# Standard or Z Score Calculator

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** z-score calculator** - step by step calculation to calculate the normalized value or relative standings of a random or individual sample of the normal distribution with population mean (μ) and standard deviation (σ). z score value is the distance between the raw score or an individual member to the population mean in terms of SD. The required inputs of this calculator are a random or individual sample of the population, population mean and the population standard deviation. In the process of normalizing, the each member value of the population can generally use the mean & standard deviation of the entire population not the samples. However, in certain experiments, getting the complete population data is not possible where the

*sample standard deviation*can be used to normalize the member values of a sample population. In statistical data analysis, Z score also known as standard score, z value, standardized score, and normal score. The standard score transformation is useful to compare the relative standings between the members of the distribution with

*population mean & standard deviation*and to determine how many standard deviations in a data set is above or below the mean. It is featured z score calculator along with formulas, solved example problem & step by step calculation for the given values.

### Formulas

The below mathematical formulas are used in this context of statistical data analysis to find population mean, standard deviation (PSD) and the z score of a member of a population. The normalized score is the signed number, therefore, the higher and lower value indicates that the z score is above and below the mean (expected value) respectively.

where

**x** is a member or raw score to be standardized

**μ** is the mean of the population

**σ** is the standard deviation of the entire population

### How to calculate z or standard score value?

Follow these below steps to find the z value of any member of the normal distribution. The solved example illustrates how the values are being used in the calculation to normalize the data.

1. Find the population mean of a data set.

2. Find the population standard deviation for the data set.

3. Pick a random sample from the population.

4. Subtract the population mean from a random sample of the population.

5. Divide the 4^{th} step answer by the population standard deviation gives the normalized value of a member of the distribution.

### Solved Example

The below solved example for z or standard score calculation illustrates how the values are being used in the formulas to normalize the value of a member of the normal distribution.

**Problem:**

The class of five students scored 68, 75, 81, 87, and 90. Find the normalized or z-score of 75 for the population mean 80.2 and standard deviation 7.98?

**Solution:**

**Step by step calculation:**

__Step 1:__ Find the population mean of a data set

Mean = (68 + 75 + 81 + 87 + 90)/5

= 401/5

= **80.2**

__Step 2:__ Find the population standard deviation for the data set

= √( (1/5) * (68 - 80.2)^{2} + ( 75 - 80.2)^{2} + ( 81 - 80.2)^{2} + ( 87 - 80.2)^{2} + ( 90 - 80.2)^{2})

= √( (1/5) * (-12.2^{2} + -5.2^{2} + 0.8^{2} + 6.8^{2} + 9.8^{2}))

= √( (1/5) * (148.84 + 27.04 + 0.64 + 46.24 + 96.04))

= √ 63.7599056

σ = **7.98498**

__Step 3:__ select the member data as 75 from the population 68, 75, 81, 87, and 90

__Step 4:__ Subtract the population mean from a random sample of the population

= 75 - 80.2

= -5.2

__Step 5:__ Divide the 4th step answer by the population standard deviation gives the normalized value of a member of the distribution

Standard or z score = (x - μ)/ σ

= -5.2/7.98

= **-0.6516**

The standardized score (z score) is important in the field of statistics data analysis to find the normalized score of an individual member of the normally distributed data. The manual calculations can be done by using the above formulas. However, when it comes to online to verify the results or to perform the such calculations, this z score calculator makes your calculations simple as possible. The normalized score formulas, solved example problem along with step by step calculation may assist users to understand how the values are being used in such calculations.