Z Score Calculator

Choose the Right Z Score Calculator

This calculator has different sections depending on what you already know and what you want to find.

• Z Score from Raw Score – Use this when you know the raw score, mean, and standard deviation.
• Z Score from Sample Mean – Use this when you want to standardize a sample mean using the sample size.
• Probability from Z Score – Use this when you already know the z score and want the probability.
• Area Between or Outside Two Z Scores – Use this when you want the probability inside or outside an interval.
• Z Score from Probability – Use this when you know the probability and want the matching z score.

For example, if your question gives a raw test score, mean, and standard deviation, use the raw score section. If your question asks for the probability below a z-score, use the probability from the z-score section. However, if your question gives a percentile or probability and asks for the z value, use the z score from the probability section.

What Is a Z Score?

A z-score is a statistical measure that tells you how many standard deviations a value is from the mean. It can take both positive and negative values, with a positive z score implying that the value is above the mean and a negative z score suggesting that the value is below the mean. However, if you ever find a z-score of 0, it means that the value is exactly the same as the mean.

For example, a z score of 2 means the value is 2 standard deviations above the mean. A z score of -1.5 means the value is 1.5 standard deviations below the mean.

Z scores are useful because they place different values on the same standard scale. This makes it easier to compare scores from different distributions. For example, you can compare exam scores from two different classes using z scores, even if the classes had different means and standard deviations.

Z Score Formula

For an individual raw score, the z-score formula is z = (x-μ)/σ

Where:

• x is the raw score
• μ is the population mean
• σ is the population standard deviation
• z is the z score, also known as the standard score

In other words, to find a z score from a single raw score, simply subtract the mean from the raw score and divide the result by the standard deviation.

Z Score Formula for a Sample Mean

Sometimes, you may need to calculate a z-score for sample data. In this case, the z-score formula changes to account for the sample size by reflecting the standard error of the mean. Thus, the formula becomes:

When you are calculating a z score for a sample mean, use:

Where:

• x̄ is the sample mean
• μ is the population mean
• σ is the population standard deviation
• n is the sample size
• $\sigma / \sqrt{n}$ is the standard error of the mean

This formula is used when the value you want to standardize is a sample mean instead of one individual observation. The denominator is called the standard error. It measures how much the sample means are expected to vary from the population mean.

Note: Use the z approach when the population standard deviation is known or when the normal approximation is appropriate. If the population standard deviation is unknown and the sample size is small, a t statistic may be more appropriate.

How to Use the Z Score Calculator

This calculator is divided into sections to help you choose the method that matches your problem. You do not need to use every section. Just select the one that matches the information given in your question.

Here’s how to use the different sections of the calculator with examples:

1. Calculate a Z Score from a Raw Score

Use this section when you want to convert a raw score (x) into a z-score. To convert the x value to z using the calculator:

1. Enter the raw score
2. Enter the mean
3. Enter the standard deviation
4. Click Calculator

The calculator will show the z-score for your raw score.

Example

A student scored 82 on a test. The class mean was 70, and the standard deviation was 8. Find the z score.

Solution

To find the z-score manually, we apply the z-score formula.

By definition, z = (x-μ)/σ

Substituting the values into the formula and solving, we have:

z = (82-70)/8

= 1.5

This means the student scored 1.5 standard deviations above the mean.

Alternatively, you can quickly calculate the z score using the calculator by following these steps:

1. Select Z Score from the Raw Score tool
2. Select the Individual value as the calculation type
3. Enter raw score, x = 82, mean, μ = 70, and standard deviation, σ = 8
4. Click Calculate

The calculator will instantly return the correct z value as 1.5

2. Calculate a Z Score from a Sample Mean

Use this section when you want to calculate how far a sample mean is from the population mean.

Enter:

• Sample mean
• Population mean
• Standard deviation
• Sample size

The calculator first finds the standard error by dividing the standard deviation by the square root of the sample size. It then uses the standard error to calculate the z score.

Example: Z Score from a Sample Mean

A researcher selects a sample of 36 students. The sample has a mean score of 74. The population mean is 70, and the population standard deviation is 12. Find the z score for the sample mean.

From the question:

• Sample mean, (\bar{x} = 74)
• Population mean, (\mu = 70)
• Standard deviation, (\sigma = 12)
• Sample size, (n = 36)

First, calculate the standard error:

[
SE = \frac{12}{\sqrt{36}}
]

[
SE = \frac{12}{6} = 2
]

Now calculate the z score:

[
z = \frac{74 – 70}{2}
]

[
z = 2.00
]

The z score is 2.00.

This means the sample mean is 2 standard errors above the population mean.

3. Find Probability from a Z Score

Use this section when you already have a z score and want to find the probability under the standard normal curve.

Enter the z score and choose the probability type:

• Left-tail probability: (P(Z < z))
• Right-tail probability: (P(Z > z))
• Area from 0 to z: used when the z score is positive
• Area from z to 0: used when the z score is negative

The calculator handles positive and negative z scores correctly. If the z score is negative, the area between the z score and 0 is read from (z) to 0.

Example: Probability from a Z Score

Find the probability that a standard normal value is less than (z = 1.25).

This is a left-tail probability:

[
P(Z < 1.25)
]

Using the calculator:

[
P(Z < 1.25) \approx 0.8944
]

The probability is 0.8944, or 89.44%.

This means about 89.44% of values in the standard normal distribution fall below a z score of 1.25.

4. Find the Area Between Two Z Scores

Use this section when you want the probability between two z scores.

Enter:

• First z score
• Second z score

Then choose the option for the area between the two z scores.

This is useful when you want to know the probability of a value falling inside a standard normal interval.

Example: Area Between Two Z Scores

Find the probability that a standard normal value falls between (z = -1.20) and (z = 1.80).

We want:

[
P(-1.20 < Z < 1.80)
]

Using the calculator, enter:

• First z score = -1.20
• Second z score = 1.80
• Probability type = between

The calculator gives:

[
P(-1.20 < Z < 1.80) \approx 0.8490
]

The probability is 0.8490, or 84.90%.

This means about 84.90% of values in the standard normal distribution fall between -1.20 and 1.80.

5. Find the Area Outside Two Z Scores

Use this section when you want the combined probability in both tails.

Enter:

• First z score
• Second z score

Then choose the option for the area outside the two z scores.

This gives the probability below the smaller z score plus the probability above the larger z score.

Example: Area Outside Two Z Scores

Find the probability outside (z = -1.96) and (z = 1.96).

We want:

[
P(Z < -1.96) + P(Z > 1.96)
]

Using the calculator, enter:

• First z score = -1.96
• Second z score = 1.96
• Probability type = outside

The calculator gives:

[
P(Z < -1.96) + P(Z > 1.96) \approx 0.0500
]

The probability outside the two z scores is 0.0500, or 5%.

This is why -1.96 and 1.96 are commonly used for a two-sided 95% confidence interval.

6. Find a Z Score from Probability

Use this section when you know the probability and want to find the matching z score.

Choose the probability type:

• Left cumulative probability
• Right-tail probability
• Central probability
• Outside probability

This section is useful for percentiles, critical values, confidence intervals, and inverse normal probability problems.

Example: Z Score from Left-Tail Probability

Find the z score that has 95% of the standard normal distribution below it.

We want:

[
P(Z < z) = 0.95
]

Using the z score from probability section:

• Probability = 0.95
• Probability type = left cumulative

The calculator gives:

[
z \approx 1.645
]

The z score is approximately 1.645.

This means 95% of values fall below this z score in the standard normal distribution.

7. Find Z Scores from a Central Probability

Use this option when you want the two z scores that contain a central area of the standard normal distribution.

This is common in confidence interval problems. For example, a central probability of 0.95 gives the z values that leave 2.5% in each tail.

Example: Z Scores from Central Probability

Find the z scores that contain the middle 95% of the standard normal distribution.

We want:

[
P(-z < Z < z) = 0.95
]

Using the calculator:

• Probability = 0.95
• Probability type = central

The calculator gives approximately:

[
z = -1.96 \text{ and } z = 1.96
]

The middle 95% of the standard normal distribution lies between -1.96 and 1.96.

This is also the reason 1.96 is commonly used as the z critical value for a two-sided 95% confidence interval.

How to Interpret a Z Score

A z score is interpreted by looking at its sign and size.

Values close to the mean have z scores near 0. Values that are far above or below the mean have larger positive or negative z scores.

For many normal distribution problems, z scores between -2 and 2 are fairly common. Z scores beyond -3 or 3 are much less common.

Z Score and Probability

A z score can be converted into a probability using the standard normal distribution. This probability is the area under the normal curve.

For example:

• (P(Z < z)) gives the area to the left of the z score.
• (P(Z > z)) gives the area to the right of the z score.
• (P(a < Z < b)) gives the area between two z scores.
• Outside probability gives the combined area in the two tails.

This is why the calculator can also be used as a z score probability calculator.

If your question says “less than,” you usually need the left-tail probability. If it says “greater than,” you usually need the right-tail probability. If it says “between,” you need the area between two z scores.

Z Score and Percentile

A percentile tells you the percentage of values below a given score. A z score can be converted into a percentile by finding the left-tail probability and multiplying it by 100.

For example, suppose a z score has a left-tail probability of 0.8413.

[
0.8413 \times 100 = 84.13%
]

So, that z score is approximately the 84th percentile.

This is useful for test scores, rankings, standardized assessments, and comparing results across different scales.

For example, if a student is at the 84th percentile, it means the student scored higher than about 84% of the comparison group.

Z Score from Probability

Sometimes you already know the probability and want the matching z score. This is called finding an inverse z score or inverse normal value.

For example, if you want the z score with 95% of the area below it, you are finding the z value where:

[
P(Z < z) = 0.95
]

The calculator can also find z scores from right-tail, central, and outside probabilities. This makes it useful for confidence intervals, critical values, percentiles, and normal distribution problems.

For example, if you enter a central probability of 0.95, the calculator returns the two z scores that contain the middle 95% of the standard normal distribution.

When Should You Use This Z Score Calculator?

Use this calculator when you need to:

• Standardize a raw score
• Compare values from different distributions
• Convert a z score to probability
• Find the percentile for a z score
• Find the probability between two z scores
• Find the area outside two z scores
• Find a z score from a known probability
• Find z critical values for common confidence levels
• Check normal distribution homework or research calculations

You can also use it when reading z tables, checking statistical results, or solving normal distribution questions that involve standard scores.

Common Z Score Values and Probabilities

The table below shows some common z scores and their approximate left-tail and right-tail probabilities.

These values are often used in normal distribution problems, confidence intervals, and hypothesis tests.

For exact values, use the calculator because rounding can vary slightly from one z table to another.

Z Score vs Raw Score

A raw score is the original value. A z score is the standardized version of that value.

For example, a raw exam score of 82 does not tell you much unless you know the mean and standard deviation. If the mean is 70 and the standard deviation is 8, the z score tells you that 82 is 1.5 standard deviations above the mean.

This makes z scores useful for comparing values that come from different scales.

For example, a score of 82 may be excellent in one class but average in another class. The z score gives more context because it shows where the score falls relative to the mean and standard deviation.

Z Score vs T Score

A z score is commonly used when the population standard deviation is known or when the normal approximation is appropriate. A t score is often used when the population standard deviation is unknown and you estimate it from the sample.

In practice:

• Use a z score when the population standard deviation is known.
• Use a t score when the population standard deviation is unknown.
• Use a t distribution for many small-sample hypothesis tests and confidence intervals.

For example, if a question gives the population standard deviation, a z statistic may be appropriate. If the question only gives the sample standard deviation, a t statistic is often the better choice, especially for smaller samples.

Related Calculators

You may also find these calculators useful:

• Standard deviation calculator
• Standard error calculator
• Normal distribution calculator
• P-value calculator
• Z critical value calculator
• T critical value calculator
• Confidence interval calculator
• Mean calculator
• Sample mean calculator

These tools are helpful when working with normal distributions, hypothesis tests, confidence intervals, and descriptive statistics.

FAQs About the Z Score Calculator

What is a z score calculator?

A z score calculator is a tool that converts a raw score into a standard score. It can also calculate probability from a z score, find the area between or outside two z scores, and find a z score from probability.

How do you calculate a z score?

To calculate a z score, subtract the mean from the raw score and divide the result by the standard deviation.

[
z = \frac{x – \mu}{\sigma}
]

For example, if (x = 82), (\mu = 70), and (\sigma = 8), then:

[
z = \frac{82 – 70}{8} = 1.50
]

What does a negative z score mean?

A negative z score means the value is below the mean.

For example, (z = -1) means the value is one standard deviation below the mean. A z score of (-2) means the value is two standard deviations below the mean.

Can I find probability from a z score?

Yes. Use the probability from z score section to find left-tail, right-tail, or mean-to-z probability.

For example, if (z = 1.25), the left-tail probability is approximately 0.8944. This means about 89.44% of values fall below that z score.

Can I find a z score from probability?

Yes. Use the z score from probability section when you know the probability and want the matching z score.

For example, if the left-tail probability is 0.95, the matching z score is approximately 1.645.

What is the z score for a 95% confidence interval?

For a two-sided 95% confidence interval, the common z critical values are approximately:

[
-1.96 \text{ and } 1.96
]

This means the middle 95% of the standard normal distribution lies between -1.96 and 1.96.

Is a z score the same as a percentile?

No. A z score tells you how many standard deviations a value is from the mean. A percentile tells you the percentage of values below a given score.

However, you can convert a z score into a percentile by finding the left-tail probability and multiplying it by 100.

What is the difference between left-tail and right-tail probability?

Left-tail probability is the area to the left of a z score. Right-tail probability is the area to the right of a z score.

For example, (P(Z < 1.25)) is a left-tail probability, while (P(Z > 1.25)) is a right-tail probability.