AAPL - Detail Report

Quick Overview

Timeframe	C-C DoR σ (Historical)	H-L DoR σ (Historical)	O-C DoR σ (Historical)	5Y ATRP (Avg)	Binomial IV (Period)	Yahoo IV (Period)	1σ Lower (68.27%)	1σ Upper	2σ Lower (95.45%)	2σ Upper
Daily	2.77%	2.23%	2.27%	2.30%	—	1.73%	260.28	268.95	256.05	273.39
Weekly	6.07%	6.11%	6.42%	5.55%	—	3.81%	255.21	274.30	246.17	284.37
Monthly	12.54%	13.18%	13.91%	11.90%	—	7.93%	245.45	285.21	227.70	307.44
Quarterly	21.79%	27.81%	23.19%	20.90%	—	13.73%	232.32	301.32	204.00	343.16
Annual	—	—	—	—	—	27.47%	204.00	343.16	157.29	445.07
Option Time Horizon	—	—	—	—	—	10.56%	239.40	292.41	216.61	323.17

Calculation Notes: Computed at 2026-02-22T14:16 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $264.58 | Strike (K): $265.00 (first OTM call ≥ spot) | Observed Mid-Price: $10.9750 (Bid $10.9000 / Ask $11.0500) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.40% | Binomial IV (Annual): 0.00% (CRR binomial tree, American, n=20 steps) | BS IV: 26.00% (European Black-Scholes) | Yahoo IV (Annual): 27.47% → period-scaled as σ_period = σ_annual·√h | Price bands: Upper = S·e^+σ√h, Lower = S·e^−σ√h

Historical Dividend Payouts

Ex-Dividend Date	Amount
2026-02-09	$0.2600
2025-11-10	$0.2600
2025-08-11	$0.2600
2025-05-12	$0.2600
2025-02-10	$0.2500
2024-11-08	$0.2500
2024-08-12	$0.2500
2024-05-10	$0.2500
2024-02-09	$0.2400
2023-11-10	$0.2400
2023-08-11	$0.2400
2023-05-12	$0.2400
2023-02-10	$0.2300
2022-11-04	$0.2300
2022-08-05	$0.2300
2022-05-06	$0.2300
2022-02-04	$0.2200
2021-11-05	$0.2200
2021-08-06	$0.2200
2021-05-07	$0.2200
2021-02-05	$0.2050
2020-11-06	$0.2050
2020-08-07	$0.2050
2020-05-08	$0.2050
2020-02-07	$0.1925
2019-11-07	$0.1925
2019-08-09	$0.1925
2019-05-10	$0.1925
2019-02-08	$0.1825
2018-11-08	$0.1825
2018-08-10	$0.1825
2018-05-11	$0.1825
2018-02-09	$0.1575
2017-11-10	$0.1575
2017-08-10	$0.1575
2017-05-11	$0.1575
2017-02-09	$0.1425
2016-11-03	$0.1425
2016-08-04	$0.1425
2016-05-05	$0.1425
2016-02-04	$0.1300
2015-11-05	$0.1300
2015-08-06	$0.1300
2015-05-07	$0.1300
2015-02-05	$0.1175
2014-11-06	$0.1175
2014-08-07	$0.1175
2014-05-08	$0.1175
2014-02-06	$0.1089
2013-11-06	$0.1089
2013-08-08	$0.1089
2013-05-09	$0.1089
2013-02-07	$0.0946
2012-11-07	$0.0946
2012-08-09	$0.0946
1995-11-21	$0.0011
1995-08-16	$0.0011
1995-05-26	$0.0011
1995-02-13	$0.0011
1994-11-18	$0.0011
1994-08-15	$0.0011
1994-05-27	$0.0011
1994-02-07	$0.0011
1993-11-19	$0.0011
1993-08-16	$0.0011
1993-05-28	$0.0011
1993-02-12	$0.0011
1992-11-30	$0.0011
1992-08-17	$0.0011
1992-06-01	$0.0011
1992-02-14	$0.0011
1991-11-18	$0.0011
1991-08-19	$0.0011
1991-05-20	$0.0011
1991-02-15	$0.0011
1990-11-16	$0.0011
1990-08-20	$0.0010
1990-05-21	$0.0010
1990-02-16	$0.0010
1989-11-17	$0.0010
1989-08-21	$0.0009
1989-05-22	$0.0009
1989-02-17	$0.0009
1988-11-21	$0.0009
1988-08-15	$0.0007
1988-05-16	$0.0007
1988-02-12	$0.0007
1987-11-17	$0.0007
1987-08-10	$0.0005
1987-05-11	$0.0005

