Z-score Calculator

The z-score calculator is designed to determine the z-value from the raw score, sample mean and size, and a data sample.

Compute Z-score Using:

z =  
(x - μ)
σ

Probabilities:

z =  
(x̄ - μ)
(σ / √n)
z =  
(x̄ - μ)
(σ / √n)
Z-score Calculator

How to Use the Z-score Calculator?

  1. First, choose how you want to calculate the z-score. The tool allows you to find the z-value using the row score or data point, sample mean and size, and a data sample.
  2. After selecting the calculation method, input all the required values based on the chosen option.
  3. For a data sample, enter the set of numbers with a comma or space separator.
  4. Press the 'Calculate' button to obtain the z-score for a normal distribution.
  5. When you select the row score method, the tool returns the probabilities (left-tailed, right-tailed, two-tailed) along with the z-score.
  6. For the new calculations, press the 'Reset' button to clear the input and output fields.

What is Z-score?

The z-score, also known as the standard score, defines how far a specific data value is from the mean (average) of a dataset. It's measured in terms of standard deviations.

It helps to understand whether a data point is unusual or typical compared to the rest of the data.

Interpretation

Z = 0: The data point is exactly at the mean.

Z > 0: The data point is above the mean.

Z < 0: The data point is below the mean.

Z-score Formula

The z-score for a single data point is calculated using the following formula:

z  =  
x - μ
σ

Where,
z = Standard score,
x = Row score or a data point,
μ = Population mean,
σ = Population standard deviation.

The following formula is used to calculate the z-score for a sample:

z  =  
x̄ - μ
σ / √n

Where,
x̄ = Sample mean,
n = Sample size,
μ = Population mean,
σ = Population standard deviation.

Example 1:

Calculate the z-score for a row score of 23, population mean of 42, and standard deviation of 17.6.

Solution:

Here,
x = 23
μ = 42
σ = 17.6

Apply the z-score formula.

z  =  
x - μ
σ
z  =  
23 - 42
17.6
z  =  
-19
17.6

z = -1.07955

P-value from the Z Table:

P(x < 23) = 0.14017

P(x > 23) = 1 - P(x < 23) = 0.85983

P(23 < x < 42) = 0.5 - P(x < 23) = 0.35983

Example 2:

Suppose the sample mean is 55, the sample size is 12, the population mean is 46, and the population standard deviation is 32. Compute the z-value for a given normal distribution.

Solution:

Here,
x̄ = 55
n = 12
μ = 46
σ = 32

Put values into the formula.

z  =  
x - μ
σ / √n
z  =  
55 - 46
32 / √12
z  =  
9
32 / 3.464102
z  =  
9
9.237603

z = 0.97428