🎓 Statistics & Data • Descriptive Statistics
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Z-Score Calculator (Standard Normal Score)
Converts raw scores into standard Z-scores (z = (x - μ) / σ) with normal distribution percentile rankings.
Descriptive Statistics Engine
Dataset Analysis & Five-Number Summary
Input comma or space separated numbers. Computes central tendency, spread, shape moments, Tukey fences, and interactive plots.
Raw Data Values10 observations parsed
Presets:
Mean (x̄)24.7Sum / N
Median (Q₂)23.550th percentile
Sample Std Dev (s)8.7819s² = 77.1222
IQR (Q₃ - Q₁)12Middle 50%
Interactive Data Visualization
IQR Box (Q₁ to Q₃) Median (23.5)✕ Mean (24.7) Outlier (Outside 1.5×IQR)
Comprehensive Metrics Table
Count (N)10
Sum (Σx)247
Mode22
Range28 (12 - 40)
Sample Variance (s²)77.1222
Pop Variance (σ²)69.41
Standard Error (SE)2.7771
Coef of Variation (CV)35.55%
10% Trimmed Mean24.375
First Quartile (Q₁)18
Third Quartile (Q₃)30
Tukey Fences [LF, UF][0, 48]
Skewness0.319 (Symmetric)
Excess Kurtosis-0.525 (Platykurtic)
Detected OutliersNone detected
Single Value Evaluator (Z-Score & Percentile)Standardized deviation from sample mean: z = (x - x̄) / s
Standardized Z-Score+0.034 σ0.034 standard deviations above mean
Percentile Rank55th PercentileBetter than or equal to 55% of sample
Step-by-Step Statistical Derivations
1. Sorted Ascending Dataset:
12, 15, 18, 22, 22, 25, 28, 30, 35, 40
2. Central Tendency:
- Mean: x̄ = Σx / n = 247 / 10 = 24.7
- Median: Ordered middle position average of items 5 & 6 = 23.5
- Mode: 22 (Frequency = 2)
3. Sum of Squared Deviations & Variance:
- SS = Σ(x_i - x̄)² = 694.1
- Sample Variance: s² = SS / (n - 1) = 694.1 / 9 = 77.1222
- Sample Standard Deviation: s = √s² = √77.1222 = 8.7819
4. Five-Number Summary & Outlier Fences:
- Min = 12, Q₁ = 18, Median = 23.5, Q₃ = 30, Max = 40
- IQR = Q₃ - Q₁ = 30 - 18 = 12
- Lower Fence = Q₁ - 1.5×IQR = 18 - 18 = 0
- Upper Fence = Q₃ + 1.5×IQR = 30 + 18 = 48
What is a Z-Score?
A Z-score (standard score) indicates how many standard deviations an observation is above or below the mean of a distribution.
Formula & Step-by-Step Calculation
z = (x - μ) / σ (or z = (x - x̄) / s)
Raw value minus mean divided by standard deviation.
Worked Step-by-Step Examples
Example 1
Student scores 85 on exam with mean 70 and std dev 10
Solution: z = +1.50 σ (93.32nd percentile)
• (85 - 70) / 10 = 15 / 10 = 1.50
Common Real-World & Academic Use Cases
- ✓ SAT/ACT standardized test comparison
- ✓ Pediatric child growth charts
- ✓ Financial credit risk Altman Z-scores
How to Use the Z-Score Calculator (Standard Normal Score)
1
Input Dataset or Mean/SD
Provide numbers or specify mean and standard deviation.
2
Input Value x
Enter score to standardize.
3
Read Z-Score
Inspect standard score and normal curve percentile.
Frequently Asked Questions
Q: What does a negative Z-score mean?
A negative Z-score indicates that the value lies below the distribution mean.