GITNUXREPORT 2026

The Empirical Rule Statistics

The empirical rule predicts data percentages within standard deviations of a normal distribution's mean.

Min-ji Park

Research Analyst focused on sustainability and consumer trends.

First published: Feb 13, 2026

Our Commitment to Accuracy

Rigorous fact-checking · Reputable sources · Regular updatesLearn more

Statistic 1

Empirical rule overestimates coverage in leptokurtic distributions like t(3df) where 3σ holds only 98.8%

Statistic 2

For uniform distribution U(-√3σ,√3σ), empirical 99.7% at 3σ covers only 100% exactly but 68% at 1σ is 57.7% less

Statistic 3

Skewed lognormal with σ=1 has only 62% within 1σ vs 68%, rule fails per QQ plots

Statistic 4

Cauchy distribution has undefined variance, empirical rule inapplicable as tails P(|X|>kσ) ~1/k

Statistic 5

Binomial n=30 p=0.5 σ≈2.74, empirical 68% covers ~27/30 successes close for large n

Statistic 6

Poisson λ=10 σ=√10≈3.16, 95% within 2σ approx 4-16 vs exact 0-22, reasonable CLT

Statistic 7

Exponential mean=1 σ=1, 68% within 1σ is 0 to 1. something only 63%, heavy tail issue

Statistic 8

Chi-square df=5 σ≈√10=3.16, 99.7% rule covers ~0-22 vs actual up to 25+, kurtosis excess

Statistic 9

Student's t df=5, P(|t|<3)=99.4% close to 99.7%, converges as df→∞

Statistic 10

Weibull shape=2 scale=1 σ≈0.886, 95% rule at 2σ≈1.77 covers 95% exactly for Rayleigh

Statistic 11

Rule 27x worse for platykurtic uniform, 3σ=100% vs normal 99.7%

Statistic 12

Financial returns leptokurtic kurtosis=25, 3σ covers 96% not 99.7%, Black swan risks

Statistic 13

Heights slightly right-skewed, empirical underestimates low tail coverage by 1-2%

Statistic 14

IQ censored at floors/ceilings, true 99.7% truncated, savant/genius beyond

Statistic 15

Rainfall lognormal skew, 68% rule misses dry spells, better gamma fit

Statistic 16

Stock vols cluster, GARCH models show conditional non-normality, rule optimistic

Statistic 17

Rule assumes μ,σ known; sample est. widens CI by √(1-1/n) factor

Statistic 18

Multimodal data like bimodal heights, rule splits coverage erroneously

Statistic 19

Heteroscedastic data violates constant σ, rule intervals fan out

Statistic 20

Small samples n<30, rule inaccurate without CLT, use t-dist

Statistic 21

Outliers inflate σ, shrinking rule coverage ironically, robust stats needed

Statistic 22

The empirical rule indicates that for a normal distribution, approximately 68.27% of all observations fall within one standard deviation of the mean, derived from the cumulative distribution function integral from -1 to 1 sigma

Statistic 23

Approximately 95.45% of data in a perfectly normal distribution lies within two standard deviations (±2σ) of the μ, as calculated precisely via z-scores of 1.96 for 95% confidence

Statistic 24

The 99.73% coverage within three standard deviations (±3σ) is a key hallmark of the empirical rule, with exact value from Φ(3) - Φ(-3) where Φ is the standard normal CDF

Statistic 25

Empirical rule approximation ignores skewness but holds best for symmetric bell-shaped curves with kurtosis near 3

Statistic 26

Named also as 68-95-99.7 rule, it provides quick estimates without tables for normal data assessment

Statistic 27

For standardized normal variable Z~N(0,1), P(-1<Z<1)=0.6827 exactly matching empirical rule's 68%

Statistic 28

The rule's percentages are erf(1/√2)≈0.6827 for 1σ, erf(2/√2)≈0.9545 for 2σ, and erf(3/√2)≈0.9973 for 3σ using error function

Statistic 29

Empirical rule applies strictly to unimodal symmetric distributions approximating normality per central limit theorem

Statistic 30

Violation occurs if data exceeds 3σ more than 0.27%, signaling non-normality like heavy tails

Statistic 31

Historical origin traces to Abraham de Moivre's 1738 approximation for binomial to normal, refined by Gauss

Statistic 32

In quality control, empirical rule sets natural process limits at ±3σ containing 99.73% of variation

