Raising the bar

Chapter 3. Describing Data: Numerical

Unit 3.6. Questions for repetition

1. Give in one block all definitions related to the five-number summary.
2. An expression for the sample variance equivalent to Eq. (3.4) (see Measures of variability) is
   $latex s_x^2 = \frac{1}{{n - 1}}\sum {x_i^2 - \frac{1}{{n(n - 1)}}{{\left( {\sum {{x_i}} } \right)}^2}}.$
   Which one is easier to apply on a hand calculator? Can you prove the equality of the two definitions of the sample variance?
3. Suppose you have a sample $latex {x_1},...,{x_n}$. If all values move to the right by a constant $latex c$, what happens to the mean, median, mode, sample variance, range and IQR?
4. Suppose you have a sample $latex {x_1},...,{x_n},{x_{n + 1}},...,{x_{2n}}$ with an even sample size $latex 2n$. If the first $latex n$ values move to the left by a constant $latex c$ and the last $latex n$ values move to the right by the same constant $latex c$, what happens to the mean, median, mode, sample variance, range and IQR?
5. Can you approximately indicate the mean of the variable in Figure 3.3 (see Shape of the distribution)? If that figure were a Pareto diagram of type Figure 3.5, would you be able to indicate the mean? Where would you place approximately the mean on a time series plot in Figure 3.6?

Figure 3.5.

Figure 3.5. Pareto diagram.

Figure 3.6

Figure 3.6. Time series plot of a stock price.

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