Area to Z Score Calculator – Find Z-Score Online

Use this calculator when you know an area or probability and need to find the corresponding z-score.

Depending on the problem, you can calculate a z-score from:

• Left-tail area
• Right-tail area
• Area between the mean and z
• Central area
• Two-tailed area

The calculator uses the standard normal distribution, where the mean is 0 and the standard deviation is 1.

How to Convert Area to a Z Score

The basic relationship between area and z-score comes from the cumulative distribution function of the standard normal distribution.

The cumulative probability to the left of z is written as:

\(\Phi(z)=P(Z\leq z)\)

If you know the cumulative area and want to find z, you use the inverse standard normal function:

\(z=\Phi^{-1}(p)\)

where:

• \(z\) = z-score
• \(p\) = cumulative area to the left
• \(\Phi\) = standard normal cumulative distribution function
• \(\Phi^{-1}\) = inverse standard normal function

The main challenge is therefore not the inverse calculation itself. The challenge is determining which probability should be entered as the cumulative left-tail probability.

Why the Location of the Area Matters

Consider an area of 0.95.

If the problem says:

Find z such that 95% of the distribution is below z.

Then:

\(P(Z\leq z)=0.95\)

and:

\(z\approx1.645\)

But if the problem says:

Find the two z-scores that contain the middle 95% of the distribution.

Then 5% is outside the central region, with 2.5% in each tail:

\(P(Z\leq-z)=0.025\)

and:

\(P(Z\leq z)=0.975\)

giving:

\(z\approx\pm1.96\)

Therefore, 0.95 alone is not enough information. You must know what the area represents.

Types of Area-to-Z Problems

Left-Tail Area to Z Score

A left-tail problem gives the area below the desired z-score.

For example:

Find z when the area to the left is 0.90.

The calculation is:

\(P(Z\leq z)=0.90\)

Therefore:

\(z=\Phi^{-1}(0.90)\)

which gives approximately:

\(z=1.2816\)

So the 90th percentile corresponds to a z-score of approximately 1.28.

Right-Tail Area to Z Score

A right-tail problem gives the probability above the desired z-score.

Suppose the right-tail area is 0.05.

The corresponding left-tail area is:

\(1-0.05=0.95\)

Therefore:

\(z=\Phi^{-1}(0.95)\)

giving:

\(z\approx1.645\)

So a right-tail probability of 5% corresponds to approximately z = 1.645.

Area Between the Mean and a Positive Z Score

Some z-tables report the area between the mean and a positive z-score rather than the cumulative area to the left.

Suppose the area between the mean and z is 0.45.

Because the area to the left of the mean is 0.50:

\(P(Z\leq z)=0.50+0.45=0.95\)

Therefore:

\(z=\Phi^{-1}(0.95)\approx1.645\)

This is an important distinction when using printed z-tables because not all tables report the same type of area. NIST, for example, provides a standard-normal table based on the area from 0 to z and explains how that table can be used to obtain cumulative probabilities.

Central Area to Z Score

A central-area problem asks for the symmetric z-scores containing a specified percentage of the distribution.

Suppose the central area is 95%.

The remaining area is:

\(1-0.95=0.05\)

Because the normal distribution is symmetric, half goes into each tail:

\(\frac{0.05}{2}=0.025\)

The cumulative area on the right side is therefore:

\(1-0.025=0.975\)

So:

\(z=\Phi^{-1}(0.975)\)

giving:

\(z\approx1.96\)

The two critical values are:

\(z\approx-1.96\quad\text{and}\quad z\approx1.96\)

Two-Tailed Area to Z Score

A two-tailed problem gives the combined probability in both tails.

For example, suppose the total area in both tails is 0.05.

The area in each tail is:

\(\frac{0.05}{2}=0.025\)

The corresponding positive critical value is:

\(z=\Phi^{-1}(1-0.025)\)

\(z=\Phi^{-1}(0.975)\)

\(z\approx1.96\)

Therefore:

\(z\approx\pm1.96\)

This is why 1.96 is commonly associated with a central 95% interval.

Area to Z Score Examples

Example 1: Left Area of 0.95

Suppose 95% of the distribution lies below z.

\(P(Z\leq z)=0.95\)

Using the inverse normal distribution:

\(z=\Phi^{-1}(0.95)\)

Therefore:

\(z\approx1.645\)

Answer: z ≈ 1.645

Example 2: Right Area of 0.05

Suppose the area to the right of z is 0.05.

