Normal distribution calculator
Everything a normal curve can tell you about a value, from the mean and standard deviation.
Probability between the two values
68.268 95%
- Probability below the lower value
- 15.865 53 %
- Probability above the upper value
- 15.865 53 %
- Z-score of the lower value
- -1.000 00
- Z-score of the upper value
- 1.000 00
- Probability density at the lower value
- 0.241 970 72
Show the working
- Mean = 0
- Standard deviation = 1
- Lower value = -1
- Upper value = 1
- Z-score of the lower value = (-1 - 0) / 1 = -1
- Z-score of the upper value = (1 - 0) / 1 = 1
- a_abs = abs(-1) = 1
- a_t = 1 / (1 + 0.2316419 * 1) = 0.811 924
- a_pdf = 0.3989422804014327 * exp(-1^2 / 2) = 0.241 971
- a_tail = 0.241 971 * (0.319381530*0.811 924 - 0.356563782*0.811 924^2 + 1.781477937*0.811 924^3 - 1.821255978*0.811 924^4 + 1.330274429*0.811 924^5) = 0.158 655
- a_cdf = (-1) >= 0 ? 1 - 0.158 655 : 0.158 655 = 0.158 655
- b_abs = abs(1) = 1
- b_t = 1 / (1 + 0.2316419 * 1) = 0.811 924
- b_pdf = 0.3989422804014327 * exp(-1^2 / 2) = 0.241 971
- b_tail = 0.241 971 * (0.319381530*0.811 924 - 0.356563782*0.811 924^2 + 1.781477937*0.811 924^3 - 1.821255978*0.811 924^4 + 1.330274429*0.811 924^5) = 0.158 655
- b_cdf = (1) >= 0 ? 1 - 0.158 655 : 0.158 655 = 0.841 345
- Probability below the lower value = 0.158 655 * 100 = 15.8655
- Probability above the upper value = (1 - 0.841 345) * 100 = 15.8655
- Probability between the two values = (0.841 345 - 0.158 655) * 100 = 68.2689
- Probability density at the lower value = 1/(1*sqrt(2*pi)) * exp(--1^2/2) = 0.241 971
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- z = (x - mean) / sd
- P(a < X < b) = Phi(z_b) - Phi(z_a)
- density = exp(-z^2/2) / (sd sqrt(2 pi))
Plus or minus one standard deviation captures 68.27% of a normal population, two captures 95.45%, three captures 99.73%.