Kalkulator ng Z-Score – Standard Score, Percentile at Probabilidad

Paano gamitin ang calculator na ito

1. Ilagay ang Calculate
2. Ilagay ang Data Point (x)
3. Ilagay ang Mean (μ)
4. Ilagay ang Standard Deviation (σ)
5. Ilagay ang Z-Score
6. I-click ang Kalkulahin na buton
7. Basahin ang resultang ipinapakita sa ibaba ng calculator

What Is a Z-Score?

A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a dataset. The formula is: z = (x − μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.

Z-scores allow comparison of values from different distributions. For example, you can compare a test score of 80 out of 100 (mean 70, SD 10) to a score of 55 out of 75 (mean 50, SD 8) by converting both to z-scores.

A z-score of 0 means the value equals the mean. A z-score of +1 means one standard deviation above the mean. A z-score of −2 means two standard deviations below the mean.

The Standard Normal Distribution

The standard normal distribution (Z distribution) has mean = 0 and standard deviation = 1. Any normal distribution can be converted to the standard normal by calculating z-scores.

Key percentages of the empirical rule:

• 68% of data falls within 1 standard deviation (z between −1 and +1)
• 95% of data falls within 2 standard deviations (z between −2 and +2)
• 99.7% of data falls within 3 standard deviations (z between −3 and +3)

Values beyond z = ±3 are rare (0.3% probability) — these are statistical outliers by conventional definitions.

Z-Score to Percentile Conversion

Applications of Z-Scores

• Education: Standardized test scores (SAT, GRE, IQ) are often expressed as z-scores or scaled equivalents
• Medical: BMI percentiles, bone density scores (T-scores, Z-scores in DEXA scans), growth charts
• Finance: Altman Z-score predicts bankruptcy probability; stock return analysis
• Quality control: Six Sigma methodology targets processes with 6 sigma (z = 6) quality
• Sports analytics: Comparing player performance across different eras, leagues, or positions
• Research: Hypothesis testing, p-values, confidence intervals all use the standard normal distribution

Hypothesis Testing with Z-Scores

In statistics, z-scores are central to hypothesis testing. A z-test compares a sample mean to a known population mean:

z = (x̄ − μ) / (σ / √n), where x̄ is the sample mean, μ is the population mean, σ is population standard deviation, and n is sample size.

The resulting z-score is compared to critical values for the chosen significance level (α):

• α = 0.05 (two-tailed): reject H₀ if |z| > 1.96
• α = 0.01 (two-tailed): reject H₀ if |z| > 2.576
• α = 0.001 (two-tailed): reject H₀ if |z| > 3.29

Z-Score vs T-Score

When should you use a z-score vs. a t-score?

• Z-score: Use when population standard deviation (σ) is known AND sample size is large (n > 30). Distribution is the standard normal.
• T-score: Use when population SD is unknown (using sample SD instead) or sample size is small (n ≤ 30). Distribution is t-distribution with n−1 degrees of freedom.

In practice, for large samples (n > 100), z and t distributions are nearly identical. The t-distribution has heavier tails, accounting for additional uncertainty from not knowing σ exactly.

Interpreting Extreme Z-Scores

Extreme z-scores have important practical implications:

• Z > 3 or Z < −3: Outlier. May indicate data entry error, measurement error, or genuinely unusual observation. Investigate before including in analysis.
• Z = 4 (p ≈ 0.003%): Extremely rare event. In particle physics, a "4 sigma" result is noteworthy but not conclusive evidence of new phenomena.
• Z = 5 (p ≈ 0.00003%): The "five sigma" standard used in physics to claim discovery of new particles (e.g., Higgs boson confirmation in 2012).