Convert it to a left-tail probability:

\(P(Z\leq z)=1-0.05=0.95\)

Then:

\(z=\Phi^{-1}(0.95)\approx1.645\)

Answer: z ≈ 1.645

Example 3: Central Area of 0.95

The central area is 0.95.

The area outside the center is:

\(1-0.95=0.05\)

Each tail contains:

\(0.05/2=0.025\)

Therefore the upper critical value corresponds to cumulative probability 0.975:

\(z=\Phi^{-1}(0.975)\approx1.96\)

Answer: z ≈ ±1.96

Example 4: Area of 0.1587 to the Left

Suppose:

\(P(Z\leq z)=0.1587\)

The corresponding z-score is approximately:

\(z=-1.00\)

This makes sense because a z-score of −1 has approximately 15.87% of the standard normal distribution below it.

Example 5: Area Between the Mean and z

Suppose the area between 0 and a positive z-score is 0.3413.

The cumulative area to the left is:

\(0.50+0.3413=0.8413\)

Therefore:

\(z=\Phi^{-1}(0.8413)\approx1.00\)

Answer: z ≈ 1.00

What Is a Z Score?

A z-score, also called a standard score, describes the position of a value relative to the mean in units of standard deviation.

For a normally distributed variable:

\(z=\frac{x-\mu}{\sigma}\)

where:

• \(x\) = observed value
• \(\mu\) = mean
• \(\sigma\) = standard deviation

A z-score of:

• 0 means the value is at the mean.
• 1 means the value is one standard deviation above the mean.
• −1 means the value is one standard deviation below the mean.
• 2 means the value is two standard deviations above the mean.
• −2 means the value is two standard deviations below the mean.

OpenStax describes z-scores as standardized values measured in standard-deviation units and uses them to work with areas and percentiles of the normal distribution.

Understanding the Standard Normal Distribution

The standard normal distribution is the normal distribution with:

\(\mu=0\)

and:

\(\sigma=1\)

Because the mean is 0 and the standard deviation is 1, the horizontal axis can be expressed directly in z-scores.

The distribution is symmetric around zero.

Negative and Positive Z Scores

A positive z-score lies above the mean.

A negative z-score lies below the mean.

For example:

\(z=1.5\)

means the value is 1.5 standard deviations above the mean.

Similarly:

\(z=-1.5\)

means the value is 1.5 standard deviations below the mean.

The corresponding probabilities are also symmetric:

\(P(Z\leq-1.5)=P(Z\geq1.5)\)

This symmetry is particularly useful when working with z-tables.

Area Under the Normal Curve

For a continuous normal distribution, probability is represented by area under the curve.

The entire area under the standard normal curve is:

\(1\)

or:

\(100%\)

The area to the left of a z-score is the cumulative probability:

\(P(Z\leq z)=\Phi(z)\)

The area to the right is:

\(P(Z\geq z)=1-\Phi(z)\)

The area between two z-scores is:

\(P(z_1\leq Z\leq z_2)=\Phi(z_2)-\Phi(z_1)\)

These relationships are the foundation of most standard-normal probability calculations.

Z Score to Area Calculator

Sometimes the problem is reversed.

Instead of being given an area and asked for z, you are given z and asked to find the probability or area.

For example:

What area lies below z = 1.96?

The cumulative standard-normal probability is approximately:

\(\Phi(1.96)=0.9750\)

Therefore:

• Area to the left = 0.9750
• Area to the right = 0.0250
• Percentile = 97.50%

Dedicated z-to-percentile calculators similarly use the standard normal CDF to convert a z-score into the area below it and its complementary right-tail area.

Z to Left-Tail Area

For a z-score of 1:

\(P(Z\leq1)=\Phi(1)\approx0.8413\)

Therefore, approximately 84.13% of the standard normal distribution lies below z = 1.

Z to Right-Tail Area

The right-tail probability is:

\(P(Z\geq z)=1-\Phi(z)\)

For z = 1:

\(P(Z\geq1)=1-0.8413=0.1587\)

So approximately 15.87% lies above z = 1.

Z Score to Percentile

For a standard normal distribution, the percentile associated with z is:

\(Percentile=\Phi(z)\times100\)

For example:

\(\Phi(1.50)\approx0.9332\)

Therefore:

\(Percentile\approx93.32%\)

A z-score of 1.50 is approximately at the 93.32nd percentile.

Area Between Two Z Scores Calculator

When two z-scores are given, the area between them represents the probability that a standard-normal observation falls within that interval.

