In statistics and probability, a Distribution is a function that shows the possible values for a variable and how often they occur. It essentially describes the "shape" of the data.
If you roll a die 1,000 times, plot the heights of everyone in a city, or track the lifespan of a lightbulb, you will see patterns emerge. These patterns are distributions. Understanding them allows us to predict future events and identify outliers.
1. Discrete vs. Continuous Distributions
The first step in understanding distributions is knowing what kind of data you have.
Discrete Distributions
These deal with data that can be counted (whole numbers). There are gaps between values.
- Example: Coin flips, rolling dice, number of students in a class.
- Key Distribution: The Binomial Distribution (Success vs. Failure scenarios).
Continuous Distributions
These deal with data that is measured. Values can be any number on a scale, including decimals.
- Example: Height, weight, temperature, time.
- Key Distribution: The Normal Distribution (The Bell Curve).
2. The Normal Distribution (Bell Curve)
This is the "King" of all distributions. In nature, most data tends to cluster around a central average, with fewer examples as you move away from the center.
[Image of standard normal distribution bell curve]Key properties of the Normal Distribution:
- Symmetry: The left side is a mirror image of the right.
- Center: The Mean, Median, and Mode are all at the exact center peak.
- 68-95-99.7 Rule: 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
3. Skewed Distributions
Real life isn't always perfect. Sometimes data "leans" to one side. This is called Skewness.
Positively Skewed (Right Skewed)
The "tail" of the graph extends to the right (towards positive numbers). The bulk of the data is on the left.
Example: Income distribution (most people earn a moderate amount, but a few billionaires pull the tail to the right).
Negatively Skewed (Left Skewed)
The "tail" extends to the left. The bulk of the data is on the right.
Example: Age at retirement (most people retire at 65+, very few retire at 20).
4. The Uniform Distribution
Imagine a situation where every outcome is equally likely. This creates a flat, rectangular shape known as a Uniform Distribution.
- Example: Rolling a single fair die. The chance of getting a 1, 2, 3, 4, 5, or 6 is exactly the same (1/6).
- There is no "peak" because no value is more popular than another.
Conclusion
Distributions are the blueprints of probability. By identifying whether data follows a Normal, Skewed, or Uniform distribution, mathematicians can calculate the likelihood of future events and make sense of chaotic data sets.