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Principles of Sampling and Statistical Errors

Principles of sampling, sampling vs non-sampling errors.

Notes

Principles of Sampling and Statistical Errors

Class 11 Statistics — Laws of Sampling, Types of Errors, Absolute and Relative Error

The Two Laws of Sampling

These principles are sometimes referred to as the 'Laws of Sampling'. They are based on Theory of Probability — a numerical measure of the possibility of occurrence or non-occurrence of an event.

Law of Statistical Regularity

If a random sample of adequate size is selected from a large population, it tends to possess the same characteristics as those of the population.

Implication: In a randomly selected sample, each and every item of population has an equal chance of being selected.

Example: If out of a population of 140 crores, a random sample of 500 persons is taken to estimate average height, according to this law the average height is likely to be approximately equal to the average of the entire population.

Law of Inertia of Large Numbers

The aggregates or averages obtained from a large group are more stable than those obtained from a small group. Larger the size of the sample, more accurate the results are likely to be.

This law is a corollary to the Law of Statistical Regularity. It is based on the psychological law of behaviour — the behaviour of a phenomenon on a large scale is generally stable.

Example: Production of mangoes in one district might show great variations year after year, but the production figures of the entire country would not vary much.

Statistical Errors — Concepts

Statistical Error
The difference between the collected data and actual value of facts. In other words, Error = Actual or True Value − Estimated Value.
In statistics, 'Error' is used in a specialised sense and should not be confused with Mistake. Mistake means wrong calculation or use of inappropriate method. On the other hand, Statistical Errors refer to difference between collected data and actual value.
Example of Mistake: If there are 500 workers in a factory and we count them as 495, that is a mistake. Example of Statistical Error: When all workers are gathered in a meeting and we approximate them as 503, that is a statistical error.

Sources of Statistical Errors

  • •Errors of Origin: Due to lack of proper definition of subject-matter, bias of investigator, defective questionnaire, improper sampling, or inherent instability of data.
  • •Error of Manipulation: Due to manipulation in counting, measurement, description and approximation of various figures.
  • •Error of Inadequacy: Due to use of incomplete or unrepresentative data or irresponsible/careless/unqualified investigators.
  • •Error of Interpretation: When data is misinterpreted.

Causes of Errors

  • ✗Selection of wrong samples.
  • ✗Incorrect information given by respondents.
  • ✗Collection of data by estimates.
  • ✗Personal prejudice of investigators.

Absolute Error and Relative Error

Absolute Error

$$\text{Absolute Error} = \text{Actual Value} - \text{Estimated Value}$$

Relative Error

$$\text{Relative Error} = \frac{\text{Actual Value} - \text{Estimated Value}}{\text{Actual Value}}$$
Relative Error is a much more useful measure than Absolute Error as it provides a useful coefficient (a pure number, independent of units of measurement) for comparing the degree of error of different sets of data.

Solved Example

Problem

A weighing machine states your weight as 75 kg, but you know your true weight is 74.5 kg. Calculate Absolute Error and Relative Error.

Solution

Absolute Error = 75 kg − 74.5 kg = 0.5 kg. Relative Error = 0.5/75 = 0.0066