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.36. Further, an Excess Kurtosis of 18.61 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.19%	58	0.51%	0.51%
-8.19% to -6.11%	119	1.05%	1.56%
-6.11% to -4.04%	386	3.40%	4.96%
-4.04% to -1.97%	1330	11.71%	16.66%
-1.97% to 0.11%	3974	34.98%	51.65%
0.11% to 2.18%	3672	32.32%	83.97%
2.18% to 4.26%	1213	10.68%	94.65%
4.26% to 6.33%	393	3.46%	98.11%
6.33% to 8.40%	124	1.09%	99.20%
Greater than 8.40%	91	0.80%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.66%	2.87%	78.06%	68.27%
±2σ	-5.42%	5.64%	94.91%	95.45%
±3σ	-8.19%	8.40%	98.69%	99.73%

Descriptive Statistics

Mean	0.108%
Standard Error	0.026%
Median	0.000%
Mode	0.000%
Standard Deviation	2.765%
Sample Variance	7.648%
Kurtosis	18.61
Skewness	-0.36
Range	85.097%
Minimum	-51.869%
Maximum	33.228%
Count	11360

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.002%	5673	49.94%	1.000%
Negative	-1.903%	5320	46.83%	-0.891%
Zero	0.000%	367	3.23%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	2.53%
20	1 Month	2.63%
60	1 Quarter	2.00%
250	1 Year	2.47%
750	3 Years	2.11%
1,250	5 Years	2.30%
2,500	10 Years	2.25%
5,000	20 Years	2.49%
12,500	50 Years	3.40%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.71. Further, an Excess Kurtosis of 17.84 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.56%	0	0.00%	0.00%
-3.56% to -1.88%	0	0.00%	0.00%
-1.88% to -0.21%	0	0.00%	0.00%
-0.21% to 1.46%	2227	19.60%	19.60%
1.46% to 3.14%	4758	41.88%	61.48%
3.14% to 4.81%	2482	21.85%	83.33%
4.81% to 6.49%	1087	9.57%	92.90%
6.49% to 8.16%	437	3.85%	96.74%
8.16% to 9.83%	185	1.63%	98.37%
Greater than 9.83%	185	1.63%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.91%	5.37%	81.05%	68.27%
±2σ	-1.33%	7.60%	95.88%	95.45%
±3σ	-3.56%	9.83%	98.37%	99.73%

Descriptive Statistics

Mean	3.138%
Standard Error	0.021%
Median	2.597%
Mode	3.448%
Standard Deviation	2.232%
Sample Variance	4.983%
Kurtosis	17.84
Skewness	2.71
Range	35.569%
Minimum	0.347%
Maximum	35.916%
Count	11361

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.138%	11361	100.00%	3.138%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.36. Further, an Excess Kurtosis of 7.93 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.76%	71	0.62%	0.62%
-6.76% to -5.06%	144	1.27%	1.89%
-5.06% to -3.35%	416	3.66%	5.55%
-3.35% to -1.65%	1271	11.19%	16.74%
-1.65% to 0.05%	4153	36.55%	53.30%
0.05% to 1.75%	3403	29.95%	83.25%
1.75% to 3.45%	1260	11.09%	94.34%
3.45% to 5.15%	414	3.64%	97.98%
5.15% to 6.86%	147	1.29%	99.28%
Greater than 6.86%	82	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.22%	2.32%	76.79%	68.27%
±2σ	-4.49%	4.59%	94.46%	95.45%
±3σ	-6.76%	6.86%	98.65%	99.73%