Statistic 33

Rule's 68% is precise for N(0,1) as ∫_{-1}^1 (1/√(2π))exp(-x²/2)dx = 0.682689492

Statistic 34

95% within 1.96σ exactly, but empirical rule rounds to 2σ for simplicity at 95.45%

Statistic 35

99.7% at 3σ versus exact 99.865% at 2.576σ for 99.7% two-tailed

Statistic 36

Empirical rule facilitates mental math for outlier detection beyond 3σ at 0.135% per tail

Statistic 37

For IQ scores N(100,15), empirical rule predicts 68% between 85-115

Statistic 38

Rule assumes independence and identical distribution for CLT convergence rate of O(1/√n)

Statistic 39

Chebyshev's inequality gives looser bound: P(|X-μ|<kσ) ≥ 1-1/k², e.g., 75% for k=2 vs empirical 95%

Statistic 40

Empirical rule's accuracy improves with sample size n>30 per Berry-Esseen theorem bound ~0.5/√n

Statistic 41

In finance, stock returns often cited as 68% daily within 1% if σ=1%, per empirical rule

Statistic 42

Rule visualizes via bell curve shading: 68% inner, 95% middle, 99.7% outer bands

Statistic 43

Extension to 4σ covers 99.9937%, 5σ=99.99994% for six sigma processes

Statistic 44

Empirical rule symmetric: equal probability in ± tails, unlike skewed distributions

Statistic 45

For height distributions N(170cm,10cm), 68% between 160-180cm by rule

Statistic 46

Rule's 99.73% leaves 0.27% outliers, critical for anomaly detection thresholds

Statistic 47

Precise values: 68.268%, 95.44997%, 99.73002% from normal table lookups

Statistic 48

Empirical rule mnemonic 68-95-99.7 aids quick recall in exams/AP stats

Statistic 49

Applies to z-scores where |z|<1 for 68%, |z|<2 for 95%, |z|<3 for 99.7%

Statistic 50

In metrology, measurement uncertainty often uses 2σ for 95% coverage intervals

Statistic 51

Rule derives from infinite series expansion of normal pdf integral

Statistic 52

Heights of US males N(69in,2.8in), 68% 66.2-71.8in per CDC data fitting empirical rule

