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	2.76%	2.23%	2.27%	2.29%	1.86%	1.94%	246.28	255.63	241.73	260.44
Weekly	6.07%	6.11%	6.41%	5.52%	4.10%	4.27%	240.82	261.42	231.14	272.37
Monthly	12.53%	13.18%	13.90%	11.86%	8.54%	8.90%	230.37	273.29	211.50	297.66
Quarterly	21.80%	27.78%	23.19%	21.08%	14.80%	15.41%	216.40	290.92	186.64	337.32
Annual	—	—	—	—	29.59%	30.81%	186.64	337.32	138.83	453.48
Option Time Horizon	—	—	—	—	11.17%	11.63%	224.39	280.56	200.68	313.72

Calculation Notes: Computed at 2026-03-23T16:30 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $250.91 | Strike (K): $255.00 (first OTM call ≥ spot) | Observed Mid-Price: $9.9500 (Bid $9.9000 / Ask $10.0000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.42% | Binomial IV (Annual): 29.59% (CRR binomial tree, American, n=20 steps) | BS IV: 29.86% (European Black-Scholes) | Yahoo IV (Annual): 30.81% → 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 -0.35. Further, an Excess Kurtosis of 18.63 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.18%	58	0.51%	0.51%
-8.18% to -6.11%	119	1.05%	1.56%
-6.11% to -4.04%	387	3.40%	4.96%
-4.04% to -1.97%	1332	11.70%	16.66%
-1.97% to 0.11%	3983	35.00%	51.66%
0.11% to 2.18%	3678	32.32%	83.98%
2.18% to 4.25%	1214	10.67%	94.65%
4.25% to 6.33%	393	3.45%	98.10%
6.33% to 8.40%	125	1.10%	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%	78.08%	68.27%
±2σ	-5.42%	5.64%	94.92%	95.45%
±3σ	-8.18%	8.40%	98.69%	99.73%

Descriptive Statistics

Mean	0.108%
Standard Error	0.026%
Median	0.000%
Mode	0.000%
Standard Deviation	2.764%
Sample Variance	7.637%
Kurtosis	18.63
Skewness	-0.35
Range	85.097%
Minimum	-51.869%
Maximum	33.228%
Count	11380

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.000%	5681	49.92%	0.998%
Negative	-1.901%	5332	46.85%	-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	1.64%
20	1 Month	2.13%
60	1 Quarter	2.16%
250	1 Year	2.40%
750	3 Years	2.10%
1,250	5 Years	2.29%
2,500	10 Years	2.26%
5,000	20 Years	2.48%
12,500	50 Years	3.39%

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 17.86 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%	2231	19.60%	19.60%
1.46% to 3.14%	4772	41.93%	61.53%
3.14% to 4.81%	2484	21.83%	83.36%
4.81% to 6.48%	1087	9.55%	92.91%
6.48% to 8.16%	437	3.84%	96.75%
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%	81.06%	68.27%
±2σ	-1.32%	7.60%	95.87%	95.45%
±3σ	-3.56%	9.83%	98.37%	99.73%

Descriptive Statistics

Mean	3.137%
Standard Error	0.021%
Median	2.594%
Mode	3.448%
Standard Deviation	2.231%
Sample Variance	4.977%
Kurtosis	17.86
Skewness	2.71
Range	35.569%
Minimum	0.347%
Maximum	35.916%
Count	11381

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.137%	11381	100.00%	3.137%
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 7.94 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.75%	71	0.62%	0.62%
-6.75% to -5.05%	145	1.27%	1.90%
-5.05% to -3.35%	415	3.65%	5.54%
-3.35% to -1.65%	1272	11.18%	16.72%
-1.65% to 0.05%	4162	36.57%	53.29%
0.05% to 1.75%	3409	29.95%	83.24%
1.75% to 3.45%	1263	11.10%	94.34%
3.45% to 5.15%	413	3.63%	97.97%
5.15% to 6.85%	149	1.31%	99.28%
Greater than 6.85%	82	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.22%	2.32%	76.81%	68.27%
±2σ	-4.49%	4.58%	94.46%	95.45%
±3σ	-6.75%	6.85%	98.66%	99.73%

