Worksheet 7: Variability

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1. What are some scenarios where you care not only where the data is centered, but also what the variability is?

2. How could we measure variability? Suggest a quantitative measure.

The variance is the average squared distance to the mean: if \(x_1,\ldots,x_n\) are the datapoints and the mean is \(\bar{x} = \frac{1}{n}(x_1+\cdots+x_n)\) then

\[ \overline{\sigma}^2 = \text{variance} = \frac{(x_1-\bar x)^2 + (x_2 - \bar x)^2 + \dots + (x_n - \bar x)^2}{n} \]

And the standard deviation is \(\sqrt{\overline{\sigma}^2} = \overline{\sigma}\).

3. For the dataset of rowers:

   a. What is the mean of the rowers’ heights? What are the units?

   b. What is the variance of the rowers’ heights? What are the units?

   c. What is the standard deviation of the rowers’ heights? What are the units?

4. Suppose someone offers you a choice between a gift of $100 or the chance to flip a fair coin for $300.

   a. What would you choose, and why?

   b. Calculate the mean and standard deviation of your earnings in each scenario.

5. Do you think the standard deviation is a satisfying measure of variability? What is it conveying? What is it not conveying?

6. How can we use quantiles to measure variability?

7. How does the information we get from the standard deviation differ from the information we get from the quantiles?

8. 80% of data points in the histogram fall between the 10th and 90th percentile. Explain why this is the case.