Normalization Formula

What is Normalization Formula?

In statistics, the term “normalization” refers to the scaling down of the data set such that the normalized data falls in the rangeRangeThe range formula computes the difference between the range's maximum and minimum values." To determine the range, the formula subtracts the minimum value from the maximum value. Range = maximum value – minimum value read more between 0 and 1. Such normalization techniques help compare corresponding normalized values from two or more different data sets in a way that eliminates the effects of the variation in the scale of the data sets i.e., a data set with large values can be easily compared with a data set of smaller values.

The equation for normalization is derived by initially deducting the minimum value from the variable to be normalized. The minimum value is deducted from the maximum value, and then the previous result is divided by the latter.

Mathematically, the Normalization equation is represented as,

x normalized = (xx minimum) / (x maximumx minimum)
Normalization-Formula

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Explanation of the Normalization Formula

The equation of calculation of normalization can be derived by using the following simple four steps:

  1. Firstly, identify the minimum and maximum value in the data set, and they are denoted by x minimum and x maximum.

  2. Next, calculate the range of the data set by deducting the minimum value from the maximum value.


    Range = x maximumx minimum

  3. Next, determine how much more in value is the variable to normalized from the minimum value by deducting the minimum value from the variable i.e., xx minimum.

  4. Finally, the formula for the calculation of normalization of the variable x is derived by dividing the expression in step 3 by the expression in step 2, as shown above.

Examples of Normalization Formula (with Excel Template)

Let’s see some simple to advanced examples of normalization equations to understand it better.

Normalization Formula – Example #1

Determine the normalized value of 11.69, i.e., on a scale of (0,1), if the data has the lowest and highest value of 3.65 and 22.78, respectively.

From the above, we have gathered the following information.

normalization formula example 1.1

Therefore the calculation of the normalization value of 11.69 is as follows,

normalization formula example 1.2
  • x (normalized)= (11.69 – 3.65) / (22.78 – 3.65)

Normalization value of 11.69 is –

normalization formula example 1.3
  • x (normalized) = 0.42

The value of 11.69 in the given data set can be converted on the scale of (0,1) as 0.42.

Normalization Formula – Example #2

Let us take another example of a data set that represents the test marks scored by 20 students during the recent science test. Present the test scores of all the students in the range of 0 to 1with the help of normalization techniques. The test scores (out of 100) are as follows:

normalization formula example 2.1

As per given test score,

The highest test mark is scored by student 11 i.e. x maximum = 95, and

The lowest test mark is scored by student 6 i.e. x minimum = 37

ex 2.2

So the calculation of the normalized score of student 1 is as follows,

ex 2.3
  • Normalized Score of student 1 = (78 – 37) / (95 – 37)

Normalized Score of student 1

ex 2.4
  • Normalized Score of student 1  = 0.71

Similarly, we have done the calculation of normalization of score for all the 20 students as follows,

  • Score of student 2 = (65– 37) / (95 – 37) = 0.48
  • Score of student 3 = (56 – 37) / (95 – 37) = 0.33
  • Score of student 4 = (87 – 37) / (95 – 37) = 0.86
  • Score of student 5 = (91 – 37) / (95 – 37) = 0.93
  • Score of student 6 = (37 – 37) / (95 – 37) = 0.00
  • Score of student 7 = (49 – 37) / (95 – 37) = 0.21
  • Score of student 8 = (77 – 37) / (95 – 37) = 0.69
  • Score of student 9 = (62 – 37) / (95 – 37) = 0.43
  • Score of student 10 = (59 – 37) / (95 – 37) = 0.38
  • Score of student 11 = (95 – 37) / (95 – 37) = 1.00
  • Score of student 12 = (63– 37) / (95 – 37) = 0.45
  • Score of student 13 = (42 – 37) / (95 – 37) = 0.09
  • Score of student 14 = (55 – 37) / (95 – 37) = 0.31
  • Score of student 15 = (72 – 37) / (95 – 37) = 0.60
  • Score of student 16 = (68 – 37) / (95 – 37) = 0.53
  • Score of student 17 = (81 – 37) / (95 – 37) = 0.76
  • Score of student 18 = (39 – 37) / (95 – 37) = 0.03
  • Score of student 19 = (45 – 37) / (95 – 37) = 0.14
  • Score of student 20 = (49 – 37) / (95 – 37) = 0.21

Now, let us draw the graph for the normalized score of the students.

graph

Normalization Formula Calculator

You can use this normalization formula calculator.

X
Xminimum
Xmaximum
Xnormalized
 

Xnormalized =
X − Xminimum
=
Xmaximum-Xminimum
00
= 0
0-0

Relevance and Use

The concept of normalization is very important because it is often used in various fields, such as ratings, where the normalization technique is used to adjust the values measured on different scales to a notionally common scale (0 to 1). The concept of normalization can also be used for more sophisticated and complicated adjustments like bringing the entire set of a probability distributionProbability DistributionProbability distribution is the calculation that shows the possible outcome of an event with the relative possibility of occurrence or non-occurrence as required. It is a mathematical function that gives results as per the possible events.read more of adjusted values into alignment or quantile normalization in which the quantiles of different measures are brought into alignment.

It also finds application in educational assessment (as shown above) to align the scores of the students to a normal distributionNormal DistributionNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more. However, the technique can’t handle outliers very well, which one of its primary limitations.

You can download this Normalization Formula Excel Template from here – Normalization Formula Excel Template

This has been a guide to Normalization Formula. Here we discuss how to normalize the given values along with examples and a downloadable excel template. You can learn more about statistical modeling from the following articles –

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