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Wednesday, January 16, 2008

Briefly explain relative frequency of occurrence in your own words

Relative Frequency of Occurrence:-

In statistics the frequency of an event i is the number ni of times the event occurred in the experiment or the study. These frequencies are often graphically represented in histograms.

We speak of absolute frequencies, when the counts ni themselves are given and of (relative) frequencies, when those are normalized by the total number of events:

f_i = \frac{n_i}{N} = \frac{n_i}{\sum_i n_i}

Taking the fi for all i and tabulating or plotting them leads to a frequency distribution.

The Relative Frequency Density of occurrence of an event is the relative frequency of i divided by the size of the bin used to classify i.

For example: If the lower extreme of the class you are measuring the density of is 15 and the upper extreme of the class you are measuring is 30, given a relative frequency of 0.0625, you would calculate the frequency density for this class to be:

Rel.freq / (Upper Extreme of Class - Lower Extreme of Class) = Density

0.0625 / (30 - 15) = 0.0625 / 15 = 0.0041666.. That is: 0.00417 to 5 S.F.

In biology, relative frequency is the occurrence of a single gene in a specific species that makes up a gene pool.

The Limiting Relative Frequency of an event over a long series of trials is the conceptual foundation of the frequency interpretation of probability. In this framework, it is assumed that as the length of the series increases without bound, the fraction of the experiments in which we observe the event will stabilize. This interpretation is often contrasted with Bayesian probability.

Frequency is the measurement of the number of occurrences of a repeated event per unit of time. It is also defined as the rate of change of phase of a sinusoidal waveform.

Relative Frequency of Occurrence:-

Question like : “What is the probability that I will live to be 85 ? or “ what are the chances that I will below one of my stereo speakers if I turn my 200 –watt amplifier up to wide open?” or “what is the probability that the location of a new paper plant on the river near our town will cause a substantial fish kill?. We quickly see that we may not be able to state in advance, without experimentation, what these probabilities are. Other approaches may be more useful.

In the 1800s, British statisticians, interested in a theoretical foundation for calculating risk of losses in life insurance and commercial insurance, began defining probabilities from statistical data collected on births and deaths. Today, this approach is called The relative frequency of Occurrence. It defines probability as either:

1. The observed relative frequency of an event in a very large number of trials, or

2. The proportion of times that an event occurs in the long run when conditions are stable.

This method used the relative frequencies of past occurrences as probabilities. We determine how often something has happened in the past and use that figure to predict the probability that it will happen again in the future.

For example. Suppose an insurance company knows from past actuarial data that of all males 40 years old, about 60 out or every 100,000 will die within a 1 – year period. Using this method, the company estimates the probability of death of that age group as:

___60____ or 0.0006

1000



A second characteristic of probabilities established by the relative frequency of occurrence method can be shown by tossing one of our fair coins 300 times. Here we can see that although the proportion of heads was far from 0.5 in the first 100 tosses, it seemed to stabilize and approach 0.5 as the number of tosses increased.

In statistical language, we would say that the relative frequency becomes stable as the number of tosses becomes large (if we are tossing the coin under uniform conditions). Thus, when we use the relative frequency approach to establish probabilities, our probability figure will gain accuracy as we increase the number of observations. Of course, this improved accuracy is not free; although more tosses of our coin will produce a more accurate probability of head occurring, we must bear the time and the cost of additional observations.

One difficulty with the relative frequency approach is that people often use it without evaluating a sufficient number of outcomes. If you heard someone say, “My aunt and uncle got the flu this year, and they are both over 65, so everyone in that age bracket will probably get the flu, “you would know that your friend did not base his assumptions on enough evidence. His observations were insufficient data for establishing a relative frequency of occurrence probability.

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