Statistic 53

IQ N(100,15), 68% score 85-115, matching WAIS standardization samples

Statistic 54

SAT scores N(1050,200), 68% 850-1250 across sections per College Board

Statistic 55

Adult female weights N(170lbs,30lbs), 68% 140-200lbs from NHANES survey

Statistic 56

Annual rainfall in Seattle N(37in,10in), 68% 27-47in per NOAA 30yr normals

Statistic 57

Machine part diameters N(50mm,0.5mm), 68% 49.5-50.5mm in SPC charts

Statistic 58

Daily stock returns N(0%,1%), 68% -1% to +1% for S&P500 historical fit

Statistic 59

Blood pressure systolic N(120,12), 68% 108-132 mmHg per AHA guidelines

Statistic 60

Exam scores N(75,10), 68% 65-85% in large university classes

Statistic 61

Shoe sizes men N(10.5,1.5), 68% 9-12 US per NPD group data

Statistic 62

CPU clock speeds N(3.5GHz,0.3GHz), 68% 3.2-3.8GHz in benchmarks

Statistic 63

House prices median N(300k,50k), 68% 250k-350k in US metro areas Zillow

Statistic 64

Reaction times N(250ms,40ms), 68% 210-290ms in psychophysics tasks

Statistic 65

Wind speeds N(10mph,3mph), 68% 7-13mph at coastal stations NOAA

Statistic 66

Battery life N(10hrs,1.5hrs), 68% 8.5-11.5hrs smartphone tests

Statistic 67

GPA undergrad N(3.2,0.4), 68% 2.8-3.6 on 4.0 scale NSSE survey

Statistic 68

Fish lengths N(20cm,3cm), 68% 17-23cm in lake populations

Statistic 69

Server response times N(200ms,30ms), 68% 170-230ms cloud metrics

Statistic 70

Crop yields N(150bu/ac,20bu), 68% 130-170bu corn belt USDA

Statistic 71

Pulse rates N(72bpm,12bpm), 68% 60-84bpm resting adults

Statistic 72

Fuel efficiency N(30mpg,4mpg), 68% 26-34mpg compact cars EPA

Statistic 73

Sleep duration N(7hrs,1hr), 68% 6-8hrs per sleep foundation polls

Statistic 74

Body temp N(98.6F,0.8F), 68% 97.8-99.4F healthy range

Statistic 75

Typing speed N(50wpm,15wpm), 68% 35-65wpm average adults

Statistic 76

Heights women N(64in,2.5in), 68% 61.5-66.5in CDC anthropometrics

Statistic 77

ACT scores N(21,5), 68% 16-26 composite national

Statistic 78

Sodium intake N(3400mg,800mg), 68% 2600-4200mg NHANES

Statistic 79

commute times N(27min,10min), 68% 17-37min US census

Statistic 80

Home values N(250k,60k), 68% 190k-310k suburban Zillow index

Statistic 81

GRE quant N(153,9), 68% 144-162 ETS norms

Statistic 82

Daily steps N(5000,1500), 68% 3500-6500 fitbit aggregates

Statistic 83

Water usage household N(300gal/d,80gal), 68% 220-380gal EPA stats

Statistic 84

Heights NBA players N(79in,3in), 68% 76-82in draft combines

Statistic 85

Energy consumption N(10kWh/d,2kWh), 68% 8-12kWh US homes EIA

Statistic 86

MCAT scores N(500,10), 68% 490-510 AAMC data

Statistic 87

Waist circumference N(36in,5in), 68% 31-41in men NHANES

Statistic 88

Heights college men N(70in,3in), 68% 67-73in campus health surveys

Statistic 89

IQ scores follow empirical rule with 99.7% between 55-145, rare profound disability below 3σ=55

Statistic 90

US men heights N(69in,2.8in), 99.7% 61.6-76.4in, extremes like tallest/shortest records fit

Statistic 91

SAT N(1050,200), 99.7% 450-1650, floor/ceiling effects noted

Statistic 92

Weights adult women N(170lbs,30lbs), 99.7% 80-260lbs obesity class III above 260lbs

Statistic 93

Seattle rain N(37in,10in), 99.7% -3-77in, but truncated at 0, multi-year extremes

