Other normal probabilities

In the previous page, we showed how z-scores could be used to find the probability of getting a value less than x in a normal distribution. Other probabilities can be similarly translated into ones involving z-scores.

Probability of higher value

The following diagram is similar to that on the previous page, but finds the probability of getting a value above the x-value rather than below it.

Confirm that ...

  • When µ = 6.1 and σ = 0.3, P(X > 6.5) = 0.091
  • When µ = 198 and σ = 1.2, P(X > 200) = 0.048

Probability of being between two values

The final diagram asks for the probability of getting a value between two x-values. This is the area under the standard normal probability density between the two corresponding z-scores.

Confirm that ...

  • When µ = 20 and σ = 1, P(19 < X < 22) = 0.82
  • When µ = 156 and σ = 8.7, P(160 < X < 165) = 0.17

Evaluating other probabilities

Statistical software and tables can easily evaluate the probability of getting a z-score less than any specified value. It takes a little more thought and work to find other probabilities.

Translating probabilities into ones about the probability of lower z-scores is relatively easy if you keep in mind the following two facts.

Probability of higher value

The probability of getting a value greater than x can be evaluated as one minus the probability of a value less than x.

This conversion can be done either before or after translating the required probability from x-values to z-scores.

Probability of value between two others

The probability of getting a value between x1 and x2 can be evaluated as the difference between the probabilities of values less than x1 and x2.

Again, the conversion can be done either before or after translating the required probability from x-values to z-scores.