Normal Distribution Calculator
The normal distribution, also known as the Gaussian distribution or "bell curve," is a fundamental concept in statistics and probability theory. It describes how the values of a variable are distributed, with most values clustering around the mean and tapering off symmetrically towards the extremes. This calculator helps you understand probabilities associated with a given normal distribution.
Single Point Probability
Calculate probability density and cumulative probabilities for a specific X value.
Range Probability
Calculate the probability that X falls between two values (X1 and X2).
Understanding the Normal Distribution
The normal distribution is characterized by two parameters: the mean (μ) and the standard deviation (σ). The mean represents the center of the distribution, while the standard deviation measures the spread or dispersion of the data. A smaller standard deviation indicates that data points are clustered closely around the mean, while a larger standard deviation suggests data points are more spread out.
Key Characteristics:
- Symmetry: The distribution is perfectly symmetrical around its mean.
- Mean, Median, Mode: For a normal distribution, the mean, median, and mode are all equal and located at the center of the curve.
- Asymptotic: The tails of the curve approach the x-axis but never quite touch it, extending infinitely in both directions.
- Empirical Rule (68-95-99.7 Rule): Approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
Applications:
Normal distributions are widely used in various fields because many natural phenomena and measurements tend to follow this pattern. Examples include:
- Heights and weights of a population
- Measurement errors in experiments
- Test scores (e.g., IQ scores)
- Financial market returns (often approximated)
- Quality control in manufacturing
How to Use the Calculator
Enter the mean (μ) and standard deviation (σ) of your normal distribution. Then, you can perform different types of probability calculations:
- Single Point Probability (X Value): Enter a specific 'X Value' to find the probability density at that point and the cumulative probabilities P(X ≤ x) and P(X ≥ x).
- Range Probability (X1 and X2 Values): Enter 'X1 Value' and 'X2 Value' to find the probability P(X1 ≤ X ≤ X2), which is the area under the curve between X1 and X2.
Interpreting the Results:
- Probability Density (PDF): This value represents the height of the curve at a specific X. It's not a probability itself, but rather a measure of how likely a value is to occur near X. Higher density means higher likelihood.
- P(X ≤ x): This is the cumulative probability that a randomly selected value from the distribution will be less than or equal to your specified X Value. It represents the area under the curve to the left of X.
- P(X ≥ x): This is the cumulative probability that a randomly selected value will be greater than or equal to your specified X Value. It represents the area under the curve to the right of X.
- P(X1 ≤ X ≤ X2): This is the probability that a randomly selected value will fall between X1 and X2. It represents the area under the curve between X1 and X2.
Examples
Let's consider a scenario where the heights of adult males in a certain population are normally distributed with a mean (μ) of 175 cm and a standard deviation (σ) of 7 cm.
Example 1: Probability of a specific height range
What is the probability that a randomly selected adult male has a height between 168 cm and 182 cm?
- Mean (μ): 175
- Standard Deviation (σ): 7
- X1 Value: 168
- X2 Value: 182
Calculation:
Z1 = (168 – 175) / 7 = -1
Z2 = (182 – 175) / 7 = 1
P(168 ≤ X ≤ 182) = P(Z ≤ 1) – P(Z ≤ -1) ≈ 0.8413 – 0.1587 = 0.6826
Result: Approximately 68.26%. This aligns with the empirical rule, as 168 cm and 182 cm are one standard deviation away from the mean.
Example 2: Probability of being taller than a certain height
What is the probability that a randomly selected adult male is taller than 190 cm?
- Mean (μ): 175
- Standard Deviation (σ): 7
- X Value: 190
Calculation:
Z = (190 – 175) / 7 ≈ 2.14
P(X ≥ 190) = 1 – P(Z ≤ 2.14) ≈ 1 – 0.9838 = 0.0162
Result: Approximately 1.62%.