How to Find Probability with Mean and Standard Deviation?

Last Updated : 22 Feb, 2024

Answer: To find probability using the mean (μ) and standard deviation (σ) in a normal distribution, use the z-score formula:

$P(X \leq x) = \Phi\left(\frac{x - \mu}{\sigma}\right)$

Mean and Standard Deviation:
- In a normal distribution, the mean (μ) represents the central tendency of the distribution, indicating the average value around which the data are centered.
- The standard deviation (σ) measures the dispersion or spread of the data points around the mean. It quantifies the variability in the data.
Z-Score:
- The formula uses the concept of a z-score, which represents the number of standard deviations a data point is from the mean.
- The z-score of a data point x is calculated as
  $z = \frac{x - \mu}{\sigma}$
- It standardizes the data to a standard normal distribution with a mean of 0 and a standard deviation of 1.
Standard Normal Cumulative Distribution Function (Φ):
- The functionΦ(z) represents the cumulative distribution function (CDF) of a standard normal distribution.
- It gives the probability that a standard normal random variable is less than or equal to a given value z.
- By plugging in the z-score calculated from the given mean and standard deviation, we can determine the probability associated with the given value x.
Interpretation:
- The equation calculates the probability that a random variable X is less than or equal to a specific value x in a normal distribution with mean μ and standard deviation σ.
- It helps assess the likelihood of observing values below a certain threshold in a normal distribution.
- The resulting probability provides insight into the relative position of the value x within the distribution and can be used for various statistical analyses and decision-making processes.
Application:
- This equation is commonly used in statistics, probability theory, and data analysis to calculate probabilities associated with normal distributions.
- It is particularly useful in areas such as hypothesis testing, confidence interval estimation, and risk assessment.