Question 1

How do you calculate a probability in the normal distribution?

Accepted Answer

Standardize the value with z = (x − μ) / σ and read the cumulative area Φ(z). For P(X < x) the answer is Φ(z); for P(X > x) it is 1 − Φ(z); for P(a < X < b) it is Φ(zb) − Φ(za). With μ = 100, σ = 15 and x = 115, z = 1 and P(X < 115) = 0.8413. In the calculator, pick the type, type the values and read the result with its step-by-step work.

Question 2

What is a z-score and how do I interpret it?

Accepted Answer

It is the distance from a value to the mean, measured in standard deviations. z = 0 sits at the mean, z = 1 is one deviation above and z = −1.5 is one and a half below. Values with |z| above 2 fall outside the central 95.45% band and are usually treated as unusual; above 3 as rare (0.27% of cases in a normal distribution). It lets you compare values from different scales.

Question 3

How do I find the x value that matches a percentile?

Accepted Answer

Use the Inverse tab: enter the probability (say 90%) and the tail type. The calculator finds z = Φ⁻¹(p) and converts with x = μ + z · σ. For the 90th percentile of an IQ with mean 100 and deviation 15, z = 1.2816 and x = 119.2. For a central 95% band the result is two values, μ − 1.96σ and μ + 1.96σ.

Question 4

What is the difference between a one-tailed and a two-tailed p-value?

Accepted Answer

The one-tailed value looks at a single tail (the statistic is larger, or smaller, than expected); the two-tailed value adds both tails and is twice the nearer tail. With z = 1.96 the right tail is 0.0250 and the two-tailed p is 0.0500. Choose the type before seeing the data: switching to one-tailed afterwards just to cross α = 0.05 distorts the test.

Question 5

What does the 68-95-99.7 empirical rule say?

Accepted Answer

In a normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. The exact figures are 68.27%, 95.45% and 99.73%. To capture exactly 95%, the band is 1.96 standard deviations, not 2. The tool's table shows the ranges already computed for your own mean and standard deviation.

Question 6

How do I read the z table?

Accepted Answer

The row gives z to the first decimal and the column gives the second: for z = 1.96, find row 1.9 and column 0.06. In the cumulative table the entry is the area to the left, Φ(1.96) = 0.9750. Other tables give the area from the mean to z (0.4750) or the right tail (0.0250). That is why you must know which type you have; all four variations are available here.

Question 7

When should I use t instead of z for the p-value?

Accepted Answer

Use z when the population standard deviation is known or the sample is very large, and t when the standard deviation is estimated from the sample itself. The t has heavier tails, so the same value gives a larger p: 2.0 yields a two-tailed p of 0.0455 with z, 0.0546 with t on 30 df and 0.0734 with 10 df. With many degrees of freedom, t approaches the normal.

Question 8

Why is P(X = x) zero?

Accepted Answer

Because the variable is continuous: probability lives in intervals, as the area under the curve between two points, and a single point has zero area. That is why P(X < 115) and P(X ≤ 115) are equal. For counted variables, such as a number of successes, use a discrete distribution like the binomial, or apply a continuity correction of 0.5.

Question 9

How accurate are the calculations?

Accepted Answer

Φ and the p-values have a relative error of 1e-12 or less, even in the tails, and quantiles are in the same range. We checked 1.959964 for 97.5%, 2.575829 for 99.5% and the NIST table t critical values of 12.706 (1 df), 2.228 (10 df) and 2.042 (30 df). Very small probabilities are shown in scientific notation.

Question 10

Can I use it for data that is not normal?

Accepted Answer

Only with care. Normal percentiles and probabilities hold if the variable is roughly normal. Skewed data (income, waiting times) call for another distribution. The average of many observations, on the other hand, tends to normal by the central limit theorem, which supports z and t in tests and confidence intervals for means.

Band	Range	Rule	Exact
μ ± 1σ	85 – 115	68%	68.2689%
μ ± 2σ	70 – 130	95%	95.45%
μ ± 3σ	55 – 145	99.7%	99.73%

Band	Range	Rule	Exact
μ ± 1σ	85 – 115	68%	68.2689%
μ ± 2σ	70 – 130	95%	95.45%
μ ± 3σ	55 – 145	99.7%	99.73%

Normal Distribution, Z-Score and P-Value Calculator