The formula is:

\(P(z_1<Z<z_2)=\Phi(z_2)-\Phi(z_1)\)

For example, suppose:

\(z_1=-1\)

and:

\(z_2=1\)

Then:

\(P(-1<Z<1)=\Phi(1)-\Phi(-1)\)

Using the standard normal distribution:

\(P(-1<Z<1)\approx0.8413-0.1587\)

\(P(-1<Z<1)\approx0.6826\)

So approximately 68.27% of the distribution lies between z = −1 and z = 1.

This is also a common standalone calculator intent in current search results. (Stat Study Hub)

Example: Between z = 0 and z = 1.50

Suppose:

\(z_1=0\)

and:

\(z_2=1.50\)

Then:

\(P(0<Z<1.50)=\Phi(1.50)-\Phi(0)\)

Using:

\(\Phi(1.50)\approx0.9332\)

and:

\(\Phi(0)=0.5000\)

we get:

\(0.9332-0.5000=0.4332\)

So the area is approximately 43.32%.

Example: Between Two Positive Z Scores

Suppose:

\(z_1=1\)

and:

\(z_2=2\)

Then:

\(P(1<Z<2)=\Phi(2)-\Phi(1)\)

\(\approx0.9772-0.8413\)

\(\approx0.1359\)

Therefore, approximately 13.59% of the distribution lies between z = 1 and z = 2.

Percentile to Z Score Calculator

A percentile can also be converted into a z-score.

If a percentile represents the cumulative area below a z-score, then:

\(z=\Phi^{-1}(p)\)

where \(p\) is the percentile expressed as a decimal.

For example, the 90th percentile means:

\(p=0.90\)

Therefore:

\(z=\Phi^{-1}(0.90)\)

giving:

\(z\approx1.2816\)

So the 90th percentile corresponds to a z-score of approximately 1.28.

Common Percentile-to-Z Conversions

Percentile	Cumulative Area	Approx. Z Score
1%	0.01	−2.326
2.5%	0.025	−1.960
5%	0.05	−1.645
10%	0.10	−1.282
25%	0.25	−0.674
50%	0.50	0
75%	0.75	0.674
90%	0.90	1.282
95%	0.95	1.645
97.5%	0.975	1.960
99%	0.99	2.326

Percentile, Area and Z Score Relationship

These three concepts are closely connected.

For the standard normal distribution:

\(Percentile=\Phi(z)\times100\)

and:

\(Area\ to\ the\ left=\Phi(z)\)

Therefore:

z-score → left-tail area → percentile

are three different ways of describing the same position on the standard normal curve.

For example:

\(z=1.00\)

corresponds to:

\(\Phi(1.00)\approx0.8413\)

which means:

• Left area = 0.8413
• Left percentage = 84.13%
• Percentile ≈ 84.13th

The right-tail area is:

\(1-0.8413=0.1587\)

or approximately 15.87%.

Why 1.645 and 1.96 Are Both Common

One of the most common sources of confusion is the difference between 1.645 and 1.96.

Both are important, but they correspond to different areas.

Z = 1.645

A z-score of approximately 1.645 has:

\(P(Z\leq1.645)\approx0.95\)

So approximately 95% of the distribution lies below it.

The remaining 5% is in the right tail.

Z = 1.96

A z-score of approximately 1.96 has:

\(P(Z\leq1.96)\approx0.975\)

Therefore, approximately 2.5% is above it.

Because of symmetry, another 2.5% is below −1.96.

Thus:

\(P(-1.96<Z<1.96)\approx0.95\)

So:

• 1.645 → 95% cumulative left area
• ±1.96 → central 95% area

The numbers are different because the questions are different.

How to Use a Z Table

A z-table can be used to convert between z-scores and probabilities.

However, not all z-tables are formatted the same way.

Some tables give:

Area to the left of z

while others give:

Area between the mean and z

NIST’s standard-normal table, for example, describes its entries as the area between 0 and z and shows how to convert those values into cumulative probabilities. (NIST)

Finding Z From a Left-Tail Area

Suppose the desired cumulative area is 0.90.

Find 0.9000, or the closest value, in a cumulative-left z-table.

The corresponding z-score is approximately:

\(z=1.28\)

Finding Z From a Mean-to-Z Area

Suppose your table gives the area from the mean to z and the problem gives:

\(0.40\)

Add the 0.50 area to the left of the mean:

\(0.50+0.40=0.90\)

Then locate 0.90 in a cumulative table.

This gives approximately:

\(z=1.28\)

Handling Negative Z Scores

The standard normal curve is symmetric.