Statistic 94

Machining N(50mm,0.5mm), 99.7% 48.5-51.5mm reject rate 0.27%

Statistic 95

Daily returns N(0,1%), 99.7% -3% to 3%, 2008 crash outliers beyond

Statistic 96

Systolic BP N(120,12), 99.7% 84-156mmHg, hypertensive crisis >180

Statistic 97

Exam N(75,10), 99.7% 45-105% capped at 100 fail <45

Statistic 98

Shoe men N(10.5,1.5), 99.7% 6.5-14.5 custom orders beyond

Statistic 99

CPU N(3.5GHz,0.3), 99.7% 2.6-4.4GHz overclock extremes

Statistic 100

House N(300k,50k), 99.7% 150-450k luxury/mansions above

Statistic 101

RT N(250ms,40ms), 99.7% 130-370ms pathology flags

Statistic 102

Wind N(10mph,3), 99.7% 1-19mph hurricane >74

Statistic 103

Battery N(10hr,1.5hr), 99.7% 4.5-15.5hr defect returns

Statistic 104

GPA N(3.2,0.4), 99.7% 2-4.4 probation/expulsion

Statistic 105

Fish N(20cm,3cm), 99.7% 11-29cm trophy sizes

Statistic 106

Server N(200ms,30ms), 99.7% 110-290ms downtime alerts

Statistic 107

Yields N(150bu,20bu), 99.7% 90-210bu disaster assistance

Statistic 108

Pulse N(72,12), 99.7% 36-108bpm arrhythmia risks

Statistic 109

MPG N(30,4), 99.7% 18-42 hypermiling records

Statistic 110

Sleep N(7hr,1hr), 99.7% 4-10hr hypersomnia/narcolepsy

Statistic 111

Temp N(98.6F,0.8F), 99.7% 95.2-102F hyperthermia >104

Statistic 112

Typing N(50wpm,15), 99.7% 5-95wpm elite competitions

Statistic 113

Women ht N(64in,2.5), 99.7% 56.5-71.5in gigantism/dwarfism

Statistic 114

ACT N(21,5), 99.7% 6-36 perfect score rarity

Statistic 115

Sodium N(3400mg,800mg), 99.7% 800-6000mg hyponatremia/hyper

Statistic 116

Commute N(27min,10min), 99.7% -3-57min remote extremes

Statistic 117

Home N(250k,60k), 99.7% 70-430k McMansions/slums

Statistic 118

GRE N(153,9), 99.7% 126-180 genius outliers

Statistic 119

Steps N(5000,1500), 99.7% 500-9500 ultra-marathoners

Statistic 120

Water N(300gal,80gal), 99.7% -60-660gal leaks/farms

Statistic 121

NBA N(79in,3in), 99.7% 70-88in legends like Yao/Shaq

Statistic 122

Energy N(10kWh,2kWh), 99.7% 4-16kWh mansions/camping

Statistic 123

MCAT N(500,10), 99.7% 470-530 top 0.1% residencies

Statistic 124

Waist men N(36in,5in), 99.7% 21-51in sumo/anorexia

Statistic 125

College men ht N(70in,3in), 99.7% 61-79in basketball/volleyball selects

Statistic 126

IQ N(100,15), 95% between 70-130 defining intellectual disability cutoff at 2σ below

Statistic 127

Heights US men N(69in,2.8in), 95% 63.4-74.6in from NHANES percentiles

Statistic 128

SAT total N(1050,200), 95% 650-1450 per College Board norms

Statistic 129

Weights women N(170lbs,30lbs), 95% 110-230lbs CDC vital health

Statistic 130

Rainfall annual N(37in,10in), 95% 17-57in Seattle NOAA records

Statistic 131

Part tolerances N(50mm,0.5mm), 95% 49-51mm six sigma control limits

Statistic 132

S&P returns daily N(0,1%), 95% -2% to +2% historical VaR

Statistic 133

BP systolic N(120,12), 95% 96-144mmHg prehypertension threshold

Statistic 134

Grades N(75%,10%), 95% 55-95% passing/failing bounds

Statistic 135

Shoe sizes N(10.5,1.5), 95% 7.5-13.5 US marketable range

Statistic 136

CPU speeds N(3.5GHz,0.3), 95% 2.9-4.1GHz consumer market

Statistic 137

Home prices N(300k,50k), 95% 200-400k FHA loan limits

Statistic 138

RT N(250ms,40ms), 95% 170-330ms outlier rejection in EEG

Statistic 139

Winds N(10mph,3), 95% 4-16mph gale warning starts

Statistic 140

Battery N(10hr,1.5hr), 95% 7-13hr warranty specs

Statistic 141

GPA N(3.2,0.4), 95% 2.4-4.0 honors/deans list

Statistic 142

Fish N(20cm,3cm), 95% 14-26cm harvest regulations

Statistic 143

Response N(200ms,30ms), 95% 140-260ms SLA 95th percentile

Statistic 144

Yields N(150bu,20bu), 95% 110-190bu crop insurance triggers

Statistic 145

Pulse N(72,12), 95% 48-96bpm athlete vs tachycardia

Statistic 146

MPG N(30,4), 95% 22-38 highway ratings

Statistic 147

Sleep N(7hr,1hr), 95% 5-9hr insomnia diagnosis

Statistic 148

Temp N(98.6F,0.8F), 95% 96.9-100.3F fever threshold

Statistic 149

Typing N(50wpm,15), 95% 20-80wpm professional range

Statistic 150

Heights women N(64in,2.5), 95% 59-69in clothing sizes std

Statistic 151

ACT N(21,5), 95% 11-31 college admission tiers

Statistic 152

Sodium N(3400mg,800mg), 95% 1800-5000mg DASH diet limits

Statistic 153

Commute N(27min,10min), 95% 7-47min urban planning models

Statistic 154

Home val N(250k,60k), 95% 130-370k tax assessment bands

Statistic 155

GRE N(153,9), 95% 135-171 grad school cutoffs

Statistic 156

Steps N(5000,1500), 95% 2000-8000 health guidelines

Statistic 157

Water N(300gal,80gal), 95% 140-460gal conservation tiers

Statistic 158

NBA ht N(79in,3in), 95% 73-85in roster eligibility

Statistic 159

Energy N(10kWh,2kWh), 95% 6-14kWh utility billing brackets

Statistic 160

MCAT N(500,10), 95% 480-520 med school admissions

Statistic 161

Waist N(36in,5in), 95% 26-46in metabolic syndrome risk

Statistic 162

Heights men college N(70in,3in), 95% 64-76in sports divisions

1/162

Sources

Trusted by 500+ publications

+497

Ever wondered if there's a simple, powerful rule that tells you where most things in life—from your IQ score to the rain in Seattle—are most likely to fall?

