04/28/2020

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- The following topics will be covered in this lecture:
- Scatter plots
- Correlation
- Linear correlation coefficient
- Computing correlation

Courtesy of Mario Triola, *Essentials of Statistics*, 6th edition

- Intuitively, we think of the idea of correlation or anti-correlation as the behavior of
**two variables****varying together or oppositely**. - To the left, we see the number of Nobel Leaureates per million persons in a country plotted in a scatter plot versus the number of kg of chocolate consumed per capita.
- At at glance, we can see that the two variables
**tend to vary together**, but**not in an exact, determinstic way**; - i.e., a \( 1 \) unit increase in chocolate consumption doesn’t automatically correspond identically to a \( 2.5 \) unit increase in the number of Nobel Laureates.
- Also, there is no reason to believe that the increase in one causes an increase in the other;

- i.e., eating more chocolate doesn’t produce more Nobel prize winning scientist.
- Indeed, a more rational explanation is that these values tend to vary together because Nobel Prizes usually go to countries with highly developed academic, cultural and industrial infrastructure.
- Likewise, inhabitants of these coutnries can better afford luxury goods like chocolate.
**Correlation should never be considered causation**, but rather that**two measurements tend to have systematic associations, which may be better explained by other latent variables**.- We may see a similar association when plotting either of the two above variables versus a third variable that acts as an economic indicator.

Courtesy of Mario Triola, *Essentials of Statistics*, 6th edition

- Understanding the
**limitations of correlation for explanatory power**, we can**use correlation as a powerful research tool for the purpose that it is intended**. **Correlation**is a statistic that we will compute from**pairs of measurements in a single sample**.- In the last example, we had a sample consisting of individual contries.
- Each observation (country) corresponded to two distinct measurements:
- The number of kg of chocolate consumed per capita.
- The number of Nobel Laureates per million inhabitants.
- This will always be a feature of computing correlation – we need
**observations which have two measurements**. - We will compute the correlation coefficient between these variables.
- Usually, we will use a scatter plot as a first check for a systematic pattern and then we wil then compute the statistic.
- Being a
**statistic**, the correlation coefficient is subject to**sampling error**; - we will also need to
**test for the significance**to**quantify the uncertainty of the value in relation to the**.**population parameter**