Descriptive Statistics

Mean	0.049%
Standard Error	0.021%
Median	0.000%
Mode	0.000%
Standard Deviation	2.269%
Sample Variance	5.150%
Kurtosis	7.93
Skewness	0.36
Range	48.346%
Minimum	-16.154%
Maximum	32.192%
Count	11361

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.698%	5398	47.51%	0.807%
Negative	-1.635%	5263	46.33%	-0.757%
Zero	0.000%	700	6.16%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.22. Further, an Excess Kurtosis of 5.38 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -17.69%	16	0.68%	0.68%
-17.69% to -13.13%	20	0.85%	1.53%
-13.13% to -8.58%	85	3.60%	5.13%
-8.58% to -4.03%	306	12.98%	18.11%
-4.03% to 0.53%	747	31.68%	49.79%
0.53% to 5.08%	749	31.76%	81.55%
5.08% to 9.63%	299	12.68%	94.23%
9.63% to 14.19%	91	3.86%	98.09%
14.19% to 18.74%	30	1.27%	99.36%
Greater than 18.74%	15	0.64%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.54%	6.60%	76.00%	68.27%
±2σ	-11.62%	12.67%	95.42%	95.45%
±3σ	-17.69%	18.74%	98.69%	99.73%

Descriptive Statistics

Mean	0.526%
Standard Error	0.125%
Median	0.556%
Mode	0.000%
Standard Deviation	6.071%
Sample Variance	36.858%
Kurtosis	5.38
Skewness	-0.22
Range	93.066%
Minimum	-54.673%
Maximum	38.393%
Count	2358

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.622%	1257	53.31%	2.464%
Negative	-4.267%	1071	45.42%	-1.938%
Zero	0.000%	30	1.27%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	6.34%
12	1 Quarter	4.54%
52	1 Year	5.87%
156	3 Years	5.07%
260	5 Years	5.55%
520	10 Years	5.51%
1,040	20 Years	6.09%
2,600	50 Years	8.19%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.44. Further, an Excess Kurtosis of 79.95 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.82%	0	0.00%	0.00%
-9.82% to -5.23%	0	0.00%	0.00%
-5.23% to -0.65%	0	0.00%	0.00%
-0.65% to 3.93%	367	15.56%	15.56%
3.93% to 8.52%	1089	46.16%	61.72%
8.52% to 13.10%	568	24.08%	85.80%
13.10% to 17.69%	196	8.31%	94.11%
17.69% to 22.27%	76	3.22%	97.33%
22.27% to 26.85%	28	1.19%	98.52%
Greater than 26.85%	35	1.48%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.41%	14.63%	87.33%	68.27%
±2σ	-3.71%	20.74%	96.48%	95.45%
±3σ	-9.82%	26.85%	98.52%	99.73%

Descriptive Statistics

Mean	8.518%
Standard Error	0.126%
Median	7.192%
Mode	9.091%
Standard Deviation	6.112%
Sample Variance	37.352%
Kurtosis	79.95
Skewness	5.44
Range	131.673%
Minimum	1.306%
Maximum	132.979%
Count	2359

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.518%	2359	100.00%	8.518%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.59. Further, an Excess Kurtosis of 55.88 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.23%	3	0.13%	0.13%
-19.23% to -14.42%	17	0.72%	0.85%
-14.42% to -9.61%	73	3.09%	3.94%
-9.61% to -4.79%	319	13.52%	17.47%
-4.79% to 0.02%	858	36.37%	53.84%
0.02% to 4.83%	724	30.69%	84.53%
4.83% to 9.64%	250	10.60%	95.13%
9.64% to 14.46%	74	3.14%	98.26%
14.46% to 19.27%	19	0.81%	99.07%
Greater than 19.27%	22	0.93%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.40%	6.44%	78.97%	68.27%
±2σ	-12.81%	12.85%	96.27%	95.45%
±3σ	-19.23%	19.27%	98.94%	99.73%

