Z-score and raw score conversions? Stuck on stats homework/papers/data analysis? This Z-score calculator is a total lifesaver🤩 Supports two-way conversion, spits out results in seconds, and includes super useful tips—students/researchers/professionals, dive right in!

📝【2 Modes! 3-Step Hack for Beginners】

  1. Pick your "Calculation Direction"➡️:
    • "Raw→Z-score": Convert raw score to Z-score (input grades/data to find how many standard deviations from the mean)
    • "Z→Raw Score": Convert Z-score to raw score (input deviation to reverse-engineer specific data)
  2. Plug in key data📥:
    • Raw Score (x) / Z-Score (z) (choose one—match your selected mode!)
    • Population Mean (μ) (e.g., class average, overall data mean)
    • Standard Deviation (σ) (data spread—pre-calculate or input directly)
  3. Select precision (default 4 decimal places, adjust as needed) and hit calculate! Results + cumulative probability pop up instantly✅

💡【Formula Breakdown + Core Hacks! No Rote Memorization】

  • Core Formulas:
    • Raw to Z-score: z = (x - μ) / σ → Positive = above mean, Negative = below mean, 0 = equal to mean
    • Z-score to Raw: x = μ + z × σ → Plug and play—calculator crunches up to 4 decimal places!
  • Example🌰: Raw=85, μ=75, σ=6.67 → Instantly get Z=1.5—know right away the data is 1.5 SD above the mean!
  • Key Pro Tips:
    • Larger absolute Z-score = more unusual data: |z|≥1.96 = "statistically significant deviation" (common stats threshold), |z|≥3 = potential "outlier"
    • Cumulative Probability = chance data is lower than the value: e.g., Z=1.5 → ~0.9332 = 93.32% of data is below that raw score
    • Standard Normal Distribution Rule: 68% of data falls between Z=±1, 95% between ±1.96, 99.7% between ±3 (68-95-99.7 Rule)

🌟【Practical Tricks to Level Up Your Z-Score Game】

  1. Grade Comparison: Compare scores across different subjects/exams (different max scores/difficulty) fairly! e.g., Math 80 (μ=70, σ=5) → Z=2; English 85 (μ=80, σ=10) → Z=0.5—Math performance is actually stronger
  2. Outlier Detection: Quick filter outliers in data analysis—e.g., a Z-score of 3.2 in survey data? Double-check for entry errors
  3. Probability Estimation: Skip the Z-table! Calculator gives cumulative probability directly—e.g., "What’s the chance of scoring above 85?" Just do 1 - cumulative probability
  4. Sample Size Note: Z-score works best with sample size ≥30; for smaller samples, pair with t-score for better analysis

✨【Why It’s a Game-Changer】

  • ✅ Two-Way Conversion Freedom: No tool-switching—toggle between modes with one click, max efficiency
  • ✅ Comprehensive Results: Gets you conversion + cumulative probability (based on standard normal distribution)—ready for papers
  • ✅ Ad-Free & No Sign-Ups: Web-based, open and use instantly. Works on phone/laptop—perfect for labs/libraries📱💻
  • ✅ Adjustable Precision: 1-4 decimal places—fits homework, research, work, and more
  • ✅ Newbie-Friendly: Formulas + tips explained simply—no textbook flipping, easy for beginners～

⚠️【Common Mistakes to Avoid!】

  1. Don’t mix up data! Raw and Z-scores must match your calculation direction—mix-up = wrong results❌
  2. Keep units consistent! If scores are percentage-based, mean/SD must be too
  3. Cumulative probability relies on standard normal distribution (μ=0, σ=1)—normalize data first for skewed distributions
  4. Only for quantitative data! Categorical data (e.g., gender, occupation) can’t be calculated～
  5. Don’t over-rely! Z-score is a tool—interpret results with data context and research goals in mind