For a negative z-score:

\(P(Z\leq-z)=1-P(Z\leq z)\)

For example:

\(P(Z\leq-1.28)=1-P(Z\leq1.28)\)

Since:

\(P(Z\leq1.28)\approx0.90\)

we get:

\(P(Z\leq-1.28)\approx0.10\)

This symmetry is useful when a z-table only provides positive z-values.

Common Z Score Reference Values

Z Score	Area to Left	Percentile	Area to Right
−3.00	0.0013	0.13%	99.87%
−2.58	0.0050	0.50%	99.50%
−2.33	0.0099	0.99%	99.01%
−1.96	0.0250	2.50%	97.50%
−1.645	0.0500	5.00%	95.00%
−1.00	0.1587	15.87%	84.13%
0	0.5000	50.00%	50.00%
1.00	0.8413	84.13%	15.87%
1.645	0.9500	95.00%	5.00%
1.96	0.9750	97.50%	2.50%
2.33	0.9901	99.01%	0.99%
2.58	0.9950	99.50%	0.50%
3.00	0.9987	99.87%	0.13%

The 68–95–99.7 Rule

The standard normal distribution also has a useful approximation known as the 68–95–99.7 rule.

Approximately:

\(P(-1<Z<1)\approx68.27%\)

\(P(-2<Z<2)\approx95.45%\)

\(P(-3<Z<3)\approx99.73%\)

This means most observations from a normal distribution fall within three standard deviations of the mean.

OpenStax gives the same 68%, 95%, and 99.7% approximation for one, two, and three standard deviations from the mean. (OpenStax)

What the Rule Does Not Tell You

The empirical rule is useful for estimating broad ranges, but it does not replace an exact z-table or calculator when precision is required.

For example, the exact central probability between −1 and +1 is approximately:

\(68.27%\)

rather than exactly 68%.

Z Score vs Raw Score

A z-score and a raw score are not the same thing.

A raw score is measured in the original units of the variable.

For example:

Test score = 85 points

A z-score expresses the same position relative to the mean and standard deviation.

Suppose:

\(x=85\)

\(\mu=70\)

\(\sigma=10\)

Then:

\(z=\frac{85-70}{10}=1.5\)

The score is therefore 1.5 standard deviations above the mean.

If you need to convert a raw score to a z-score, use:

\(z=\frac{x-\mu}{\sigma}\)

This is a related but distinct calculation from converting an area to a z-score.

Z Score vs Percentile

A z-score tells you how far a value is from the mean in standard-deviation units.

A percentile tells you what percentage of the distribution lies below that value, assuming the standard-normal model.

For example:

\(z=1\)

corresponds to approximately:

\(84.13\text{th percentile}\)

while:

\(z=-1\)

corresponds to approximately:

\(15.87\text{th percentile}\)

A percentile is therefore often easier to interpret for someone who is unfamiliar with standard deviations.

Area, Probability and Percentage

Area and probability are often expressed in different forms.

For example:

\(0.95=95%=P(Z\leq z)\)

These represent the same quantity.

When using a calculator:

• 0.95 means probability/area
• 95% means percentage
• 95th percentile means the point with 95% of the distribution below it

Always check whether the calculator expects a decimal or percentage.

Common Mistakes When Finding Z From Area

Using 0.95 automatically as the answer for every 95% problem

The area must first be interpreted.

A 95% left-tail problem gives:

\(z\approx1.645\)

while a central 95% problem gives:

\(z\approx\pm1.96\)

Forgetting to convert right-tail area

If the right-tail area is 0.05, the cumulative left area is:

\(1-0.05=0.95\)

Forgetting to split a two-tailed area

If the total two-tailed probability is 0.05:

\(0.05/2=0.025\)

must be placed in each tail.

Using the wrong type of z-table

Check whether your table reports:

• cumulative area to the left,
• area from the mean to z, or
• another area convention.

Losing the negative sign

A z-score below the mean is negative.

If the left-tail probability is less than 0.50, the corresponding z-score is negative.

Confusing percentile with percentage score

A 90th percentile does not mean a person scored 90% on a test.

It means approximately 90% of the reference distribution lies at or below that position.

How to Choose the Right Calculator

What you know	What you need	Calculator
Left-tail area	Z-score	Area to Z
Right-tail area	Z-score	Area to Z
Central area	Critical z-values	Area to Z
Two-tailed area	Critical z-values	Area to Z
Z-score	Area/probability	Z to Area
Two z-scores	Area between them	Between Two Z Scores
Percentile	Z-score	Percentile to Z
Raw score, mean and SD	Z-score	Z Score Calculator

This distinction keeps the calculations simple and avoids using the wrong formula for the problem.

