Z Score Calculator

Table of contents:
What on earth is a Z-score? While it might seem strange, it’s straightforward to understand. The Z-score is the standard score. The standard score is the number of standard deviations that a data point is above the mean. 

A Z-Score calculator will help you work out the value. Below, you can also learn how to calculate and use the Z-score and Z-score table.

How to Calculate a Z-Score?

The Z-score defines the distance between a data point and a mean score. You use it to describe normal distribution. You would also explain it in standard deviation (SD) terms. Many people call it a Z-score, but you can also call it the standardized score, standard, z-value, or average score. 

You need to identify the SD and mean of a data set before you find a Z-score. The mean (μ) is the sum of values in a data set. You then divide them by the number of data points. 

The equation looks like this: 

μ = ∑x / n

And you find SD according to this expression: 

σ = √ [∑ (x - μ) ² / n]

x = raw value
n = number of data points 

Apply the following formula to find the Z-score: 

z = (x - μ) / σ

An Example of Calculating the Z-Score

We’ll posit that you are playing darts with four friends. Your scores are 49, 56, 60, and 100. What is the Z-score of the friend who scored 60? 

  1.	Find the mean of all the results. You can use an average calculator, or use the formula below.
μ = (49 + 56 + 60 + 100) / 4 = 66.25

  2.	Calculate the values of (x - μ) ² for each dart score. 
(49 – 66.25) ² = 297.56
(56 – 66.25) ² = 105.062
(60 – 66.25) ² = 39.063
(100 – 66.25) ² = 1,139.063

  3.	Calculate the standard deviation. 
√ [(297.56 + 105.062 + 39.063 + 1,139.063) / 4] = √ (1580.748 / 4) = 19.88

  4.	Input the results. 
z = (60 – 66.25) / 19.88 = 0.31

  5.	Get your answer. 
With that formula, you worked out the Z-score of 60. The Z-score calculator can also help you find your SD or mean, once you know your Z-score.

What is a Z-Score Table?

A Z-score table lets you find the area of a Z-score in the SD graph. The first column would have Z- values. In the first row, identify the number with the same second decimal place as your Z-score. 

Previously,we found the Z-score for 60. It was 0.31. You would have to locate 0.3 in the first column and 0.01 in the first row. 

The area in the SD graph, left of the Z-score, offers a value. The area = 1. With this information, you can determine the probability of scoring less than 60 in darts.

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