Descriptive Statistics

Mean	0.049%
Standard Error	0.021%
Median	0.000%
Mode	0.000%
Standard Deviation	2.268%
Sample Variance	5.143%
Kurtosis	7.94
Skewness	0.36
Range	48.346%
Minimum	-16.154%
Maximum	32.192%
Count	11381

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.697%	5408	47.52%	0.806%
Negative	-1.633%	5273	46.33%	-0.757%
Zero	0.000%	700	6.15%	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 5.39 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -17.68%	16	0.68%	0.68%
-17.68% to -13.13%	20	0.85%	1.52%
-13.13% to -8.58%	85	3.60%	5.12%
-8.58% to -4.03%	306	12.96%	18.08%
-4.03% to 0.52%	751	31.80%	49.87%
0.52% to 5.07%	749	31.71%	81.58%
5.07% to 9.62%	299	12.66%	94.24%
9.62% to 14.17%	91	3.85%	98.09%
14.17% to 18.72%	30	1.27%	99.36%
Greater than 18.72%	15	0.64%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.54%	6.59%	75.99%	68.27%
±2σ	-11.61%	12.66%	95.39%	95.45%
±3σ	-17.68%	18.72%	98.69%	99.73%

Descriptive Statistics

Mean	0.523%
Standard Error	0.125%
Median	0.550%
Mode	0.000%
Standard Deviation	6.067%
Sample Variance	36.807%
Kurtosis	5.39
Skewness	-0.22
Range	93.066%
Minimum	-54.673%
Maximum	38.393%
Count	2362

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.623%	1257	53.22%	2.460%
Negative	-4.258%	1075	45.51%	-1.938%
Zero	0.000%	30	1.27%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	4.83%
12	1 Quarter	5.19%
52	1 Year	5.72%
156	3 Years	5.06%
260	5 Years	5.52%
520	10 Years	5.51%
1,040	20 Years	6.07%
2,600	50 Years	8.18%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.45. Further, an Excess Kurtosis of 80.02 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.81%	0	0.00%	0.00%
-9.81% to -5.23%	0	0.00%	0.00%
-5.23% to -0.65%	0	0.00%	0.00%
-0.65% to 3.93%	366	15.49%	15.49%
3.93% to 8.51%	1094	46.30%	61.79%
8.51% to 13.09%	568	24.04%	85.82%
13.09% to 17.68%	196	8.29%	94.12%
17.68% to 22.26%	76	3.22%	97.33%
22.26% to 26.84%	28	1.18%	98.52%
Greater than 26.84%	35	1.48%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.41%	14.62%	87.35%	68.27%
±2σ	-3.70%	20.73%	96.49%	95.45%
±3σ	-9.81%	26.84%	98.52%	99.73%

Descriptive Statistics

Mean	8.513%
Standard Error	0.126%
Median	7.190%
Mode	9.091%
Standard Deviation	6.108%
Sample Variance	37.308%
Kurtosis	80.02
Skewness	5.45
Range	131.673%
Minimum	1.306%
Maximum	132.979%
Count	2363

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.513%	2363	100.00%	8.513%
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 55.96 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.21%	3	0.13%	0.13%
-19.21% to -14.40%	17	0.72%	0.85%
-14.40% to -9.60%	73	3.09%	3.94%
-9.60% to -4.79%	319	13.50%	17.44%
-4.79% to 0.02%	858	36.31%	53.75%
0.02% to 4.83%	728	30.81%	84.55%
4.83% to 9.64%	250	10.58%	95.13%
9.64% to 14.45%	74	3.13%	98.26%
14.45% to 19.25%	19	0.80%	99.07%
Greater than 19.25%	22	0.93%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.39%	6.43%	78.93%	68.27%
±2σ	-12.80%	12.84%	96.23%	95.45%
±3σ	-19.21%	19.25%	98.94%	99.73%

