Understanding Normal Distribution
📌 What is Normal Distribution (Gaussian Distribution)?
The normal distribution (or Gaussian distribution) is a type of continuous probability distribution for a real-valued random variable. It describes how many natural phenomena and errors in measurements are distributed. The graph is symmetric and bell-shaped.
📚 This post is part of the "Intro to Statistics" series
🔙 Previously: Mean, Variance, and Standard Deviation of Random Variables
📐 The Probability Density Function (PDF) for Normal Distribution
The equation for the PDF of a normal distribution is:
\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp \left( -\frac{(x - \mu)^2}{2\sigma^2} \right) \]
Where:
- \( \mu \) is the mean (location parameter) of the distribution, which defines where the peak of the bell curve is located.
- \( \sigma \) is the standard deviation (shape parameter), which controls the width of the bell curve.
- \( \exp \) is the exponential function, describing how particles or phenomena distribute themselves in nature (e.g., diffusion).
This equation connects the statistical world to real-world distributions.
📊 Understanding the Equation
This equation is an exponential function and, after standardization, it describes how the values are distributed symmetrically around the mean.
- The area under the curve represents the total probability, and the sum of all probabilities equals 1.
- The variable \( x \) can take any value from \( -\infty \) to \( +\infty \), meaning the distribution extends infinitely in both directions.
🔄 Important Characteristics of Normal Distribution
- \( \mu \) describes the location of the distribution, i.e., where the center of the bell curve lies.
- \( \sigma \) defines the shape of the distribution, i.e., how spread out the values are around the mean.
- The probability for any given range can be found using the cumulative distribution function (CDF).
🧮 Example of Normal Distribution
For any normal distribution:
- 68% of values lie between \( \mu - \sigma \) and \( \mu + \sigma \).
- 95% of values lie between \( \mu - 2\sigma \) and \( \mu + 2\sigma \).
- 99.7% of values lie between \( \mu - 3\sigma \) and \( \mu + 3\sigma \).
📈 Visualizing the 68%, 95%, and 99.7% Rule
Here’s a visual showing the 68%, 95%, and 99.7% areas under the curve:
📝 How to Calculate Probabilities Using Normal Distribution
To calculate the probability that a variable \( X \) lies within a specific range:
- We use the Cumulative Distribution Function (CDF), which gives the area under the curve from \( -\infty \) to a specified \( x \).
🤖 Why It Matters for Machine Learning
- In Linear Regression, residuals are ideally normally distributed — this ensures valid confidence intervals and hypothesis tests.
- Gaussian Naive Bayes classifier assumes features are normally distributed within each class.
- Many statistical tests (like t-tests or ANOVA) assume normality — often used in feature selection.
- The Central Limit Theorem justifies normal approximations in ensemble learning, bootstrapping, and model evaluation.
🧠 Level Up: Understanding the Normal Distribution in Detail
- The normal distribution is foundational in statistics. It is used in hypothesis testing, confidence intervals, and in many natural and social sciences.
- The 68-95-99.7 rule: This empirical rule highlights the percentage of data that falls within 1, 2, and 3 standard deviations from the mean.
- The central limit theorem suggests that, regardless of the original distribution of data, the sampling distribution of the sample mean will approximate a normal distribution as the sample size increases.
- In practice, many natural phenomena and errors in measurement follow a normal distribution because of the law of large numbers.
✅ Best Practices for Normal Distribution
- Check if your data is approximately symmetric before assuming normality.
- Use Q-Q plots or histograms to assess normality visually.
- Apply normal distribution when dealing with large samples (thanks to CLT).
- Understand when standardizing (Z-scores) is appropriate.
⚠️ Common Pitfalls
- ❌ Assuming data is normal without checking (especially for small samples).
- ❌ Using normal distribution with categorical or non-continuous data.
- ❌ Confusing the normal distribution with the uniform distribution.
- ❌ Misinterpreting the standard deviation as covering 100% of data.
📌 Try It Yourself: Normal Distribution
Q1: What is the normal distribution also called?
💡 Show Answer
✅ It’s also known as the Gaussian distribution.
Named after Carl Friedrich Gauss, who helped develop the mathematical theory behind it.
Q2: What does the standard deviation \( \sigma \) control in a normal distribution?
💡 Show Answer
✅ It controls the spread (width) of the bell curve.
A larger \( \sigma \) means a wider curve; a smaller \( \sigma \) results in a tighter, narrower shape.
Q3: What percentage of values fall between \( \mu - 3\sigma \) and \( \mu + 3\sigma \)?
💡 Show Answer
✅ Approximately 99.7% of values fall within this range.
This is part of the Empirical Rule for normal distributions.
Q4: How about the range \( \mu - 2\sigma \) to \( \mu + 2\sigma \)?
💡 Show Answer
✅ Around 95% of values fall in this range.
This is commonly used for confidence intervals in statistics.
Q5: What does the cumulative distribution function (CDF) tell us?
💡 Show Answer
✅ The CDF gives the probability that a random variable is less than or equal to a certain value.
It's useful for computing probabilities over ranges instead of exact points.
Q6: How much of the distribution lies within one standard deviation from the mean?
💡 Show Answer
✅ About 68% of the data lies between \( \mu - \sigma \) and \( \mu + \sigma \).
This forms the central region of the normal curve.
📏 Summary of Key Points
- The normal distribution is symmetric and bell-shaped.
- The mean \( \mu \) determines the location of the peak.
- The standard deviation \( \sigma \) controls the spread.
- 68% of values lie within one standard deviation (\( \mu \pm \sigma \)).
- 95% of values lie within two standard deviations (\( \mu \pm 2\sigma \)).
- 99.7% of values lie within three standard deviations (\( \mu \pm 3\sigma \)).
🔜 Up Next
Next, we’ll explore the Z-Distribution — a standardized version of the normal distribution that is used to calculate probabilities and percentiles.
Stay tuned!
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