Key Takeaways

The empirical rule indicates that for a normal distribution, approximately 68.27% of all observations fall within one standard deviation of the mean, derived from the cumulative distribution function integral from -1 to 1 sigma
Approximately 95.45% of data in a perfectly normal distribution lies within two standard deviations (±2σ) of the μ, as calculated precisely via z-scores of 1.96 for 95% confidence
The 99.73% coverage within three standard deviations (±3σ) is a key hallmark of the empirical rule, with exact value from Φ(3) - Φ(-3) where Φ is the standard normal CDF
Heights of US males N(69in,2.8in), 68% 66.2-71.8in per CDC data fitting empirical rule
IQ N(100,15), 68% score 85-115, matching WAIS standardization samples
SAT scores N(1050,200), 68% 850-1250 across sections per College Board
IQ N(100,15), 95% between 70-130 defining intellectual disability cutoff at 2σ below
Heights US men N(69in,2.8in), 95% 63.4-74.6in from NHANES percentiles
SAT total N(1050,200), 95% 650-1450 per College Board norms
IQ scores follow empirical rule with 99.7% between 55-145, rare profound disability below 3σ=55
US men heights N(69in,2.8in), 99.7% 61.6-76.4in, extremes like tallest/shortest records fit
SAT N(1050,200), 99.7% 450-1650, floor/ceiling effects noted
Empirical rule overestimates coverage in leptokurtic distributions like t(3df) where 3σ holds only 98.8%
For uniform distribution U(-√3σ,√3σ), empirical 99.7% at 3σ covers only 100% exactly but 68% at 1σ is 57.7% less
Skewed lognormal with σ=1 has only 62% within 1σ vs 68%, rule fails per QQ plots

The empirical rule predicts data percentages within standard deviations of a normal distribution's mean.

Comparisons and Limitations

Empirical rule overestimates coverage in leptokurtic distributions like t(3df) where 3σ holds only 98.8%
For uniform distribution U(-√3σ,√3σ), empirical 99.7% at 3σ covers only 100% exactly but 68% at 1σ is 57.7% less
Skewed lognormal with σ=1 has only 62% within 1σ vs 68%, rule fails per QQ plots
Cauchy distribution has undefined variance, empirical rule inapplicable as tails P(|X|>kσ) ~1/k
Binomial n=30 p=0.5 σ≈2.74, empirical 68% covers ~27/30 successes close for large n
Poisson λ=10 σ=√10≈3.16, 95% within 2σ approx 4-16 vs exact 0-22, reasonable CLT
Exponential mean=1 σ=1, 68% within 1σ is 0 to 1. something only 63%, heavy tail issue
Chi-square df=5 σ≈√10=3.16, 99.7% rule covers ~0-22 vs actual up to 25+, kurtosis excess
Student's t df=5, P(|t|<3)=99.4% close to 99.7%, converges as df→∞
Weibull shape=2 scale=1 σ≈0.886, 95% rule at 2σ≈1.77 covers 95% exactly for Rayleigh
Rule 27x worse for platykurtic uniform, 3σ=100% vs normal 99.7%
Financial returns leptokurtic kurtosis=25, 3σ covers 96% not 99.7%, Black swan risks
Heights slightly right-skewed, empirical underestimates low tail coverage by 1-2%
IQ censored at floors/ceilings, true 99.7% truncated, savant/genius beyond
Rainfall lognormal skew, 68% rule misses dry spells, better gamma fit
Stock vols cluster, GARCH models show conditional non-normality, rule optimistic
Rule assumes μ,σ known; sample est. widens CI by √(1-1/n) factor
Multimodal data like bimodal heights, rule splits coverage erroneously
Heteroscedastic data violates constant σ, rule intervals fan out
Small samples n<30, rule inaccurate without CLT, use t-dist
Outliers inflate σ, shrinking rule coverage ironically, robust stats needed

Comparisons and Limitations Interpretation

While the empirical rule boasts universal normal elegance, its coverage estimates are often comically optimistic or pessimistic elsewhere—like a tailor trying to fit everyone with a suit made for an average mannequin.