Descriptive Statistics

Mean	0.021%
Standard Error	0.132%
Median	-0.353%
Mode	0.000%
Standard Deviation	6.411%
Sample Variance	41.103%
Kurtosis	55.96
Skewness	3.59
Range	143.716%
Minimum	-23.871%
Maximum	119.845%
Count	2363

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.612%	1097	46.42%	2.141%
Negative	-4.063%	1233	52.18%	-2.120%
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 -35.37%	2	0.37%	0.37%
-35.37% to -25.97%	8	1.47%	1.84%
-25.97% to -16.58%	17	3.13%	4.97%
-16.58% to -7.18%	82	15.10%	20.07%
-7.18% to 2.22%	165	30.39%	50.46%
2.22% to 11.62%	164	30.20%	80.66%
11.62% to 21.01%	69	12.71%	93.37%
21.01% to 30.41%	26	4.79%	98.16%
30.41% to 39.81%	9	1.66%	99.82%
Greater than 39.81%	1	0.18%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-10.31%	14.75%	72.19%	68.27%
±2σ	-22.84%	27.28%	94.66%	95.45%
±3σ	-35.37%	39.81%	99.45%	99.73%

Descriptive Statistics

Mean	2.219%
Standard Error	0.538%
Median	2.028%
Mode	-10.760%
Standard Deviation	12.530%
Sample Variance	157.011%
Kurtosis	1.38
Skewness	-0.06
Range	103.122%
Minimum	-57.744%
Maximum	45.378%
Count	543

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.763%	300	55.25%	5.946%
Negative	-8.364%	242	44.57%	-3.728%
Zero	0.000%	1	0.18%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	10.09%
12	1 Year	11.13%
36	3 Years	10.48%
60	5 Years	11.86%
120	10 Years	12.46%
240	20 Years	13.59%
600	50 Years	18.21%

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 24.13 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.02%	0	0.00%	0.00%
-19.02% to -9.14%	0	0.00%	0.00%
-9.14% to 0.75%	0	0.00%	0.00%
0.75% to 10.63%	85	15.62%	15.62%
10.63% to 20.52%	252	46.32%	61.95%
20.52% to 30.40%	117	21.51%	83.46%
30.40% to 40.28%	52	9.56%	93.01%
40.28% to 50.17%	24	4.41%	97.43%
50.17% to 60.05%	9	1.65%	99.08%
Greater than 60.05%	5	0.92%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	7.34%	33.70%	82.35%	68.27%
±2σ	-5.84%	46.87%	96.69%	95.45%
±3σ	-19.02%	60.05%	99.08%	99.73%

Descriptive Statistics

Mean	20.516%
Standard Error	0.565%
Median	17.255%
Mode	5.990%
Standard Deviation	13.179%
Sample Variance	173.685%
Kurtosis	24.13
Skewness	3.43
Range	148.971%
Minimum	3.738%
Maximum	152.709%
Count	544

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	20.516%	544	100.00%	20.516%
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 19.10 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -42.09%	0	0.00%	0.00%
-42.09% to -31.67%	0	0.00%	0.00%
-31.67% to -21.25%	19	3.49%	3.49%
-21.25% to -10.82%	75	13.79%	17.28%
-10.82% to -0.40%	199	36.58%	53.86%
-0.40% to 10.02%	165	30.33%	84.19%
10.02% to 20.44%	59	10.85%	95.04%
20.44% to 30.86%	14	2.57%	97.61%
30.86% to 41.28%	6	1.10%	98.71%
Greater than 41.28%	7	1.29%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.30%	13.49%	77.76%	68.27%
±2σ	-28.20%	27.39%	97.24%	95.45%
±3σ	-42.09%	41.29%	98.71%	99.73%

