Area Under Normal Curve Z Score Calculator

This calculator finds the area (probability) under the standard normal curve to the left of a given z-score, and also gives the area to the right and the total area between two z-scores. You enter the z-score (or two z-scores for a range), and the calculator returns the corresponding probabilities using the standard normal distribution.

Formula

The area under the standard normal curve to the left of a z-score is given by the cumulative distribution function (CDF) of the standard normal distribution:

Φ(z) = 0.5 × [1 + erf(z / √2)]

where:

• Φ(z) = area (probability) to the left of z (unitless, between 0 and 1)
• z = z-score (number of standard deviations from the mean, unitless)
• erf = the error function, a standard mathematical function

The area to the right is simply 1 − Φ(z). For a range between two z-scores, the area is |Φ(z1) − Φ(z2)|.

Worked Example

Using the default values: z = 1.96 and z2 = −1.96.

1. Compute Φ(1.96): 1.96 / √2 ≈ 1.3859. erf(1.3859) ≈ 0.9500. Then Φ(1.96) = 0.5 × (1 + 0.9500) = 0.9750.
2. Compute Φ(−1.96): −1.96 / √2 ≈ −1.3859. erf(−1.3859) ≈ −0.9500. Then Φ(−1.96) = 0.5 × (1 − 0.9500) = 0.0250.
3. Area to the left of 1.96 = 0.9750 (97.50%).
4. Area to the right of 1.96 = 1 − 0.9750 = 0.0250 (2.50%).
5. Area between −1.96 and 1.96 = |0.9750 − 0.0250| = 0.9500 (95.00%).

These match the well-known 95% confidence interval bounds (±1.96).

What Is a Z-Score?

A z-score tells you how many standard deviations a data point is from the mean of a normal distribution. If a score is 2.5, it means the value is 2.5 standard deviations above the mean. Negative z-scores are below the mean. The standard normal distribution has a mean of 0 and a standard deviation of 1, so every z-score maps directly to a point on that curve.

The total area under the entire curve is 1 (100%). The area under the curve between any two z-scores represents the proportion of observations that fall in that range. For example, about 68% of values lie between z = −1 and z = 1, 95% between −1.96 and 1.96, and 99.7% between −3 and 3 (the empirical rule).

How to Use the Calculator

• Enter a z-score in the first field. The calculator immediately shows the area to the left (the cumulative probability) and the area to the right.
• Optionally enter a second z-score to get the area between the two values. If you leave the second field blank or equal to the first, the “between” output will be 0.
• Read the outputs as decimals or percentages: 0.9750 means 97.5% of the distribution is to the left of that z-score.
• For one-tailed tests, use the area to the right or left directly. For two-tailed tests, double the smaller tail area.

Common Mistakes and Misconceptions

• Confusing left and right: The “area to the left” is the cumulative probability from negative infinity up to your z-score. Many people mistakenly read it as the area between the mean and z. To get the area between 0 and z, subtract 0.5 from the left area.
• Using the wrong tail: For a z-score of 1.96, the area to the right is 0.025, not 0.975. Check whether your test is one-tailed or two-tailed.
• Rounding too early: Rounding z to 2 decimal places before looking up the area can shift the result by a few hundredths. Keep full precision until the final step.
• Assuming all data is normal: The z-score table only applies to normally distributed data. For skewed distributions, the area under the curve does not correspond to probabilities in the same way.

Why Use the Standard Normal Curve?

Many real-world variables—heights, test scores, measurement errors, and biological traits—are approximately normally distributed. Converting raw data to z-scores allows you to compare values from different distributions and to compute probabilities without integrating the normal density function by hand. The standard normal table (or calculator) is a shortcut for the integral of the bell curve, which has no simple closed-form antiderivative.

In statistics, z-scores are used in hypothesis testing (z-tests), confidence intervals, and quality control. For example, if a process produces parts with a mean length of 10 cm and a standard deviation of 0.2 cm, a part measuring 10.5 cm has a z-score of (10.5 − 10) / 0.2 = 2.5. The area to the right of 2.5 is about 0.0062, meaning only 0.62% of parts are expected to be longer than 10.5 cm.

Limitations and Alternatives

The standard normal curve applies only to data that is truly normally distributed. For small samples (n < 30) from a normal population, the t-distribution is more appropriate when the population standard deviation is unknown. For non-normal data, use nonparametric methods or transformations. Also, the z-score table gives areas for the standard normal; if your data has a different mean and standard deviation, you must standardize first (z = (x − μ) / σ).

Alternatives to this calculator include printed z-tables, statistical software (like R or Python), and spreadsheet functions such as NORM.S.DIST in Excel. Those tools give the same results, but this calculator is quick and requires no installation.

Frequently Asked Questions

What does the area under the normal curve represent?

It represents the probability that a randomly selected value from the distribution falls within that range of z-scores. For example, the area between z = −1 and z = 1 is about 0.68, meaning 68% of values lie within one standard deviation of the mean.

How do I find the area to the right of a z-score?

Subtract the area to the left from 1. For z = 2.0, the left area is about 0.9772, so the right area is 1 − 0.9772 = 0.0228.

Can I use this for a two-tailed test?

Yes. For a two-tailed test, you typically want the total area in both tails. That is twice the area in one tail. For z = 1.96, the right tail is 0.025, so both tails together are 0.05.

What if my z-score is negative?

Negative z-scores work the same way. The area to the left of z = −1.96 is 0.025, and the area to the right is 0.975. The curve is symmetric, so the area to the left of −z equals the area to the right of +z.

Why does the calculator show 0.9750 for z = 1.96?

Because 97.5% of the distribution lies to the left of 1.96. That leaves 2.5% in the right tail, which is why ±1.96 corresponds to a 95% confidence interval (the middle 95% of the distribution).

Is the area the same as the p-value?

In hypothesis testing, the p-value is the probability of observing a result as extreme as, or more extreme than, the one you got, assuming the null hypothesis is true. For a one-tailed test, the p-value equals the area in the tail beyond your z-score. For a two-tailed test, it is twice that area.

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in z | Z-score | - | 1.96 | -10
in z2 | Second Z-score (optional) | - | -1.96 | -10
out area_left | Area to the left | - | 0.5 * (1 + erf(z / sqrt(2))) | 4
out area_right | Area to the right | - | 1 - area_left | 4
out area_between | Area between z and z2 | - | abs(area_left - (0.5 * (1 + erf(z2 / sqrt(2))))) | 4