Calculating the Probability of a Number Being More or Less than the Mean in a Normal Distribution

What is the probability that any random number is more/less than the mean for normal distribution?

For a normal distribution, about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99

For a normal distribution, about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.

To calculate the probability of a randomly chosen number being more or less than the mean, we need to consider the standard deviation.

If we assume a standard normal distribution (mean = 0, standard deviation = 1), then the probability of a randomly chosen number being greater than the mean is the same as the probability of a standard normal variable being greater than zero. This probability is 0.5, or 50%.

In general, if we have a normal distribution with mean μ and standard deviation σ, we can calculate the probability of a randomly chosen number being more/less than the mean using Z-scores.

Let’s say we want to calculate the probability P(X > μ), where X is a normally distributed random variable with mean μ and standard deviation σ. We can calculate the Z-score using the formula:

Z = (X – μ) / σ

Once we have the Z-score, we can look up the corresponding probability in a standard normal distribution table or use statistical software.

For example, if we have a normal distribution with mean μ = 10 and standard deviation σ = 2, and we want to calculate P(X > 10), we can calculate the Z-score as follows:

Z = (X – μ) / σ
Z = (X – 10) / 2

Suppose we find that the Z-score is 0.75. To find the probability using a standard normal distribution table, we would look up the area to the right of 0.75. Let’s assume the table value is 0.7734.

We can subtract this probability from 1 to find the probability of the number being more than the mean:

P(X > 10) = 1 – 0.7734 = 0.2266

Therefore, the probability that a randomly chosen number from this specific normal distribution is more than the mean of 10 is approximately 0.2266, or 22.66%.

Similarly, we can also calculate the probability of a number being less than the mean using the same approach.

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