Z-Score Deviation in Small Samples: Is -0.07 Significant?

A z-score quantifies a number’s deviation from the mean in standard deviation units, making it easy to gauge how unusual x=22 is compared to [27, 43, 17, 5]. Here, the calculated z-score of -0.07 signals a tiny shift—just 0.07 standard deviations below the mean—negligible in any context, especially a small sample where standard deviation swings wildly. Your JavaScript nails the math for descriptive purposes, but for small datasets like this (n=4), treat z-scores cautiously; they’re unstable and better suited to larger sets.

Contents

• What is a Z-Score and Standard Deviation?
• Step-by-Step: Calculating Z-Score for This Dataset
• Is a Z-Score of -0.07 Significant or Negligible?
• Z-Score Interpretation: Thresholds and Tables
• Challenges of Z-Scores in Small Samples
• Z-Score vs. T-Test: Better Choices for Tiny Datasets
• Reviewing and Improving the JavaScript Code
• Sources
• Conclusion

What is a Z-Score and Standard Deviation?

Picture this: you’ve got a bunch of numbers, and one stands out—or does it? A z-score tells you exactly how far that number strays from the group average, measured in units of standard deviation. It’s like asking, “Is 22 way off from 27, 43, 17, and 5, or just hanging out nearby?”

Standard deviation (often just “std dev”) captures the spread. High std dev? Numbers are all over the place. Low? They’re clustered tight. The formula’s simple: z = (x - mean) / std dev. Why bother? It standardizes everything, so you compare apples to oranges—say, test scores or stock prices—on the same scale.

In your case, it flags whether 22 is an outlier or business as usual. But hold up—with only four others, things get tricky fast. We’ll unpack that.

Step-by-Step: Calculating Z-Score for This Dataset

Let’s crunch the numbers for x=22 against [27, 43, 17, 5]. No calculator needed; we’ll do it by hand.

First, the mean (μ): (27 + 43 + 17 + 5) / 4 = 92 / 4 = 23.

Delta? x - mean = 22 - 23 = -1. Already small.

Now variance: sum of squared differences from mean, divided by n (population style, like your code). Differences: 27-23=4 (16), 43-23=20 (400), 17-23=-6 (36), 5-23=-18 (324). Total squared: 776. Variance = 776 / 4 = 194. Std dev = √194 ≈ 13.93.

Z-score: -1 / 13.93 ≈ -0.072. Matches your JS output spot‑on.

Your code uses population variance (/n), perfect for describing this exact set. (For samples estimating a bigger population, some swap to /n-1, bumping std dev to ~16.1 and z to ~-0.062. Minor tweak.)

// Quick verify in console
const others = [27, 43, 17, 5];
const mean = others.reduce((s, v) => s + v, 0) / others.length; // 23
console.log(mean); // Exact match

Solid start. But does -0.07 scream “outlier”? Spoiler: nope.

Is a Z-Score of -0.07 Significant or Negligible?

Short answer: negligible. A z-score near zero means 22 sits comfortably inside the pack. Under the normal curve’s 68‑95‑99.7 rule, |z| < 1 covers 68% of data—totally typical. Yours? Laughably close to zero.

Significant? Statisticians eyeball |z| > 1.96 (95% confidence) or >2.58 (99%). -0.07 doesn’t even register. In hypothesis testing, you’d need p < 0.05, but this p-value hovers near 1—boring as watching paint dry.

Why so small? That std dev of 13.93 dwarfs the 1‑unit delta. Change one number (say, swap 5 for 10), mean shifts to 24.25, std to 14.8, z to -0.15. Still tiny. That’s the game.

Z-Score Interpretation: Thresholds and Tables

Z-scores shine with context. Here’s a quick table from standard normal distribution stats:

| |z| Range | Interpretation | Probability (two‑tailed) |
|---------|---------------|-----------------------|--------------------------|
| 0 to 1 | Normal variation | ~68% of data |
| 1 to 1.96 | Mildly unusual | 68‑95% |
| 1.96 to 2.58 | Notable (p<0.05) | 95‑99% |
| >3 | Extreme outlier | <0.3% |

Source: LibreTexts on z-scores. Your -0.07? Dead center in “normal.”

For outliers, some flag |z| > 3. Others adjust: in finance, |z| > 2 might ping alerts. Always ask: what’s “extreme” here? Machine learning might use it for anomaly detection, per GeeksforGeeks.

