Today, let’s explore the fascinating combination of AI and statistics through visual examples generated by the AI analysis tool Bayeslab, with a focus on the most common statistical concept: the normal distribution.
Following the five questions about the normal distribution introduced in the previous template,
1.why normal distribution is important?
2.When normal distribution is used?
3.Normal distribution looks like
4. Normal distribution where is the mean?
5.Normal distribution with standard deviation
we will continue to introduce the following issues in this template:
6. What is the density function?
7. Normal distribution with percentages
8. What is the 68–95–99.7 rule?
9. Normal distribution to standard normal distribution
10. Normal distribution with z-scores
6. What is density function？
In a normal distribution, probabilities in specific areas are expressed as percentages.
(1) Discrete Variables vs. Continuous Variables
The normal distribution is a probability distribution for continuous variables. For example, flipping a coin, which has only two possible outcomes, is considered a discrete variable. Rolling a die, even with six outcomes, is also a discrete variable because the results are limited.
In contrast, continuous variables have an infinite number of possible values between any two numbers. For example, height can have intermediate values between 180 cm and 180.00001 cm, making it a continuous variable.
Summary:
Discrete variables: Have a limited number of possible outcomes, like flipping a coin or rolling a die.
Continuous variables: Have an infinite number of possible outcomes, like height, weight, scores, or time.
For continuous random variables, the probability of falling within a certain interval — such as having a probability above 90% in a food-eating contest — is represented as P{X ≥ 90}.
(2) Probability Density Function — Definition
If the area under the curve of function f(x) and the x-axis equals 1 (or 100%), then f(x) is the probability density function (PDF) of a continuous random variable X. The PDF for a normal distribution is given by:
If the density function of a random variable ( X ) is given by the above formula, then ( X ) is said to follow a normal distribution
denoted as X~N(μ,σ²) ,
8. what is the 68 95 99.7 rule？
Assuming X~N(μ,σ²), the normal distribution’s 68–95–99.7 rule describes the distribution of data around the mean:
- Approximately 68% of data points lie within ±1 standard deviation of the mean,
- About 95% lie within ±2 standard deviations,
- About 99.7% lie within ±3 standard deviations.
These percentages help assess the likelihood of observing data points within specific ranges.
Visualization Example:
Different colored areas below represent the regions formed by ±1σ, ±2σ, and ±3σ with the x-axis, corresponding to 68%, 95%, and 99.7%, respectively, illustrating the 68–95–99.7 rule.
μ ± σ² Rule: For normally distributed variables, confidence intervals are determined by the mean μ and variance σ². We will discuss confidence intervals in detail in later templates.
9. Normal distribution to standard normal distribution
Converting a normal distribution to a standard normal distribution is achieved through a process called standardization
Previously, we mentioned that the density function of a normal distribution is given by the formula shown below, denoted as X~N(μ,σ²).
By setting the mean ( μ = 0 ) and the standard deviation (σ= 1 ), the normal distribution becomes a standard normal distribution, denoted as X~N(0,¹²).
Its density function formula can be further simplified as shown below:
10.Normal distribution with z scores
The process of converting a normal distribution to a standard normal distribution involves standardization.
Specifically, each data point is subtracted by the mean and then divided by the standard deviation, resulting in a new distribution with a mean of 0 and a standard deviation of 1.
This is called standardization. The benefit of standardization is that it allows probability calculations and other standardized analyses on the standard normal table, regardless of the original data.
The standardization process uses the Z-score.
Z-scores measure the relative position of data points within a normal distribution.
Z-scores indicate how many standard deviations a data point is from the mean.
The calculation formula is: (X — μ) / σ
where:
X is the data point,
μ is the mean,
σ is the standard deviation.
Example: Often, raw data cannot reflect an individual’s position within a group. For instance, a height of 170 cm needs comparison to the height distribution of different groups, such as those based on ethnicity or region.
Thus, we convert the raw score to a Z-score, or standard score.
After conversion, the sign and magnitude of the Z-score can reflect an individual’s position within the group. For example, with a height of 170 cm:
If Z=0, the height is exactly at the average level.
If Z=1, the height is one standard deviation above the mean, indicating that about 84.134% of people are shorter (0.5 + 0.34134 = 0.84134).
If Z=2, the height is two standard deviations above the mean, indicating that about 97.725% of people are shorter (0.5 + 0.47725 = 0.97725).
What about negative values?
If Z=-1, the height is one standard deviation below the mean, indicating that about 15.866% of people are shorter (0.5–0.34134 = 0.15866).
If Z=-2, the height is two standard deviations below the mean, indicating that about 2.275% of people are shorter (0.5–0.47725 = 0.02275).
📌 That’s right, do you remember the 68–95–99.7 rule we introduced earlier?
Z-score conversion helps standardize data, allowing for the use of the standard normal table (Z-table) to find probabilities or to make comparisons across different distributions.
See the illustration below:
(Note: The Z-score standardization process assumes that the data distribution is normal. Otherwise, consulting the Z-table for group position is meaningless. For how to test the normality of data, please refer to the later sections.)
In the next template, we will specifically introduce other related issues about the normal distribution:
11. Which test for normal distribution?
12.Can normal distribution be skewed?
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