Determining Probabilities and Percents using z-scores

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This graph can be used to determine the probability of picking a data point, x, within a certain number of standard deviations from the mean. But what is the probability of choosing a random data point within, say, less than half of one standard deviation of the mean?

This probability is unclear. Here, the standard normal distribution, which is a normal distribution with a mean $\mu =0$ and standard deviation $\sigma =1,$ can be useful. The corresponding curve can be drawn.

For a standard normal distribution, the numbers under the horizontal axis are denoted z.

The left-hand column gives the whole number portion of z, while the top row gives the decimal part of z.
Consider finding the probability that a randomly chosen data point is less than or equal to z=-0.5. First, locate the whole number of the z-score in the left-hand column. In this case that is -0.

The probability that corresponds to a z-score of $\text{-}0\dots$ appears in the shaded row above. To determine exactly which cell, consider the decimal portion of z in the top row. Here, that is .5.

Note that there are other standard normal tables. For instance, one might give the probability of a value being greater than a specific z-score.