Descriptive Statistics

Mean	-0.403%
Standard Error	0.596%
Median	-1.717%
Mode	0.000%
Standard Deviation	13.896%
Sample Variance	193.089%
Kurtosis	19.10
Skewness	2.52
Range	169.321%
Minimum	-31.214%
Maximum	138.107%
Count	544

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.539%	238	43.75%	4.611%
Negative	-9.061%	301	55.33%	-5.014%
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 -58.70%	1	0.55%	0.55%
-58.70% to -42.35%	1	0.55%	1.10%
-42.35% to -26.00%	11	6.08%	7.18%
-26.00% to -9.65%	21	11.60%	18.78%
-9.65% to 6.70%	59	32.60%	51.38%
6.70% to 23.05%	55	30.39%	81.77%
23.05% to 39.40%	19	10.50%	92.27%
39.40% to 55.74%	9	4.97%	97.24%
55.74% 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	-15.10%	28.50%	72.38%	68.27%
±2σ	-36.90%	50.30%	95.58%	95.45%
±3σ	-58.70%	72.09%	98.34%	99.73%

Descriptive Statistics

Mean	6.697%
Standard Error	1.620%
Median	5.308%
Mode	-61.501%
Standard Deviation	21.799%
Sample Variance	475.192%
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	19.189%	111	61.33%	11.768%
Negative	-13.112%	70	38.67%	-5.071%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	19.27%
12	3 Years	21.42%
20	5 Years	21.08%
40	10 Years	22.40%
80	20 Years	25.09%
200	50 Years	33.10%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.76. Further, an Excess Kurtosis of 24.45 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -42.88%	0	0.00%	0.00%
-42.88% to -22.05%	0	0.00%	0.00%
-22.05% to -1.21%	0	0.00%	0.00%
-1.21% to 19.63%	27	14.84%	14.84%
19.63% to 40.47%	89	48.90%	63.74%
40.47% to 61.30%	45	24.73%	88.46%
61.30% to 82.14%	10	5.49%	93.96%
82.14% to 102.98%	5	2.75%	96.70%
102.98% to 123.81%	4	2.20%	98.90%
Greater than 123.81%	2	1.10%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	12.68%	68.25%	86.81%	68.27%
±2σ	-15.10%	96.03%	96.70%	95.45%
±3σ	-42.88%	123.81%	98.90%	99.73%

Descriptive Statistics

Mean	40.465%
Standard Error	2.059%
Median	33.925%
Mode	9.350%
Standard Deviation	27.783%
Sample Variance	771.876%
Kurtosis	24.45
Skewness	3.76
Range	257.078%
Minimum	9.350%
Maximum	266.428%
Count	182

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	40.465%	182	100.00%	40.465%
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 12.31 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -71.34%	0	0.00%	0.00%
-71.34% to -53.95%	0	0.00%	0.00%
-53.95% to -36.55%	5	2.75%	2.75%
-36.55% to -19.15%	27	14.84%	17.58%
-19.15% to -1.76%	73	40.11%	57.69%
-1.76% to 15.64%	53	29.12%	86.81%
15.64% to 33.03%	10	5.49%	92.31%
33.03% to 50.43%	10	5.49%	97.80%
50.43% to 67.82%	2	1.10%	98.90%
Greater than 67.82%	2	1.10%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-24.95%	21.43%	78.57%	68.27%
±2σ	-48.15%	44.63%	96.70%	95.45%
±3σ	-71.34%	67.82%	98.90%	99.73%

Descriptive Statistics

Mean	-1.759%
Standard Error	1.719%
Median	-4.853%
Mode	-45.320%
Standard Deviation	23.194%
Sample Variance	537.959%
Kurtosis	12.31
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	18.089%	71	39.01%	7.057%
Negative	-14.455%	111	60.99%	-8.816%
Zero	0.000%	0	0.00%	0.000%

AAPL Deep Analytical Report