Standard z‑tables give cumulative probs—e.g., z=-0.07 lands at ~0.472, meaning 47.2% of data below it. Useless for inference here, though.

Challenges of Z-Scores in Small Samples

Small datasets like n=4? Z-scores wobble. Standard deviation’s volatile—one wild value (43 here) inflates it, shrinking z-scores. Drop 43: new mean=16.33, std=10.41, z=22-16.33/10.41=+0.54. Bigger deviation suddenly.

Why? Variance formula assumes normality, which tiny sets rarely have. Wikipedia on z-tests warns: z needs large n or known population std dev. Neither applies.

Result: overconfident calls. -0.07 looks negligible, but is it? Can’t say reliably. You might wonder, “So, ditch z-scores?” Not entirely— for rough descriptions, they’re fine. Just don’t bet the farm.

Z-Score vs. T-Test: Better Choices for Tiny Datasets

Z-score assumes known std dev and normality. Small samples? Switch to t‑test, which adjusts for uncertainty.

Rule of thumb: n>30? Z ok. n<30? T‑test, per Investopedia and Statistics How To.

For one‑sample t: t = (x - mean) / (s / √n), where s is sample std (/n-1). Your data: s≈16.1, t=(22-23)/(16.1/√4) ≈ -0.124. Degrees of freedom=3, still insignificant (critical t~3.18 for 95%).

DataCamp compares them cleanly: t’s wider confidence intervals protect against small‑sample lies. For non‑normal? Non‑parametrics like Wilcoxon.

Bottom line: z descriptive, t inferential. Your JS? Add t‑calc for robustness.

Reviewing and Improving the JavaScript Code

Your function? Clean, efficient. Filters x out (smart, avoids self‑inclusion), computes population stats correctly. Output {delta: -1, z: -0.07}—flawless.

Tweaks for small samples:

function deviationStats(data, x) {
 const others = data.filter(v => v !== x);
 if (others.length < 2) return { error: 'Need at least 2 others for std dev' };
 
 const n = others.length;
 const mean = others.reduce((s, v) => s + v, 0) / n;
 const delta = x - mean;
 
 // Population std (your original)
 const popVariance = others.reduce((s, v) => s + Math.pow(v - mean, 2), 0) / n;
 const popStd = Math.sqrt(popVariance);
 const z = delta / popStd;
 
 // Sample std (/n-1) + t‑score
 const sampleVariance = others.reduce((s, v) => s + Math.pow(v - mean, 2), 0) / (n - 1);
 const sampleStd = Math.sqrt(sampleVariance);
 const t = delta / (sampleStd / Math.sqrt(n));
 
 return { delta, z, t, popStd, sampleStd, n };
}

console.log(deviationStats([27, 43, 17, 5], 22));
// {delta: -1, z: -0.0718, t: -0.124, popStd: 13.93, sampleStd: 16.1, n: 4}

Now it’s production‑ready: handles edge cases, adds t‑score. Throw in normality checks (Shapiro‑Wilk lib) for extra polish.

Sources

1. Z‑Scores — Core explanation of z‑scores, standard deviation, and small‑sample limitations: https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Statistics:_Open_for_Everyone_(Peter)/05:_The_Normal_Curve_z-Scores_and_Probability/5.03:_z-Scores
2. Z‑Test — Differences between z‑test and alternatives for small samples: https://www.investopedia.com/terms/z/z-test.asp
3. Z‑test Wikipedia — Assumptions, formula, and risks with small datasets: https://en.wikipedia.org/wiki/Z-test
4. Z‑Score Tutorial — Step‑by‑step calculation and interpretation guidelines: https://www.datacamp.com/tutorial/z-score
5. Z‑Score Basics — Definition and use in measuring deviations: https://www.investopedia.com/terms/z/zscore.asp
6. T‑Score vs Z‑Score — When to use each based on sample size: https://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/
7. Z‑Score Outlier Detection — Thresholds for identifying extremes: https://www.geeksforgeeks.org/machine-learning/z-score-for-outlier-detection-python/
8. T‑Test vs Z‑Test — Practical comparison for small datasets: https://www.datacamp.com/tutorial/t-test-vs-z-test

Conclusion

Your z‑score of -0.07 marks a negligible deviation—22 blends right in with [27, 43, 17, 5], no red flags. But small samples amplify standard deviation’s fickleness, so lean on z‑scores for quick descriptions, not big decisions; grab t‑tests or bigger data for real insight. Enhanced JS code above equips you to handle both—run it, tweak, and spot those subtle shifts confidently.