Solving for the Area Under the Curve

Learn how to use the normal distribution to answer questions about proportion, percentage, probability, and percentile rank. Use the controls below to modify the Question Asked. Then observe the resulting changes in the graph and the step-by-step solution guide. This demonstration helps with both visualizing and understanding the logic for finding the area under the curve

Keyboard shortcuts: Tab to move, Space to choose radios, and arrow keys to adjust the slider.

Normal Curve Visualization

Normal curve with mean and chosen value markers, dynamic shaded area, and accessible axes. A bell curve centered at the mean μ. A vertical line marks μ, and another marks the chosen value x. The shaded region shows the area above or below x. Labels show x, z, and which z-table applies. Proportion μ x = 40 z = 0.00 .500

Steps

Question: What proportion of the distribution N(50, 10) is below the value of 40?

Solution: For x ≤ 41, the corresponding proportion under the curve is 0.184.

All controls are keyboard reachable. Use Tab to move focus; arrow keys adjust the slider; Space selects radio items.

Value (x)
40
Mean μ defaults to 50 and standard deviation σ to 10. Use arrow keys for fine changes.
Shading choice
“ Below ” shades left of x; “ Above ” shades right of x.
Output type
Percentages/percentiles show with one decimal (e.g., 84.1%). Proportions/probabilities show three decimals (e.g., 0.841).