1. Meaning of a Normal Distribution
A normal distribution is a continuous probability distribution whose graph looks like a smooth, symmetric bell-shaped curve. It shows how values are spread around the mean.
Many natural measurements such as height, weight, marks, and errors tend to follow this pattern.
1.1. Key Features
- Symmetric about the mean.
- Mean = Median = Mode.
- Most values lie close to the center.
- Very few values are extremely low or extremely high.
2. Bell Curve Shape
The curve rises gradually, reaches a peak at the mean, and then falls smoothly. The two tails extend endlessly but never touch the horizontal axis.
2.1. Interpretation
Values near the mean are most common, and values far from the mean are rare. This matches many real-world data patterns.
3. Probability Density Function of Normal Distribution
The PDF of a normal distribution with mean \( \mu \) and standard deviation \( \sigma \) is:
\( f(x) = \dfrac{1}{\sigma \sqrt{2\pi}} e^{-\dfrac{(x-\mu)^2}{2\sigma^2}} \)
This function gives the bell-shaped curve.
3.1. Meaning of Parameters
- \( \mu \): The center (mean) of the distribution.
- \( \sigma \): The spread or dispersion.
- Larger \( \sigma \) = wider curve; smaller \( \sigma \) = narrower curve.
4. Standard Normal Distribution
A standard normal distribution is a special case where:
\( \mu = 0, \quad \sigma = 1 \)
Any normal variable can be converted into this form using a z-score.
4.1. Z-Score
The z-score tells how many standard deviations a value is from the mean:
\( z = \dfrac{x - \mu}{\sigma} \)
Positive z → value above mean; negative z → value below mean.
5. Empirical Rule (68–95–99.7 Rule)
A normal distribution follows a well-known pattern for how data is spread around the mean:
- About 68% of values fall within \( \pm 1\sigma \)
- About 95% fall within \( \pm 2\sigma \)
- About 99.7% fall within \( \pm 3\sigma \)
This rule gives a quick idea of how spread-out the distribution is.
6. Real-Life Examples
- Heights of people in a region.
- Measurement errors in experiments.
- Marks in a test where many students perform around average.
- Blood pressure readings in a healthy population.