1 Probabilities, Frequencies, And The Chi Squared Goodness Of Fit Test

1.1 Relative frequency and probability

If one flips a normal coin, it is equally likely that one will obtain heads or tails. One way of expressing this is to say that the ratio of heads to tails is 1:1. Another way of expressing the relationship is to describe the relative frequency of each outcome. The relative frequency is the fraction of times each outcome is achieved. Relative frequency can be calculated by taking the count of an individual kind of outcome and divide by the total counts for all kinds of outcomes. For a ratio of 1:1, there are two total outcomes, so the relative frequency of heads is ½ or 0.5 and the relative frequency of tails is the same. It is normal practice to express relative frequencies as decimal fractions.

One can also express this relationship using probability. If a system behaves consistently over time, it is reasonable to expect that the relative frequency at which we observe a certain event is related to the probability of occurrence of that event. If the probability of obtaining heads is 0.5, then if we flip a coin many times, we would expect to obtain heads with a relative frequency of 0.5. Based on this assumption, we can state that the expected relative frequency of an outcome is equal to the probability of that outcome. Based on the 1:1 ratio of heads to tails, the probability of obtaining tails is also 0.5 and the expected relative frequency is 0.5 as well.

Note that the two probabilities add up to one, which makes sense since the only possible outcomes are heads and tails. The sum of relative frequencies is also equal to one, since the sum of all fractional parts must equal the whole.

These ideas are summarized in Table 1:

Table 1. Relationship between ratio, frequency, and probability for a penny