Why Unequal Observation Times Require Care
When a rate is recorded at several times, it is tempting to combine the recorded values as if each one described the same amount of time. That is only appropriate when the time intervals are equal. If observations are unevenly spaced, a rate associated with a long interval must contribute for longer than a rate associated with a short interval.
For instance, a speed of \(5\) metres per second maintained for \(3\) seconds corresponds to more displacement than the same speed maintained for \(1\) second. A useful estimate of accumulated change therefore has two ingredients: a rate sample for each interval and the duration of that interval.
Intervals, Samples, and Durations
Suppose observations are made at increasing times \(t_0,t_1,\ldots,t_n\). Consecutive observations divide the time span into intervals: from \(t_0\) to \(t_1\), from \(t_1\) to \(t_2\), and so on. The duration of the interval from \(t_{i-1}\) to \(t_i\) is \(t_i-t_{i-1}\). These durations need not be equal.
Let \(r(t)\) be the rate being observed. To estimate accumulation on each interval, choose a representative time within it and use the rate there. Three common choices are:
- Left sample: use the rate at the interval’s starting time, \(r(t_{i-1})\).
- Right sample: use the rate at the interval’s ending time, \(r(t_i)\).
- Midpoint sample: use the rate halfway through the interval, at time \((t_{i-1}+t_i)/2\).
If the chosen sample time in interval \(i\) is \(s_i\), the interval’s estimated contribution is \(r(s_i)(t_i-t_{i-1})\). The estimate over the full span is the sum of these separate contributions:
The sample time \(s_i\) can be a left endpoint, a right endpoint, or a midpoint, depending on the method and the available data. The duration paired with it must always be the length of that same interval.
Why Each Rate Must Be Weighted by Its Own Duration
The rule comes from what a rate means. If a rate is constant over an interval, the change over that interval is exactly rate multiplied by duration. For example, a steady flow of \(4\) litres per minute for \(2\) minutes delivers \(4\cdot2=8\) litres.
For rates that vary, a rectangle estimate treats the selected sample as if it represented the rate over its interval. The same multiplication then gives an estimate of the interval’s contribution. The logic can be stated directly:
Each pair of consecutive observation times determines one duration.
Use the left endpoint, right endpoint, or midpoint rate as specified.
Multiply that rate sample by the duration of its own interval.
The result estimates the total change across the full time span.
When the rate really is constant on each interval, the procedure gives the exact total: each product is the exact change on that interval, and adding the changes gives the change over the whole span. When rates vary within intervals, the products are estimates because a single sample stands in for the rates at other times in that interval.
Worked Example: Left and Right Samples at Uneven Times
Worked Example: Left and Right Samples at Uneven Times
A vehicle’s velocity is recorded as shown. Estimate its displacement from \(0\) to \(6\) seconds using left samples and then right samples.
| Time (s) | Velocity (m/s) |
|---|---|
| 0 | 2 |
| 1 | 5 |
| 4 | 1 |
| 6 | 3 |
The intervals are from \(0\) to \(1\), from \(1\) to \(4\), and from \(4\) to \(6\) seconds. Their durations are \(1-0=1\) second, \(4-1=3\) seconds, and \(6-4=2\) seconds.
For the left-sample estimate, use the velocity at the start of each interval: \(2\), \(5\), and \(1\) metres per second. Multiply each by its own duration:
For the right-sample estimate, use the velocity at the end of each interval: \(5\), \(1\), and \(3\) metres per second:
Check the pairings: the first interval has duration \(1\) second, the second has duration \(3\) seconds, and the third has duration \(2\) seconds in both estimates. Each product has units \((\text{m/s})(\text{s})=\text{m}\), as displacement should. The two estimates differ because they use different samples; neither calculation assigns equal duration to all three observations.
Choosing a Midpoint Sample
A midpoint sample is taken halfway through each interval in time. With unequal intervals, the midpoint times are different distances apart: each must be calculated from the endpoints of its own interval. If rates are actually measured at those midpoint times, those measurements can be used directly.
If only endpoint rates are available, a midpoint rate is not automatically known. It must be measured, supplied, or estimated using a stated model. For example, if the rate is assumed to vary linearly on an interval, the midpoint rate is the average of the two endpoint rates. Without such an assumption, averaging endpoint rates does not in general establish the rate at the midpoint.
