Inferential Statistics
We’ve learned how to use descriptive statistics to characterize or summarize some set of measures. For example, we can count the number of measures in (say) each of Beethoven’s 32 piano sonatas, and express the average number of bars or measures.
Where statistics becomes really useful is when we can expand beyond describing properties of a known sets of measures, and infer some properties about an unknown set of measures. In particular, inferential statistics is process by which the we infer properties of a population given a sample.
For example, we might count the number of measures in only half (16) of Beethoven’s piano sonatas. On the basis of this sample, how well can we infer the true average number of measures in all 32 of the sonatas?
Some useful terminology:
External validity: Where the sample is truly representative of a population and so the sample measure correctly generalizes to the population.
Internal validity: Where the measures of the sample are correct. That is, the results generalize to the sample.
Statistic: Some value derived from a sample.
Parameter: A true population value.
Inferential statistics: The process by which population parameters are inferred from sample statistics.
We use inferential statistics to infer a population parameter based on a statistic which we calculate from sample measures.