Estimating

Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data. The parameters describe an underlying physical setting in such a way that the value of the parameters affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.

For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the unobservable parameter; the estimate is based on a small random sample of voters.

Or, for example, in radar the goal is to estimate the location of objects (airplanes, boats, etc.) by analyzing the received echo and a possible question to be posed is "where are the airplanes?" To answer where the airplanes are, it is necessary to estimate the distance the airplanes are at from the radar station, which can provide an absolute location if the absolute location of the radar station is known.

In estimation theory, it is assumed that the desired information is embedded in a noisy signal. Noise adds uncertainty, without which the problem would be deterministic and estimation would not be needed.




Example Questions

(a) Estimate the cost of 21 packs of screws each costing £2.90.
The actual calculation is 21 × 2.9.
If we round the numbers to 1 significant figure we get 20 × 3, which we can do without a calculator.
Our estimated answer is £60 (which is quite close to the actual answer of £60.90).

(b) Estimate the length of 29 pieces of wood layed end-to-end if each is 1.48m long.
The actual calculation is 29 × 1.5.
If we round the numbers to 1 significant figure we get 30 × 1.
The estimated answer using 1 significant figure is 30m (which is quite different to the actual answer of 42.92m).
In this case, we need to be sensible about the rounding, for example we could calculate 30 × 1.5.
The estimated answer using more sensible rounding is 45m (which is much closer to the actual answer).

(c) James has worked out 31.5 × 49.6 and got the answer 156.24.
Use estimation to check whether his answer is likely to be correct.
If we round the numbers to 1 significant figure we get 30 × 50.
The estimated answer using 1 significant figure is 1500m (which is very different to the answer James got).
James has probably got the decimal point in the wrong place - the answer should be 1562.4


Practice Questions

(a) Estimate the weight of 38 boxes each weighing 26 kg.

(b) Jane has calculated 3.88 × 16.1 and got 62.468. Does her answer seem correct?




Answer :?