Descriptive Statistics

Mean	0.019%
Standard Error	0.132%
Median	-0.368%
Mode	0.000%
Standard Deviation	6.416%
Sample Variance	41.167%
Kurtosis	55.88
Skewness	3.59
Range	143.716%
Minimum	-23.871%
Maximum	119.845%
Count	2359

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.623%	1093	46.33%	2.142%
Negative	-4.062%	1233	52.27%	-2.123%
Zero	0.000%	33	1.40%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.06. Further, an Excess Kurtosis of 1.38 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -35.38%	2	0.37%	0.37%
-35.38% to -25.97%	8	1.48%	1.85%
-25.97% to -16.57%	17	3.14%	4.98%
-16.57% to -7.17%	82	15.13%	20.11%
-7.17% to 2.23%	164	30.26%	50.37%
2.23% to 11.64%	165	30.44%	80.81%
11.64% to 21.04%	68	12.55%	93.36%
21.04% to 30.44%	26	4.80%	98.15%
30.44% to 39.85%	9	1.66%	99.82%
Greater than 39.85%	1	0.18%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-10.30%	14.77%	72.14%	68.27%
±2σ	-22.84%	27.31%	94.65%	95.45%
±3σ	-35.38%	39.85%	99.45%	99.73%

Descriptive Statistics

Mean	2.235%
Standard Error	0.539%
Median	2.106%
Mode	-10.760%
Standard Deviation	12.537%
Sample Variance	157.172%
Kurtosis	1.38
Skewness	-0.06
Range	103.122%
Minimum	-57.744%
Maximum	45.378%
Count	542

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.764%	300	55.35%	5.958%
Negative	-8.373%	241	44.47%	-3.723%
Zero	0.000%	1	0.18%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	10.08%
12	1 Year	11.71%
36	3 Years	10.66%
60	5 Years	11.90%
120	10 Years	12.51%
240	20 Years	13.63%
600	50 Years	18.23%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.43. Further, an Excess Kurtosis of 24.14 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.00%	0	0.00%	0.00%
-19.00% to -9.12%	0	0.00%	0.00%
-9.12% to 0.77%	0	0.00%	0.00%
0.77% to 10.65%	84	15.47%	15.47%
10.65% to 20.54%	252	46.41%	61.88%
20.54% to 30.43%	117	21.55%	83.43%
30.43% to 40.31%	52	9.58%	93.00%
40.31% to 50.20%	24	4.42%	97.42%
50.20% to 60.08%	9	1.66%	99.08%
Greater than 60.08%	5	0.92%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	7.36%	33.72%	82.32%	68.27%
±2σ	-5.82%	46.90%	96.69%	95.45%
±3σ	-19.00%	60.08%	99.08%	99.73%

Descriptive Statistics

Mean	20.539%
Standard Error	0.566%
Median	17.280%
Mode	5.990%
Standard Deviation	13.181%
Sample Variance	173.731%
Kurtosis	24.14
Skewness	3.43
Range	148.971%
Minimum	3.738%
Maximum	152.709%
Count	543

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	20.539%	543	100.00%	20.539%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.52. Further, an Excess Kurtosis of 19.09 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -42.13%	0	0.00%	0.00%
-42.13% to -31.70%	0	0.00%	0.00%
-31.70% to -21.27%	19	3.50%	3.50%
-21.27% to -10.84%	75	13.81%	17.31%
-10.84% to -0.41%	199	36.65%	53.96%
-0.41% to 10.02%	164	30.20%	84.16%
10.02% to 20.44%	59	10.87%	95.03%
20.44% to 30.87%	14	2.58%	97.61%
30.87% to 41.30%	6	1.10%	98.71%
Greater than 41.30%	7	1.29%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.32%	13.49%	77.90%	68.27%
±2σ	-28.23%	27.40%	97.24%	95.45%
±3σ	-42.13%	41.30%	98.71%	99.73%

Descriptive Statistics

Mean	-0.415%
Standard Error	0.597%
Median	-1.763%
Mode	0.000%
Standard Deviation	13.906%
Sample Variance	193.375%
Kurtosis	19.09
Skewness	2.52
Range	169.321%
Minimum	-31.214%
Maximum	138.107%
Count	543