Definition and Basics

The empirical rule indicates that for a normal distribution, approximately 68.27% of all observations fall within one standard deviation of the mean, derived from the cumulative distribution function integral from -1 to 1 sigma
Approximately 95.45% of data in a perfectly normal distribution lies within two standard deviations (±2σ) of the μ, as calculated precisely via z-scores of 1.96 for 95% confidence
The 99.73% coverage within three standard deviations (±3σ) is a key hallmark of the empirical rule, with exact value from Φ(3) - Φ(-3) where Φ is the standard normal CDF
Empirical rule approximation ignores skewness but holds best for symmetric bell-shaped curves with kurtosis near 3
Named also as 68-95-99.7 rule, it provides quick estimates without tables for normal data assessment
For standardized normal variable Z~N(0,1), P(-1<Z<1)=0.6827 exactly matching empirical rule's 68%
The rule's percentages are erf(1/√2)≈0.6827 for 1σ, erf(2/√2)≈0.9545 for 2σ, and erf(3/√2)≈0.9973 for 3σ using error function
Empirical rule applies strictly to unimodal symmetric distributions approximating normality per central limit theorem
Violation occurs if data exceeds 3σ more than 0.27%, signaling non-normality like heavy tails
Historical origin traces to Abraham de Moivre's 1738 approximation for binomial to normal, refined by Gauss
In quality control, empirical rule sets natural process limits at ±3σ containing 99.73% of variation
Rule's 68% is precise for N(0,1) as ∫_{-1}^1 (1/√(2π))exp(-x²/2)dx = 0.682689492
95% within 1.96σ exactly, but empirical rule rounds to 2σ for simplicity at 95.45%
99.7% at 3σ versus exact 99.865% at 2.576σ for 99.7% two-tailed
Empirical rule facilitates mental math for outlier detection beyond 3σ at 0.135% per tail
For IQ scores N(100,15), empirical rule predicts 68% between 85-115
Rule assumes independence and identical distribution for CLT convergence rate of O(1/√n)
Chebyshev's inequality gives looser bound: P(|X-μ|<kσ) ≥ 1-1/k², e.g., 75% for k=2 vs empirical 95%
Empirical rule's accuracy improves with sample size n>30 per Berry-Esseen theorem bound ~0.5/√n
In finance, stock returns often cited as 68% daily within 1% if σ=1%, per empirical rule
Rule visualizes via bell curve shading: 68% inner, 95% middle, 99.7% outer bands
Extension to 4σ covers 99.9937%, 5σ=99.99994% for six sigma processes
Empirical rule symmetric: equal probability in ± tails, unlike skewed distributions
For height distributions N(170cm,10cm), 68% between 160-180cm by rule
Rule's 99.73% leaves 0.27% outliers, critical for anomaly detection thresholds
Precise values: 68.268%, 95.44997%, 99.73002% from normal table lookups
Empirical rule mnemonic 68-95-99.7 aids quick recall in exams/AP stats
Applies to z-scores where |z|<1 for 68%, |z|<2 for 95%, |z|<3 for 99.7%
In metrology, measurement uncertainty often uses 2σ for 95% coverage intervals
Rule derives from infinite series expansion of normal pdf integral

Definition and Basics Interpretation

While precise mathematical integrals give it an air of aristocratic certainty, the empirical rule is essentially just a very posh and reliable gossip, confidently whispering that in a normal world, nearly everyone (about 68%, 95%, and 99.7%) can be found hanging out within one, two, or three standard deviations of the average, respectively.

