Using Chebyshev's Theorem
Chebyshev's theorem (also called Chebyshev's inequality) gives a guaranteed lower bound on how much of a data set clusters near its mean, without assuming anything about the shape of the distribution. For any data set — skewed, bimodal, spiky, or bell-shaped — at least 1 − 1/k² of the observations lie within k standard deviations of the mean, for any k > 1. Enter k above to get that minimum fraction; add a mean and standard deviation to see the actual interval it covers.
The formula
If a data set has mean μ and standard deviation σ, then for any k > 1:
Proportion within [μ − kσ, μ + kσ] ≥ 1 − 1/k²
Equivalently, at most 1/k² of the values fall more than k standard deviations from the mean. The bound is a floor, not an exact percentage: a specific data set usually has more than 1 − 1/k² of its values inside the interval, but never fewer.
Why it is useful
The familiar empirical rule (68% within 1σ, 95% within 2σ, 99.7% within 3σ) only applies to roughly normal distributions. When you cannot assume normality — income data, insurance claims, reliability lifetimes, exam scores with a heavy tail — Chebyshev still holds. It lets you make a rigorous "at least this much" statement about outliers using nothing but the mean and standard deviation, which is why it appears in the proofs behind the weak law of large numbers.
Common reference values
- k = 1.5: at least 1 − 1/2.25 = 55.56% of values within 1.5 SDs.
- k = 2: at least 1 − 1/4 = 75% within 2 SDs (at most 25% outside).
- k = 2.5: at least 1 − 1/6.25 = 84% within 2.5 SDs.
- k = 3: at least 1 − 1/9 = 88.89% within 3 SDs (at most 11.11% outside).
- k = 4: at least 1 − 1/16 = 93.75% within 4 SDs.
Frequently Asked Questions
What does Chebyshev's Theorem say?
Why must k be greater than 1?
How is this different from the empirical (68-95-99.7) rule?
Can the actual proportion be higher than the bound?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |