Z-Score Calculator

Q: What is the z-score formula?

For a population, the z-score formula is z = (x - μ) / σ, where x is the data point, μ is the population mean, and σ is the population standard deviation. For a sample, the formula is z = (x - x̄) / s, where x̄ is the sample mean and s is the sample standard deviation.

Q: How do you use the z-score calculator without the standard deviation?

Use the 'From Data Set' tab on our calculator. Paste your entire list of numbers into the text box and enter the specific data point you want to test. The calculator will automatically compute the mean and standard deviation from your data before calculating the z-score.

Q: What does the p-value from the z-score mean?

The p-value is the probability of observing a value as extreme as, or more extreme than, your data point, assuming the null hypothesis is true. A small p-value (typically < 0.05) suggests that your result is statistically significant. Our calculator provides the p-value for left-tailed, right-tailed, and two-tailed tests.

Calculate the Z-score from a raw data set or a known distribution.

Data Point (x)

Mean (μ or x̄)

Standard Deviation (σ or s)

Data Set (comma, space, or new line separated)

Data Point to Test (x)

Data Type

The Ultimate Guide to the Z-Score

In statistics, a Z-score is a powerful tool that tells you how many standard deviations a specific data point is from the mean of a distribution. It's a way of standardizing values from different distributions to compare them on a common scale. This guide explains the Z-score formula, how to interpret the results, and how our comprehensive calculator can find the Z-score for you, either from a known distribution or directly from a raw data sample, complete with steps and a graph of the standard normal distribution.

How to Use the Z-Score Calculator

Our tool offers two convenient methods for calculation:

Direct Z-Score Calculation: If you already know the mean and standard deviation of your data, use the first tab. Simply enter your data point (x), the mean, and the standard deviation.
Calculate from a Data Set: If you have a list of numbers but don't know the mean or standard deviation, use the second tab. Paste your data set, enter the data point you want to test, and specify whether your data is a sample or a whole population. The calculator handles the rest.

After calculating, the tool provides the Z-score, the associated P-value (probability), a step-by-step breakdown of the calculation, and an interactive graph.

What is the Z-Score Formula?

The formula depends on whether you are working with a population or a sample.

Population Z-Score: z = (x - μ) / σ
Sample Z-Score: z = (x - x̄) / s

Where:

x = The data point you are testing.
μ = The population mean.
σ = The population standard deviation.
x̄ = The sample mean.
s = The sample standard deviation.

Our "From Data Set" calculator will compute the mean and standard deviation for you, making it an ideal z-score calculator without standard deviation known beforehand.

Interpreting the Z-Score and P-Value

A positive Z-score means the data point is above the mean.
A negative Z-score means the data point is below the mean.
A Z-score of 0 means the data point is exactly the mean.

The P-value is the probability of finding a value as extreme as your data point. Our calculator provides three types:

Left-tailed (P(Z < z)): The probability of finding a value less than your data point.
Right-tailed (P(Z > z)): The probability of finding a value greater than your data point.
Two-tailed (2 * P(Z < -|z|)): The probability of finding a value as extreme in either direction (above or below the mean).

In hypothesis testing, a P-value less than a significance level (usually 0.05) indicates a statistically significant result.

Practical Applications

Z-scores are used across many fields to standardize and compare data.

Education: Comparing a student's score on two different tests with different means and standard deviations.
Finance: Assessing the volatility of a stock compared to the market average.
Medical Fields: In pediatric anthropometry, Z-scores are used to assess a child's growth (e.g., height-for-age, weight-for-age) relative to a standardized population. A pediatric Z-score calculator helps doctors determine if a child's measurements are within a normal range.

How to Calculate Z-Score in Excel

Excel has a built-in function to calculate the Z-score called STANDARDIZE.

The syntax is: =STANDARDIZE(x, mean, standard_dev). For example, to find the Z-score for the value 75 from a distribution with a mean of 60 and a standard deviation of 10, you would use: =STANDARDIZE(75, 60, 10), which would return 1.5.

Frequently Asked Questions (FAQ)

What is the z-score formula?

For a population, the z-score formula is z = (x - μ) / σ, where x is the data point, μ is the population mean, and σ is the population standard deviation. For a sample, the formula is z = (x - x̄) / s, where x̄ is the sample mean and s is the sample standard deviation.

How do you use the z-score calculator without the standard deviation?

Use the 'From Data Set' tab on our calculator. Paste your entire list of numbers into the text box and enter the specific data point you want to test. The calculator will automatically compute the mean and standard deviation from your data before calculating the z-score.

What does the p-value from the z-score mean?

The p-value is the probability of observing a value as extreme as, or more extreme than, your data point. A small p-value (typically < 0.05) suggests that your result is statistically significant. Our calculator provides the p-value for left-tailed, right-tailed, and two-tailed tests.

To analyze the properties of your raw data, such as mean and median, visit our Mean, Median, Mode, Range Calculator.

Disclaimer: This calculator provides statistical analysis for educational purposes. For medical or financial decisions, consult a qualified professional.