How to Find a Critical Z Value

A critical z-value is a z-score that marks a specified probability boundary.

For example, for a two-tailed 5% significance level:

\(\alpha=0.05\)

The total tail area is 0.05, so each tail contains:

\(\frac{\alpha}{2}=0.025\)

The cumulative area below the positive critical value is:

\(1-\frac{0.05}{2}=0.975\)

Therefore:

\(z^*=\Phi^{-1}(0.975)\approx1.96\)

The two critical values are:

\(z^*=\pm1.96\)

For a one-tailed 5% upper-tail test:

\(z^*=\Phi^{-1}(0.95)\approx1.645\)

The correct critical value therefore depends on whether the probability is one-tailed or two-tailed.

Area to Z Score on a TI-84

You can also use a TI-84 or similar graphing calculator to find a z-score from a cumulative probability.

For a left-tail problem, the relevant function is generally:

invNorm

For example, if the area to the left is 0.95:

\(z=\operatorname{invNorm}(0.95,0,1)\)

which gives approximately:

\(z=1.645\)

For a central 95% interval, the upper cumulative probability is 0.975:

\(z=\operatorname{invNorm}(0.975,0,1)\approx1.96\)

The exact calculator menu and syntax can vary by model, so verify the instructions for your particular calculator.

Frequently Asked Questions

What is the formula for converting area to z-score?

If the area represents cumulative probability to the left of z:

\(z=\Phi^{-1}(p)\)

where p is the cumulative probability.

What z-score corresponds to 95%?

It depends on what the 95% represents.

For 95% cumulative area to the left:

\(z\approx1.645\)

For a central 95% interval:

\(z\approx\pm1.96\)

What z-score corresponds to the 90th percentile?

The 90th percentile corresponds to:

\(z=\Phi^{-1}(0.90)\approx1.282\)

What is the z-score for the 95th percentile?

The 95th percentile corresponds to approximately:

\(z=1.645\)

What is the z-score for the 97.5th percentile?

The 97.5th percentile corresponds to approximately:

\(z=1.96\)

What is the z-score for the 5th percentile?

The 5th percentile corresponds to approximately:

\(z=-1.645\)

What is the area to the left of z = 1?

Approximately:

\(P(Z\leq1)=0.8413\)

So z = 1 corresponds to approximately the 84.13th percentile.

What is the area to the right of z = 1?

Approximately:

\(P(Z\geq1)=0.1587\)

or about 15.87%.

What is the area between z = −1 and z = 1?

Approximately:

\(P(-1<Z<1)=0.6827\)

or 68.27%.

Why is z = 1.96 associated with 95%?

Because approximately 95% of the standard normal distribution lies between −1.96 and +1.96:

\(P(-1.96<Z<1.96)\approx0.95\)

Can an area produce a negative z-score?

Yes.

If the cumulative area to the left is less than 0.50, the corresponding z-score is negative.

For example:

\(\Phi(-1)\approx0.1587\)

so approximately 15.87% lies below z = −1.

What does a negative z-score mean?

A negative z-score means the value lies below the mean.

For example:

\(z=-2\)

means the value is two standard deviations below the mean.

Is a z-score the same as a percentile?

No.

A z-score describes distance from the mean in standard-deviation units, while a percentile describes the percentage of the distribution below that position.

Is area the same as probability?

For a continuous probability distribution, the area under the probability density curve over an interval represents the probability of falling within that interval.

Which z-table should I use?

Check the table’s heading and instructions first. Some tables report cumulative area to the left of z, while others report area between 0 and z. NIST’s standard-normal table is an example of the latter format. (NIST)

Final Takeaway

Converting an area into a z-score is fundamentally an inverse normal-distribution problem:

\(z=\Phi^{-1}(p)\)

But before applying that formula, determine exactly what the area represents.

A left-tail area, right-tail area, central area, and two-tailed area can produce different z-score results even when they involve the same numerical percentage.

The most useful relationships are:

\(P(Z\leq z)=\Phi(z)\)

\(P(Z\geq z)=1-\Phi(z)\)

\(P(z_1<Z<z_2)=\Phi(z_2)-\Phi(z_1)\)

and:

\(z=\Phi^{-1}(p)\)

For example:

• 95% below z → z ≈ 1.645
• 5% above z → z ≈ 1.645
• central 95% → z ≈ ±1.96
• 90th percentile → z ≈ 1.282
• 95th percentile → z ≈ 1.645
• 97.5th percentile → z ≈ 1.960