Worked Example: Midpoint Measurements on Unequal Intervals
Worked Example: Midpoint Measurements on Unequal Intervals
A tank receives water at the following measured midpoint rates. Estimate the volume delivered from time \(0\) to time \(9\) minutes using midpoint samples.
| Interval (min) | Midpoint time (min) | Measured rate (L/min) |
|---|---|---|
| 0 to 2 | 1 | 4 |
| 2 to 5 | 3.5 | 3 |
| 5 to 9 | 7 | 6 |
The interval durations are \(2-0=2\) minutes, \(5-2=3\) minutes, and \(9-5=4\) minutes. The midpoint sample for each is already given, so multiply each measured rate by the duration of its interval:
Check the arithmetic: \(4\cdot2=8\), \(3\cdot3=9\), and \(6\cdot4=24\); \(8+9+24=41\). Each contribution is measured in litres because the minutes cancel. The estimated total is \(41\) litres. In particular, the \(6\) L/min measurement is associated with a \(4\)-minute interval, not with the same duration as the \(4\) L/min measurement.
Signed Rates and Net Change
Rates can be negative as well as positive. For example, velocity in a chosen direction is negative when motion is in the opposite direction. The products retain their signs, so adding them estimates net change, with positive and negative contributions able to cancel. Taking absolute values instead would estimate a different quantity.
Worked Example: Signed Velocity with Unequal Intervals
Worked Example: Signed Velocity with Unequal Intervals
A particle has the recorded velocities below. Estimate its displacement from \(0\) to \(7\) seconds using left samples.
| Time (s) | Velocity (m/s) |
|---|---|
| 0 | 3 |
| 2 | -1 |
| 5 | 2 |
| 7 | 1 |
The interval durations are \(2-0=2\) seconds, \(5-2=3\) seconds, and \(7-5=2\) seconds. The left samples are \(3\), \(-1\), and \(2\) m/s. Therefore:
Check the signed arithmetic: \(6-3+4=3+4=7\). All products have units of metres. The negative contribution represents estimated motion in the direction opposite to the chosen positive direction; it reduces the net displacement. The estimate is \(7\) metres, not the sum of the absolute contributions.
A Common Mistake: Giving Every Observation Equal Weight
Suppose the left-sample velocities in the first example are \(2\), \(5\), and \(1\) m/s. Adding them gives \(2+5+1=8\), but \(8\) is not a displacement: it is a sum of rates, with units of m/s. Even averaging them and multiplying by the total time would wrongly treat the three intervals as equally long.
For that example, the interval durations are \(1\), \(3\), and \(2\) seconds. The correct left-sample estimate is \(2\cdot1+5\cdot3+1\cdot2=2+15+2=19\) metres. The middle sample contributes for \(3\) seconds, while the first and last contribute for \(1\) and \(2\) seconds. Equal weighting discards this essential information.
How to Set Up an Unequal-Time Estimate
Before calculating, write down the intervals and their durations. Then identify which rate belongs to each interval under the chosen left, right, or midpoint rule. This makes it harder to accidentally use a time value or rate from a neighbouring interval.
- For a left estimate, use the rate at the start of every interval, including the first observation and excluding the final endpoint as a sample.
- For a right estimate, use the rate at the end of every interval, including the final observation and excluding the initial endpoint as a sample.
- For a midpoint estimate, find the time halfway through each individual interval and use a rate known or justified at that time.
- Multiply each sample by its own interval duration; do not give all observations the same weight unless the durations actually match.
- Keep negative signs when the rate represents signed change, and check that the final units are appropriate.
These steps turn unevenly spaced data into a transparent accumulation estimate. The central question is not just “Which rates were recorded?” but also “For how long does each selected rate represent the process?”
Check Your Understanding
Use the interval-specific durations in each setup, and pay attention to the meaning and units of the accumulated quantity.
- Observations occur at times \(0\), \(2\), and \(7\) seconds. What are the two interval durations?
- For a left-sample estimate on those intervals, which observation times supply the two rate samples?
- A rate is \(3\) L/min on a \(4\)-minute interval and \(5\) L/min on a \(1\)-minute interval. What is the corresponding two-interval rectangle estimate?
- When can a midpoint rate be used directly, and what additional assumption might be needed if only endpoint rates are known?
- Why does a sum of signed velocity contributions estimate net displacement rather than total distance traveled?
