TSLA 3σ tail frequency vs Gaussian expectation (last ~3 years)
TSLA produced seven daily moves at or beyond a 3σ threshold over the ~36-month window we studied—about 3.45 times the count a Gaussian would predict (7 observed vs ~2.03 expected). That gap is not trivial: the two-sided excess has a binomial p ≈ 0.0048, and the distribution shows clear non‑Gaussian shape (positive skew +0.465 and excess kurtosis 3.33).
We looked at 751 trading days (minute bars resampled to daily closes from mid‑2023 to mid‑2026), standardized returns, and counted ≥+3σ and ≤−3σ occurrences versus Normal(0,1) probabilities. The detailed tables, charts, and binomial tests below show the asymmetric tail structure (a richer upside tail but meaningful downside risk) and quantify why Gaussian-based tail models understate TSLA’s true frequency of violent moves.
For TSLA over the past ~3 years, how badly does the bell curve underprice its tail risk — how many more ≥3σ daily moves actually landed than a normal distribution says should occur, and which tail is fatter? Thesis: TSLA printed several times the count of ≥3σ days a Gaussian allows, with heavy excess kurtosis and negative skew, so any risk model assuming normal returns dramatically understates how violent and how downside-skewed the real moves are.
How this was measured
Resampled minute bars to daily closes, computed close-to-close returns over the trailing ~36 months (2023-06-30 to 2026-06-30). Standardized returns using the sample mean and standard deviation (z-scores), then counted ≥+3σ and ≤−3σ daily moves. Compared observed counts to a Normal(0,1) benchmark with per-day probabilities sf(3)=1−Φ(3) for one tail and 2×sf(3) for two-sided. Reported observed/expected ratios and binomial p-values for observing at least as many ≥3σ days under the Gaussian assumption. Skewness and excess kurtosis quantify asymmetry and tail heaviness.
The key numbers
Reading the numbers
Over 751 trading days there were 7 days with |z| ≥ 3σ versus a Gaussian expectation of about 2.03 — roughly 3.45× as many. The positive tail had 4 such days (~3.95× expected) and the negative tail had 3 (~2.96× expected).
The charts
The histogram is centered at zero (mean 0.0) but stretches into extreme values: the sample min is −4.8846 and the max is 6.191. Focus on the bars beyond z = ±3 on the x-axis — those correspond to the 7 days with |z| ≥ 3σ noted in the stats, far above what a normal curve would produce. That handful of extreme bars is exactly why a Gaussian assumption misses how often TSLA posts very large moves.
This bar chart compares raw counts: observed 3 negative, 4 positive, 7 two-sided versus Gaussian expectations of about 1.01 negative, 1.01 positive, and 2.03 two-sided. The observed/expected ratios are roughly 2.96 (neg), 3.95 (pos) and 3.45 (two-sided), so the upside tail shows the larger excess in this sample window while the downside is also meaningfully elevated. The take-away is clear: the bell curve underestimates tail frequency by multiple-fold, with the positive tail doing the heavier lifting here.
3σ exceedances — observed vs expected (Gaussian)
| bucket | observed_count | expected_count | observed_over_expected | binomial_p_ge_obs |
|---|---|---|---|---|
| Negative (≤ -3σ) | 3 | 1.01 | 2.96 | 0.0827 |
| Positive (≥ +3σ) | 4 | 1.01 | 3.95 | 0.0198 |
| Two-sided (|z| ≥ 3) | 7 | 2.03 | 3.45 | 0.0048 |
Top 15 absolute-move days by |z|
| date | daily_return | z_score | tail |
|---|---|---|---|
| 2025-04-09 | 0.231 | 6.19 | positive |
| 2025-03-10 | -0.1799 | -4.88 | negative |
| 2024-04-23 | 0.1591 | 4.25 | positive |
| 2024-04-29 | 0.1445 | 3.86 | positive |
| 2025-03-24 | 0.132 | 3.52 | positive |
| 2025-06-05 | -0.1273 | -3.47 | negative |
| 2025-04-04 | -0.1225 | -3.34 | negative |
| 2024-11-11 | 0.109 | 2.9 | positive |
| 2024-11-06 | 0.1083 | 2.88 | positive |
| 2024-10-23 | 0.101 | 2.69 | positive |
| 2024-07-02 | 0.1008 | 2.68 | positive |
| 2025-04-22 | 0.1007 | 2.68 | positive |
| 2024-07-23 | -0.0968 | -2.64 | negative |
| 2024-07-11 | -0.0951 | -2.6 | negative |
| 2025-06-23 | 0.0966 | 2.57 | positive |
The takeaway
Short answer: TSLA had 7 ≥3σ daily moves in the last ~3 years versus ~2.03 expected under a Normal — about 3.45× more — and the upside was actually the richer tail (4 positive ≥3σ days vs 1.01 expected, O/E ≈ 3.95; negative had 3 vs 1.01, O/E ≈ 2.96). That two-sided excess is unlikely to be luck (binomial p ≈ 0.0048, roughly a 5-in-1,000 chance under Gaussian assumptions); the positive-tail excess is also statistically meaningful (p ≈ 0.0198, ~2%), while the negative-tail excess is weaker (p ≈ 0.0827, ~8%). The moments confirm heavy, asymmetric tails: skewness is +0.465 (leaning toward large gains) and excess kurtosis is 3.33 (much fatter tails than Gaussian). Practical takeaway: a normal-return risk model would materially understate how often violent TSLA moves occur in this window — roughly 3–4× too few ≥3σ days — so Gaussian-based tail estimates are not credible here.
The fine print
- Standardization used the in-sample mean and std over the full ~3-year window; regime shifts can inflate 3σ counts versus a regime-aware model.
- Gaussian expectations assume i.i.d. returns; volatility clustering and autocorrelation in |returns| violate that and can produce excess 3σ counts.
- Close-to-close daily sampling ignores intraday extremes; high-low or intraday sampling would likely show even thicker tails.
- Results are window-dependent; a different 36-month slice can change skew, kurtosis, and tail counts materially.