To normalize data using standard deviation, you subtract the mean of the data from each data point and then divide the result by the standard deviation of the data.
This process is also commonly known as Standardization or Z-score normalization. It transforms the data so that it has a mean of 0 and a standard deviation of 1.
The Standardization Process
The fundamental steps for normalizing data using standard deviation are:
-
Calculate the mean (µ) of the data you want to normalize.
-
Calculate the standard deviation (σ) of the same data.
-
For each individual data point ($x$) in your dataset, apply the following formula to get the normalized value ($x_{normalized}$):
$x_{normalized} = \frac{x - µ}{σ}$
As stated in the provided reference, the data can be normalized by subtracting the mean (µ) of each feature and a division by the standard deviation (σ). This way, each feature has a mean of 0 and a standard deviation of 1.
Why Normalize Data This Way?
Standardizing data using the mean and standard deviation is a powerful technique used in many fields, particularly in statistics, machine learning, and data analysis.
- Scalability: It brings all features onto a similar scale, regardless of their original units.
- Improved Algorithm Performance: Many machine learning algorithms perform better or converge faster when features are on a similar scale. The reference notes that this process results in faster convergence.
- Comparability: Normalized features can be compared more easily.
- Centering Data: The process centers the data around zero, which is beneficial for algorithms sensitive to the scale and distribution of the data.
Practical Application Example
A common application mentioned in the reference is in machine vision, where each image channel is normalized this way. For instance, in an RGB image, the pixel values for the Red, Green, and Blue channels might be normalized separately using the mean and standard deviation of the values within each respective channel across the image or a dataset of images.
Consider a simple dataset representing the heights of a group of people:
| Original Height (cm) |
|---|
| 160 |
| 165 |
| 170 |
| 175 |
| 180 |
Let's calculate the mean (µ) and standard deviation (σ) for this simple data:
- µ = (160 + 165 + 170 + 175 + 180) / 5 = 170 cm
- σ ≈ 7.07 cm
Now, normalize each data point:
- 160: $(160 - 170) / 7.07 \approx -1.41$
- 165: $(165 - 170) / 7.07 \approx -0.71$
- 170: $(170 - 170) / 7.07 = 0$
- 175: $(175 - 170) / 7.07 \approx 0.71$
- 180: $(180 - 170) / 7.07 \approx 1.41$
The normalized heights would be approximately: -1.41, -0.71, 0, 0.71, 1.41. These normalized values now have a mean close to 0 and a standard deviation close to 1, centered around the average height.
This standardization process is a fundamental step in preparing data for many analytical and machine learning tasks.
