Signed Error in a Rectangle Estimate

Why the Sign of an Error Matters

A rectangle sum is an estimate of an accumulated quantity. Knowing only how far the estimate is from the actual amount may not be enough: it can also matter whether the estimate is too large or too small. For example, an estimate of accumulated distance that is too high may lead to a different decision from an estimate that is too low.

The sign records this direction. To use it consistently, we first choose an order for subtracting the quantities. In this tutorial, the error is defined as estimate minus actual accumulation. With that convention, a positive error means an overestimate, and a negative error means an underestimate.

This convention applies whether the rate is positive or negative. For a rate below zero, the accumulated contribution is negative. The comparison still concerns whether the rectangle's signed contribution is greater or less than the graph's signed contribution.

Comparing One Rectangle with the Graph

Suppose a subinterval has width \(\Delta x\), and the rectangle uses the sample value \(f(c)\) as its height. The rectangle contributes \(f(c)\Delta x\). Let \(A\) denote the exact signed accumulation represented by the graph over that same subinterval. The local error is therefore

Geometrically, the difference is the signed area between the horizontal top of the rectangle and the graph. If the rectangle is above the graph throughout the subinterval, its contribution is greater, so \(e\) is positive. If the graph is above the rectangle throughout, \(e\) is negative. If the graph and rectangle cross, parts of the difference have opposite signs; their effects may cancel, so a visual comparison at just one point does not determine the local error.

These comparisons concern the signed vertical values, not the distance from the graph to the horizontal axis. For example, a rectangle at height \(-2\) is above a graph at height \(-3\), even though both are below the axis. The rectangle's signed contribution is then greater, because \(-2\Delta x>-3\Delta x\) when \(\Delta x>0\).

From Local Errors to Total Error

Divide the interval into adjacent subintervals. On each one, the rectangle contribution and actual accumulation can be compared separately. The estimated total is the sum of the rectangle contributions, and the actual total is the sum of the actual contributions. Subtracting the actual total from the estimate gives the sum of the local differences.

Here \(R_i\) is the rectangle contribution, \(A_i\) is the exact accumulation, and \(e_i=R_i-A_i\) is the signed local error on the \(i\)th subinterval. This identity explains why it is important to retain signs. Positive and negative local errors can cancel; adding only their magnitudes would not give the error of the total estimate.

Worked Example: A Rectangle Below an Increasing Graph

Worked Example: Local Errors Can Partly Cancel

Worked Example: A Rectangle Above a Graph Below the Axis

A Practical Method

For a rectangle estimate over several subintervals, organize the comparison one interval at a time. This prevents an error on one subinterval from hiding an error of the opposite sign elsewhere.

Common Mistakes

A frequent mistake is to reverse the subtraction without changing the interpretation of the sign. In the first worked example, the estimate is \(2\) and the actual amount is \(4\). The convention here gives

Writing \(4-2=2\) calculates actual minus estimate instead. That number is not the signed error defined here. It describes the same difference in the opposite order, so it has the opposite sign. Always state the subtraction order before interpreting the result.

Another mistake is to assume that a particular sampling rule always produces an overestimate or always produces an underestimate. The direction depends on the graph's position relative to each rectangle, not just on whether a sample is at the left, right, or midpoint. A left-endpoint rectangle underestimates on the increasing line in the first example, but overestimates on the decreasing segment in the second example.

Finally, if the graph crosses the rectangle height within a subinterval, do not decide the sign by checking only the sample point or one endpoint. The part of the graph above the rectangle contributes error in one direction, and the part below contributes in the other. Their signed effects must be compared or calculated before the local sign is known.

Why Signed Errors Are Useful

Signed errors retain information that absolute differences discard. In the second example, the first rectangle is \(1\) unit too high and the second is \(\frac{1}{2}\) unit too low. The total estimate is not wrong by \(1+\frac{1}{2}=\frac{3}{2}\); the two local errors partly cancel, leaving a total error of \(\frac{1}{2}\). This distinction matters whenever an estimate combines contributions from multiple time periods, intervals, or pieces of data.

When only the direction of the total error is needed, a useful first check is whether every rectangle lies on the same side of the graph over its subinterval. If so, the local errors have the same sign, and their sum has that sign too. When the rectangles lie on different sides, local comparisons still help, but the signed amounts must be combined before the total estimate can be classified.