How to Find Probability with Mean and Standard Deviation? Last Updated : 22 Feb, 2024 Improve Improve Like Article Like Save Share Report Answer: To find probability using the mean (μ) and standard deviation (σ) in a normal distribution, use the z-score formula:[Tex]P(X \leq x) = \Phi\left(\frac{x – \mu}{\sigma}\right) [/Tex]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 [Tex]z = \frac{x – \mu}{\sigma} [/Tex]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. Like Article Suggest improvement Next How To Find Mean of Probability Distribution? Share your thoughts in the comments Add Your Comment Please Login to comment...