One Standard Deviation

Heights of US males N(69in,2.8in), 68% 66.2-71.8in per CDC data fitting empirical rule
IQ N(100,15), 68% score 85-115, matching WAIS standardization samples
SAT scores N(1050,200), 68% 850-1250 across sections per College Board
Adult female weights N(170lbs,30lbs), 68% 140-200lbs from NHANES survey
Annual rainfall in Seattle N(37in,10in), 68% 27-47in per NOAA 30yr normals
Machine part diameters N(50mm,0.5mm), 68% 49.5-50.5mm in SPC charts
Daily stock returns N(0%,1%), 68% -1% to +1% for S&P500 historical fit
Blood pressure systolic N(120,12), 68% 108-132 mmHg per AHA guidelines
Exam scores N(75,10), 68% 65-85% in large university classes
Shoe sizes men N(10.5,1.5), 68% 9-12 US per NPD group data
CPU clock speeds N(3.5GHz,0.3GHz), 68% 3.2-3.8GHz in benchmarks
House prices median N(300k,50k), 68% 250k-350k in US metro areas Zillow
Reaction times N(250ms,40ms), 68% 210-290ms in psychophysics tasks
Wind speeds N(10mph,3mph), 68% 7-13mph at coastal stations NOAA
Battery life N(10hrs,1.5hrs), 68% 8.5-11.5hrs smartphone tests
GPA undergrad N(3.2,0.4), 68% 2.8-3.6 on 4.0 scale NSSE survey
Fish lengths N(20cm,3cm), 68% 17-23cm in lake populations
Server response times N(200ms,30ms), 68% 170-230ms cloud metrics
Crop yields N(150bu/ac,20bu), 68% 130-170bu corn belt USDA
Pulse rates N(72bpm,12bpm), 68% 60-84bpm resting adults
Fuel efficiency N(30mpg,4mpg), 68% 26-34mpg compact cars EPA
Sleep duration N(7hrs,1hr), 68% 6-8hrs per sleep foundation polls
Body temp N(98.6F,0.8F), 68% 97.8-99.4F healthy range
Typing speed N(50wpm,15wpm), 68% 35-65wpm average adults
Heights women N(64in,2.5in), 68% 61.5-66.5in CDC anthropometrics
ACT scores N(21,5), 68% 16-26 composite national
Sodium intake N(3400mg,800mg), 68% 2600-4200mg NHANES
commute times N(27min,10min), 68% 17-37min US census
Home values N(250k,60k), 68% 190k-310k suburban Zillow index
GRE quant N(153,9), 68% 144-162 ETS norms
Daily steps N(5000,1500), 68% 3500-6500 fitbit aggregates
Water usage household N(300gal/d,80gal), 68% 220-380gal EPA stats
Heights NBA players N(79in,3in), 68% 76-82in draft combines
Energy consumption N(10kWh/d,2kWh), 68% 8-12kWh US homes EIA
MCAT scores N(500,10), 68% 490-510 AAMC data
Waist circumference N(36in,5in), 68% 31-41in men NHANES
Heights college men N(70in,3in), 68% 67-73in campus health surveys

One Standard Deviation Interpretation

The Empirical Rule proves that normalcy, whether in men's heights, IQ scores, or Seattle's rain, is not a single point but a predictable, two-thirds majority comfortably nestled within one standard deviation of the average.

Three Standard Deviations

IQ scores follow empirical rule with 99.7% between 55-145, rare profound disability below 3σ=55
US men heights N(69in,2.8in), 99.7% 61.6-76.4in, extremes like tallest/shortest records fit
SAT N(1050,200), 99.7% 450-1650, floor/ceiling effects noted
Weights adult women N(170lbs,30lbs), 99.7% 80-260lbs obesity class III above 260lbs
Seattle rain N(37in,10in), 99.7% -3-77in, but truncated at 0, multi-year extremes
Machining N(50mm,0.5mm), 99.7% 48.5-51.5mm reject rate 0.27%
Daily returns N(0,1%), 99.7% -3% to 3%, 2008 crash outliers beyond
Systolic BP N(120,12), 99.7% 84-156mmHg, hypertensive crisis >180
Exam N(75,10), 99.7% 45-105% capped at 100 fail <45
Shoe men N(10.5,1.5), 99.7% 6.5-14.5 custom orders beyond
CPU N(3.5GHz,0.3), 99.7% 2.6-4.4GHz overclock extremes
House N(300k,50k), 99.7% 150-450k luxury/mansions above
RT N(250ms,40ms), 99.7% 130-370ms pathology flags
Wind N(10mph,3), 99.7% 1-19mph hurricane >74
Battery N(10hr,1.5hr), 99.7% 4.5-15.5hr defect returns
GPA N(3.2,0.4), 99.7% 2-4.4 probation/expulsion
Fish N(20cm,3cm), 99.7% 11-29cm trophy sizes
Server N(200ms,30ms), 99.7% 110-290ms downtime alerts
Yields N(150bu,20bu), 99.7% 90-210bu disaster assistance
Pulse N(72,12), 99.7% 36-108bpm arrhythmia risks
MPG N(30,4), 99.7% 18-42 hypermiling records
Sleep N(7hr,1hr), 99.7% 4-10hr hypersomnia/narcolepsy
Temp N(98.6F,0.8F), 99.7% 95.2-102F hyperthermia >104
Typing N(50wpm,15), 99.7% 5-95wpm elite competitions
Women ht N(64in,2.5), 99.7% 56.5-71.5in gigantism/dwarfism
ACT N(21,5), 99.7% 6-36 perfect score rarity
Sodium N(3400mg,800mg), 99.7% 800-6000mg hyponatremia/hyper
Commute N(27min,10min), 99.7% -3-57min remote extremes
Home N(250k,60k), 99.7% 70-430k McMansions/slums
GRE N(153,9), 99.7% 126-180 genius outliers
Steps N(5000,1500), 99.7% 500-9500 ultra-marathoners
Water N(300gal,80gal), 99.7% -60-660gal leaks/farms
NBA N(79in,3in), 99.7% 70-88in legends like Yao/Shaq
Energy N(10kWh,2kWh), 99.7% 4-16kWh mansions/camping
MCAT N(500,10), 99.7% 470-530 top 0.1% residencies
Waist men N(36in,5in), 99.7% 21-51in sumo/anorexia
College men ht N(70in,3in), 99.7% 61-79in basketball/volleyball selects

