# Z-Score in Statistics

• What’s a Z-Score?
• Formula for Z-Score
• How to Calculate Z-Score?
• Interpretation of Z-score

What’s a Z-Score ?
Z-score also known as standard score gives us an idea of how far a data point is from the mean. It indicates how many standard deviations an element is from the mean. In order to use a z-score, we need to know the population mean (μ) and also the population standard deviation (σ).

Formula for Z-Score :

A z-score can be calculated using the following formula.

#### z = (X – μ) / σ

where,
z = Z-Score,
X = The value of the element,
μ = The population mean, and
σ = The population standard deviation

#### How to Calculate Z-Score?

Example:

Question:
You take the GATE examination and score 500. The mean score for the GATE is 390 and the standard deviation is 45. How well did you score on the test compared to the average test taker?

Solution:
Given data in the above question are
X = 500
μ = 390
σ = 45

By applying the formula of z-score,

z = (X – μ) / σ
z = (500 – 390) / 45
z = 110 / 45 = 2.4

This means that your score is 2.4 standard deviation above the mean.

Interpretation of Z-score :

• An element having a z-score less than 0 represents that the element is less than the mean.
• An element having a z-score greater than 0 represents that the element is greater than the mean.
• An element having a z-score equal to 0 represents that the element is equal to the mean.
• An element having a z-score equal to 1 represents that the element is 1 standard deviation greater than the mean; a z-score equal to 2, 2 standard deviations greater than the mean, and so on.
• An element having a z-score equal to -1 represents that the element is 1 standard deviation less than the mean; a z-score equal to -2, 2 standard deviations less than the mean, and so on.
• If the number of elements in a given set is large, then about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2; about 99% have a z-score between -3 and 3.

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