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.559%	237	43.65%	4.609%
Negative	-9.062%	301	55.43%	-5.023%
Zero	0.000%	5	0.92%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.45. Further, an Excess Kurtosis of 1.62 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -58.63%	1	0.55%	0.55%
-58.63% to -42.29%	1	0.55%	1.10%
-42.29% to -25.95%	11	6.08%	7.18%
-25.95% to -9.61%	21	11.60%	18.78%
-9.61% to 6.73%	59	32.60%	51.38%
6.73% to 23.07%	55	30.39%	81.77%
23.07% to 39.41%	19	10.50%	92.27%
39.41% to 55.75%	9	4.97%	97.24%
55.75% to 72.09%	3	1.66%	98.90%
Greater than 72.09%	2	1.10%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.05%	28.52%	72.93%	68.27%
±2σ	-36.84%	50.30%	95.58%	95.45%
±3σ	-58.63%	72.09%	98.34%	99.73%

Descriptive Statistics

Mean	6.732%
Standard Error	1.619%
Median	5.308%
Mode	-61.501%
Standard Deviation	21.786%
Sample Variance	474.634%
Kurtosis	1.62
Skewness	0.45
Range	149.463%
Minimum	-61.501%
Maximum	87.962%
Count	181

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	19.036%	112	61.88%	11.779%
Negative	-13.239%	69	38.12%	-5.047%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	18.36%
12	3 Years	21.12%
20	5 Years	20.90%
40	10 Years	22.31%
80	20 Years	25.04%
200	50 Years	33.08%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.75. Further, an Excess Kurtosis of 24.37 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -42.98%	0	0.00%	0.00%
-42.98% to -22.12%	0	0.00%	0.00%
-22.12% to -1.27%	0	0.00%	0.00%
-1.27% to 19.59%	27	14.84%	14.84%
19.59% to 40.44%	89	48.90%	63.74%
40.44% to 61.30%	45	24.73%	88.46%
61.30% to 82.15%	10	5.49%	93.96%
82.15% to 103.01%	5	2.75%	96.70%
103.01% to 123.86%	4	2.20%	98.90%
Greater than 123.86%	2	1.10%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	12.64%	68.25%	86.26%	68.27%
±2σ	-15.17%	96.05%	96.70%	95.45%
±3σ	-42.98%	123.86%	98.90%	99.73%

Descriptive Statistics

Mean	40.442%
Standard Error	2.061%
Median	33.925%
Mode	9.350%
Standard Deviation	27.806%
Sample Variance	773.201%
Kurtosis	24.37
Skewness	3.75
Range	257.078%
Minimum	9.350%
Maximum	266.428%
Count	182

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	40.442%	182	100.00%	40.442%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.36. Further, an Excess Kurtosis of 12.34 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -71.36%	0	0.00%	0.00%
-71.36% to -53.97%	0	0.00%	0.00%
-53.97% to -36.58%	5	2.75%	2.75%
-36.58% to -19.19%	27	14.84%	17.58%
-19.19% to -1.79%	72	39.56%	57.14%
-1.79% to 15.60%	53	29.12%	86.26%
15.60% to 32.99%	11	6.04%	92.31%
32.99% to 50.38%	10	5.49%	97.80%
50.38% to 67.77%	2	1.10%	98.90%
Greater than 67.77%	2	1.10%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-24.98%	21.39%	78.57%	68.27%
±2σ	-48.17%	44.58%	96.70%	95.45%
±3σ	-71.36%	67.77%	98.90%	99.73%

Descriptive Statistics

Mean	-1.795%
Standard Error	1.719%
Median	-4.852%
Mode	-45.320%
Standard Deviation	23.189%
Sample Variance	537.716%
Kurtosis	12.34
Skewness	2.36
Range	202.509%
Minimum	-45.320%
Maximum	157.189%
Count	182

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	18.278%	70	38.46%	7.030%
Negative	-14.342%	112	61.54%	-8.826%
Zero	0.000%	0	0.00%	0.000%

AAPL Deep Analytical Report