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Normal Distribution: Areas to the Left and Right of x

Module by: Dr. Barbara Illowsky, Susan Dean

The arrow in the graph below points to the area to the left of xx. This area is represented by the probability P ( X < x ) P ( X x ) . Normal tables, computers, and calculators provide or calculate the probability P ( X < x ) P ( X x ) .

Normal distribution curve with a x value on the x-axis. The x-axis is equal to X. A vertical upward line extends from point x to the curve and the probability area occurs from the beginning of the curve to point x.

The area to the right is then P ( X > x ) = 1 - P ( X < x ) P ( X x ) = 1 - P ( X x ) .

Remember, P ( X < x ) = P ( X x ) = Area to the left of the vertical line through xx.

P ( X > x ) = 1 - P ( X < x ) = P ( X x ) = 1 - P ( X x ) =. Area to the right of the vertical line through xx

P ( X < x ) P ( X x ) is the same as P ( X ≤ x ) P ( X x ) and P ( X > x ) P ( X x ) is the same as P ( X ≥ x ) P ( X x ) for continuous distributions.

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