The formula for calculating probability given the mean and standard deviation is:
P(x) = (1 / (σ√(2π))) * e^(-((x-μ)^2) / (2σ^2))
Where P(x) is the probability, σ is the standard deviation, μ is the mean, and e is the base of natural logarithm.
The probability value represents the likelihood of a particular event occurring. A higher probability value means that the event is more likely to occur, while a lower probability value means that the event is less likely to occur.
The mean and median are both measures of central tendency in a probability distribution. The mean represents the average value of all the data points, while the median represents the middle value when the data is arranged in ascending or descending order. In a symmetrical distribution, the mean and median will be the same. In a skewed distribution, the mean will be affected by extreme values, while the median will not be affected.
The standard deviation measures the spread of data around the mean. A larger standard deviation indicates that the data points are more spread out, resulting in a wider and flatter probability distribution. A smaller standard deviation means that the data points are closer to the mean, resulting in a narrower and taller probability distribution.
The Empirical Rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. This rule is significant because it allows us to make predictions about the likelihood of events occurring and to understand the spread of data in a probability distribution.