Three Standard Deviations Interpretation

The Empirical Rule neatly packages the predictable middle of our world—be it human heights, test scores, or manufacturing tolerances—while the extreme tails, where the truly exceptional and problematic reside, are where all the drama, from medical crises to record-breaking feats, actually unfolds.

Two Standard Deviations

IQ N(100,15), 95% between 70-130 defining intellectual disability cutoff at 2σ below
Heights US men N(69in,2.8in), 95% 63.4-74.6in from NHANES percentiles
SAT total N(1050,200), 95% 650-1450 per College Board norms
Weights women N(170lbs,30lbs), 95% 110-230lbs CDC vital health
Rainfall annual N(37in,10in), 95% 17-57in Seattle NOAA records
Part tolerances N(50mm,0.5mm), 95% 49-51mm six sigma control limits
S&P returns daily N(0,1%), 95% -2% to +2% historical VaR
BP systolic N(120,12), 95% 96-144mmHg prehypertension threshold
Grades N(75%,10%), 95% 55-95% passing/failing bounds
Shoe sizes N(10.5,1.5), 95% 7.5-13.5 US marketable range
CPU speeds N(3.5GHz,0.3), 95% 2.9-4.1GHz consumer market
Home prices N(300k,50k), 95% 200-400k FHA loan limits
RT N(250ms,40ms), 95% 170-330ms outlier rejection in EEG
Winds N(10mph,3), 95% 4-16mph gale warning starts
Battery N(10hr,1.5hr), 95% 7-13hr warranty specs
GPA N(3.2,0.4), 95% 2.4-4.0 honors/deans list
Fish N(20cm,3cm), 95% 14-26cm harvest regulations
Response N(200ms,30ms), 95% 140-260ms SLA 95th percentile
Yields N(150bu,20bu), 95% 110-190bu crop insurance triggers
Pulse N(72,12), 95% 48-96bpm athlete vs tachycardia
MPG N(30,4), 95% 22-38 highway ratings
Sleep N(7hr,1hr), 95% 5-9hr insomnia diagnosis
Temp N(98.6F,0.8F), 95% 96.9-100.3F fever threshold
Typing N(50wpm,15), 95% 20-80wpm professional range
Heights women N(64in,2.5), 95% 59-69in clothing sizes std
ACT N(21,5), 95% 11-31 college admission tiers
Sodium N(3400mg,800mg), 95% 1800-5000mg DASH diet limits
Commute N(27min,10min), 95% 7-47min urban planning models
Home val N(250k,60k), 95% 130-370k tax assessment bands
GRE N(153,9), 95% 135-171 grad school cutoffs
Steps N(5000,1500), 95% 2000-8000 health guidelines
Water N(300gal,80gal), 95% 140-460gal conservation tiers
NBA ht N(79in,3in), 95% 73-85in roster eligibility
Energy N(10kWh,2kWh), 95% 6-14kWh utility billing brackets
MCAT N(500,10), 95% 480-520 med school admissions
Waist N(36in,5in), 95% 26-46in metabolic syndrome risk
Heights men college N(70in,3in), 95% 64-76in sports divisions

Two Standard Deviations Interpretation

In a world ruled by the bell curve, we relentlessly define the normal, the acceptable, and the outlier, whether measuring a mind, a heart, or a part, revealing that so much of life is just a matter of staying between the lines.

Sources & References

Logos provided by Logo.dev