The position of a raw score in terms of its distance from the mean when measured in standard deviation units is described by the Z- score. We represent Z-score in terms of standard deviations from their means. Z scores have a distribution with a mean as zero and a standard deviation as 1. Let us learn the Z score formula in detail.

**What is a Z Score?**

A Z-score is a numerical measurement used in statistics, which is measured in terms of standard deviations from the mean. The standard score is another name of z score and it can be placed on a normal distribution curve. If a Z-score is 0, it indicates that the data point score is identical to the mean score. A Z-score of 1, it indicates a value that is one standard deviation from the mean.

Z-scores may be positive or negative. If the value of the z score is positive, then the z score is above the mean and if the value of the z score is negative, then z score indicates it is below the mean.

We calculate Z- score by subtracting the population mean from an individual raw score and thereafter dividing the difference by the standard deviation of the population. This process of conversion is standardizing or normalizing.

## The Z Score Formula

**\(z = \frac{(x – \mu)}{\sigma}\)**

x | Data points or observation |

\(\mu\) | Mean |

\(\sigma\) | Standard deviation |

**Derivation of The Z Score Formula**

The equation or the formula for z-score of a data point can be derived by using the steps below:

**Step 1:** Firstly, determine the mean of the data set based on the data points or observation and the total number of data points in the data set.

Mean \(\mu\) = \(\frac{\sum_ {i}^{n} xi}{N}\)

x_{i} |
data points or observation |

N | the total number of data points in the data set |

**Step 2: **Next, determine the standard deviation of the population on the basis of the population mean, data points and the number of data points in the population.

Standard Deviation \(\sigma = \sqrt{ \frac{\sum_ i^n (x_i – \mu)^2}{N}}\)

x_{i} |
data points or observation |

N | the total number of data points in the data set |

\(\mu\) | Mean |

**Step 3: **Finally, the formula for z-score is derived by subtracting the mean from the data point and then the result is divided by the standard deviation as shown below.

\(z = \frac{(x – \mu)}{\sigma}\)

## Solved Example for Z Score Formula

Q.1: The grades on a history midterm at a school have a mean of \mu is 85, and a standard deviation of \(\sigma\) is 2. Michael scored 86 in the exam. Find the z-score for Michael’s exam grade.

Solution: \(z = \frac{(x – \mu)}{\sigma}\)

\(z = \frac{(86 – 85)}{2}\)

\(z = \frac{1}{2}\) = 0.5

Michael’s z-score is 0.50.

Q.2 : In a class of 30 students who appeared for a class test. Determine the z-test score for the 4^{th} student of based on the marks scored by the students out of 100 – 55, 67, 84, 65, 59, 68, 77, 95, 88, 78, 53, 81, 73, 66, 65, 52, 54, 83, 86, 94, 85, 72, 62, 64, 74, 82, 58, 57, 51, 91.

Solution: Given, x = 65, Number of data points, N = 30.

Mean = (55 + 67 + 84 + 65 + 59 + 68 + 77 + 95 + 88 + 78 + 53 + 81 + 73 + 66 + 65 + 52 + 54 + 83 + 86 + 94 + 85 + 72 + 62 + 64 + 74 + 82 + 58 + 57 + 51 + 91) / 30

Mean = 71.30

Standard Deviation \(\sigma = \sqrt{ \frac{\sum_ i^n (x_i – \mu)^2}{N}}\)

ơ = 13.44

\(z = \frac{(x – \mu)}{\sigma}\)

\(z = \frac{(65 – 71.30)}{13.44}\)

= -0.47

Therefore, the 4^{th} student’s score is – 0.47

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