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Z-Score
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What is Z score? 

If you talk about the definition of Z score, it is the length/distance between the mean of a sample distribution and its corresponding Standard Deviation (SD) value. There are various other terms used apart from Z score including normal score and z-value. It is important to have a clear idea of the formula used to determine the Z Score. This helps in getting the calculations right.

It takes a good time span to understand the formulae and use them without committing mistakes. Even though users get a basic understanding of the concept but still get stuck while performing. A lot of values have to be inserted when the formulae are being used. Users should be sure that each parameter has been entered properly.

Z Score Formula

\(\Large z = \dfrac{(x - \mu)}{\sigma}\)

In accordance with the literal definition of Z Score, it is the distance between the mean of a data sample and the standard deviation. When you drill down further, the mean “denoted by μ” is calculated when the sum of data sets is divided by the number of data points. Here the data set is shown by the term “x” and “n” is used to identify the number of data points.

\(\mu = \dfrac{\sum x}{n}\)

Similarly, the formula of SD (Standard Deviation) is given as 

Standard Deviation \(= σ = \sqrt{ \dfrac{\sum (x - \mu)^2}{n}}\)

Eventually, the Z Score will be calculated by using the formula given above.

Z Score Value \(= \dfrac{(x - \mu)}{\sigma}\)

In the above formula, “x” is used to show the raw value

How to find Z Score?

Calculating the Z score for a data sample is a step wise process. It is important to understand all the steps so that the correct value is determined without any inconvenience. Let us consider the following example to gain more clarity and understanding.,

•    Consider that you have a data sample that comprises of five values. These values are \(48, 65, 56, 72  \&  38\). In addition to that, the z score has to be determined at \(56\). This means that the raw value (x) is \(56\). To start with, you need to determine the value of μ which is the mean.

•    The mean is another term for average. Hence, the average of all five numbers given above would be calculated as.

\(\mu = \dfrac{48+65+56+72+38}{5}\)
Mean \(= 55.8\)

•    The next step is related to calculating the value of Standard Deviation. However, before that, we need to determine to the value of \((x- \mu)^2\).  Here, we have five values of x which are \(48, 65, 56, 72  \&  38\). Thus, the following calculations would be performed in this relation.

For \(x = 48\)
\((x- μ)^2  = (48-55.8)^2 = 60.84\)

Similarly, when we calculate the value of \((x- \mu)^2\)  for \(65,56,72  \&  38\), the results would be \(84.64, 0.04, 262.44  \&  316.84\) respectively.

 

 

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Lena B. Martinez | 02/09/2019

Your calculator has completely simplifies my calculation of Z score. Preety Good.

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