Area Under Z Score Calculator: Find Probability

This calculator finds the area (probability) under the standard normal distribution curve to the left of a given z-score. You enter a z-score, and the calculator returns the cumulative probability, which tells you the percentage of data below that point. It also gives the area to the right and the area between two z-scores for quick comparisons.

Formula

The standard normal distribution has a mean of 0 and a standard deviation of 1. The cumulative probability Φ(z) is the area under the curve to the left of z. It is calculated using the error function (erf):

Φ(z) = 0.5 × (1 + erf(z / √2))

• z – the z-score (standardized value), unitless
• erf – the Gaussian error function, a mathematical function that gives the probability integral
• Φ(z) – the cumulative probability, a value between 0 and 1

The area to the right is simply 1 − Φ(z). The area between two z-scores is Φ(z₂) − Φ(z�) when z₂ > z�.

Worked Example

Using the default values from the calculator above: z = 1.5 and zâ‚‚ = 2.0.

1. Compute Φ(1.5): 0.5 × (1 + erf(1.5 / √2)) = 0.5 × (1 + erf(1.06066))
2. erf(1.06066) ≈ 0.866386
3. Φ(1.5) = 0.5 × (1 + 0.866386) = 0.933193
4. So the area to the left of z = 1.5 is 0.9332 (rounded to 4 decimals).
5. Area to the right: 1 − 0.933193 = 0.066807, rounded to 0.0668.
6. Compute Φ(2.0): 0.5 × (1 + erf(2.0 / √2)) = 0.5 × (1 + 0.995322) = 0.977250.
7. Area between 1.5 and 2.0: 0.977250 − 0.933193 = 0.044057, rounded to 0.0441.

The calculator shows exactly these values: 0.9332 (left), 0.0668 (right), and 0.0441 (between).

What Is a Z-Score and Why Use It?

A z-score measures how many standard deviations a data point is from the mean of a distribution. It is calculated as (x − μ) / σ, where x is the raw value, μ is the mean, and σ is the standard deviation. Converting raw data to z-scores lets you compare values from different normal distributions and find probabilities using the standard normal table.

For example, if test scores have a mean of 70 and a standard deviation of 10, a score of 85 has a z-score of (85 − 70) / 10 = 1.5. The area to the left (0.9332) means about 93.3% of students scored below 85.

How to Read the Result

The primary output is the area to the left, which is a probability between 0 and 1. Multiply by 100 to get a percentage. A z-score of 0 gives 0.5000 (50% below, 50% above). A negative z-score gives an area less than 0.5, and a positive z-score gives an area greater than 0.5.

The area to the right is the complement. It is useful for one-tailed tests in statistics. The area between two z-scores is the probability that a value falls in that range. For instance, the area between z = −1 and z = 1 is about 0.6827, which matches the empirical rule that 68% of data falls within one standard deviation of the mean.

Common Mistakes and Misconceptions

• Using the wrong tail – Many people look up the wrong side of the table. Always decide whether you need “less thanâ€� (left tail) or “greater thanâ€� (right tail) before calculating.
• Confusing z-score with raw score – The calculator expects a standardized z-score, not a raw measurement. Convert raw data to z-scores first.
• Rounding too early – If you round the z-score to 1.5 instead of 1.53, the probability can change noticeably. Keep at least two decimals in the z-score.
• Assuming normality – The z-table only works for normally distributed data. If your data is skewed, the probabilities are wrong.
• Forgetting the continuity correction – When using a normal approximation for discrete data (like counts), add or subtract 0.5 to the value before converting to a z-score.

When to Use This Calculator

Use this calculator when you need a cumulative probability from a standard normal distribution. Typical situations include:

• Finding percentiles for normally distributed data (e.g., “what percentage scored below 120?â€�)
• Performing hypothesis tests where you need the p-value from a z-statistic
• Calculating confidence intervals for large samples (z-based intervals)
• Checking if a data point is unusual (typically |z| > 2 is considered unusual)

For non-normal data, consider using the empirical rule only as a rough guide, or use a non-parametric method. The calculator assumes a perfect normal distribution; real-world data is often approximately normal, so results are approximations.

Limitations and Alternatives

The standard normal table only covers z-scores from about −3.49 to 3.49, but this calculator handles any value. However, for extreme z-scores (|z| > 5), the area to the left is essentially 1 (or 0) to many decimal places, so the result may be limited by floating-point precision.

If you have a t-statistic instead of a z-statistic, use a t-distribution calculator. If you have raw data, you can compute the z-score first with a separate calculator, then use this one. For two-tailed tests, double the area to the right (or left) to get the p-value.

Frequently Asked Questions

What does the area under the z curve mean?

The area under the standard normal curve between two points equals the probability that a randomly selected value from a standard normal distribution falls between those points. The total area under the entire curve is 1, or 100%.

How do I find the z-score from a percentage?

You need the inverse normal function. If you have a cumulative probability (e.g., 0.95), find the z-score that gives that area to the left. This calculator does not do the inverse, but many statistical tables and software provide it. For 0.95, the z-score is approximately 1.645.

What is the difference between a z-score and a t-score?

A z-score uses the population standard deviation and is used when the sample size is large (n > 30) or the population standard deviation is known. A t-score uses the sample standard deviation and is used for smaller samples. The t-distribution is wider and flatter, but approaches the normal distribution as sample size increases.

Can I use this for non-normal data?

No, the z-table is specifically for the standard normal distribution. For non-normal data, the probabilities will be inaccurate. If your data is skewed, consider transforming it or using a different distribution (like the log-normal or exponential).

Why is the area to the left of z = 0 exactly 0.5?

Because the standard normal distribution is perfectly symmetric around zero. The mean, median, and mode are all zero, so exactly half of the area lies to the left and half to the right.

How do I calculate the area between two z-scores?

Subtract the area to the left of the smaller z-score from the area to the left of the larger z-score. For example, the area between z = −1 and z = 1 is Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6826.

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in z | Z-score | - | 1.5 | 0
in z2 | Second Z-score (for between) | - | 2.0 | 0
out area_left | Area to the left | - | 0.5 * (1 + erf(z / sqrt(2))) | 4
out area_right | Area to the right | - | 1 - (0.5 * (1 + erf(z / sqrt(2)))) | 4
out area_between | Area between z and z2 | - | abs((0.5 * (1 + erf(z2 / sqrt(2)))) - (0.5 * (1 + erf(z / sqrt(2))))) | 4