Three scenarios, each with the inputs stated and the result computed. Every figure was calculated with the tools linked at the foot of the study, using the formula each tool documents, so it can be reproduced by entering the same values. They are illustrative examples, not client results, and they are not advice.
Two numerical methods on the same integral
A student has to approximate the integral of sin(x) between 0 and pi, where the exact answer is 2, and can choose between the trapezoidal rule and Simpson's rule. Both runs use the same number of subintervals.
Inputs: f(x) = sin(x) on the interval 0 to pi. Four subintervals first, then eight.
Result: At four subintervals the trapezoidal rule returns 1.896119, an error of 0.103881, while Simpson's rule returns 2.004560, an error of 0.004560. At eight subintervals the values are 1.974232 and 2.000269.
What it means: At the same node count Simpson's rule lands more than twenty times closer to the answer, so the choice of method buys more here than doubling the work does.
Worked with: Simpson's Rule Calculator, Trapezoidal Rule Calculator
A plot of land with three measured sides
A surveyor measures the three sides of a triangular plot as 13 m, 14 m and 15 m. There is no right angle to work from and no convenient height to measure.
Inputs: Sides a = 13 m, b = 14 m, c = 15 m.
Result: Heron's formula gives an area of 84 square metres and a perimeter of 42 m. The three angles are 53.13, 59.49 and 67.38 degrees.
What it means: An awkward-looking triangle comes out on an exact whole number of square metres, which is the reason Heron's formula earns its place when nothing can be measured from a right angle.
Worked with: Triangle Calculator, Area Calculator
Reading a poll that puts a candidate at 47%
A poll of 1,000 respondents puts one candidate at 47% and the report calls it a lead. The question is whether the sample supports that word.
Inputs: 1,000 respondents, a proportion of 47%, and 95% confidence. Then 500 and 1,500 respondents for comparison.
Result: The standard error is 1.58 percentage points, so the margin is 3.09 points and the interval runs from 43.91% to 50.09%. At 500 respondents the margin widens to 4.37 points, and at 1,500 it narrows to 2.53 points.
What it means: At 1,000 responses the interval crosses 50%, so the poll does not establish a lead, and the extra precision from a bigger sample arrives slowly.
Worked with: Margin of Error Calculator, Confidence Interval Calculator, Sample Size Calculator