Understanding Normal Distribution: Calculating Probability for Numbers Greater or Less Than the Mean

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

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

In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and around 99.7% falls within three standard deviations. This distribution follows a bell-shaped curve.

To determine the probability of any random number being more or less than the mean, we can look at the area under the curve outside of one standard deviation from the mean.

If we assume that the normal distribution is centered at zero (mean = 0) and has a standard deviation of 1, we can use a standard normal distribution table (also known as a Z-table) to find the probability.

Finding the probability of a random number being more than the mean (greater than zero) involves finding the area under the curve to the right of zero. The Z-table provides values for the area to the left of a given Z-score, so we need to find the Z-score for zero and subtract it from 1.

In the Z-table, the area to the left of a Z-score of zero is 0.5000. To find the area to the right of zero, we subtract 0.5000 from 1:

1 – 0.5000 = 0.5000

Therefore, the probability of a random number being greater than the mean (in a standard normal distribution) is 0.5000 or 50%.

Similarly, the probability of a random number being less than the mean (less than zero) is the same since the normal distribution is symmetrical. Therefore, the probability of a random number being less than the mean is also 0.5000 or 50%.

Please note that the specific probabilities may vary depending on the mean and standard deviation of the normal distribution being considered.

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