BellCurve

Normal Distribution Calculator

Φ(z) — the classic bell-curve probability.

z-score

Φ(z) — probability below: 0.841
1 − Φ(z) — probability above: 0.159

Shaded region = area to the left of z (standard normal CDF)

What you entered

z-score: 1

1
Standard normal CDF at z (area to the left)
Φ(1)= 0.8413
2
Area to the right of z
1 − 0.8413= 0.1587

Result

Probability below z: 0.841

For z = 1, 84.13% of a standard normal distribution falls below this point, and 15.87% falls above.

Standard normal probabilities by z-score

For a standard normal distribution, the area to the left of z is the probability of a value falling below it. The two columns always sum to 1.

Z-score	Probability below	Probability above	Percentile
-3.00	0.0013	0.9987	0.13
-2.58	0.0049	0.9951	0.49
-1.96	0.0250	0.9750	2.50
-1.00	0.1587	0.8413	15.87
0.00	0.5000	0.5000	50.00
1.00	0.8413	0.1587	84.13
1.96	0.9750	0.0250	97.50
2.58	0.9951	0.0049	99.51
3.00	0.9987	0.0013	99.87

The 1.96 and 2.58 rows are the ones worth memorising: they cut off 2.5% and 0.5% in each tail, which is exactly why they are the critical values for 95% and 99% confidence intervals. The table is perfectly symmetric about zero - the probability below -1 equals the probability above +1 - which is why published z-tables often list only positive values. The 68-95-99.7 rule falls straight out of it: the area between -1 and +1 is 0.8413 minus 0.1587, or 68.3%.

The area under the bell curve

Φ(z) gives the probability that a randomly drawn value from a normal distribution falls below a given z-score — visually, it's the shaded area to the left of that point on the bell curve.

The famous 68-95-99.7 rule

About 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 — a quick mental shortcut this calculator makes precise for any z-score.

Frequently asked questions

Adult male height averages 5'9" with SD of 3 inches. What percentage of men are over 6'3"?

z = (75 − 69) / 3 = 2.0. From the bell curve, P(z > 2) = 2.3%. Only about 1 in 43 men is taller than 6'3". This is why the normal distribution is powerful — one formula answers any 'what percentage above/below X?' question.

How is the normal distribution different from the z-score calculator?

The z-score calculator converts a raw value into standard deviations from the mean. This calculator takes that z-score and tells you the probability (area under the curve). Use them together: z-score first to standardize, then this calculator to find the percentile.

What is the 68-95-99.7 rule?

In any normal distribution: 68% of data falls within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD. Quick mental math: if the mean is 100 and SD is 15, then 95% of values are between 70 and 130.

Is my data normally distributed?

Check: is it roughly bell-shaped and symmetric? Are mean ≈ median ≈ mode? Do about 68% of values fall within 1 SD of the mean? If the data is skewed (like income or reaction times), the normal distribution doesn't fit well — consider a log-normal or use non-parametric methods.

What is the standard normal distribution?

A normal distribution with mean = 0 and SD = 1. Any normal distribution can be converted to it using z-scores: z = (x − μ) / σ. This standardization is why one table or calculator handles all normal distributions, regardless of their original mean and SD.