invest_3ed.pdf

# Definition another graph is based on the five number

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Definition: Another graph is based on the five-number summary, called a boxplot (invented by John Tukey in 1970). The box extends from the lower quartile to the upper quartile with a vertical line inside the box at the location of the median. Whiskers then typically extend to the min and max values. (e) Create by hand a boxplot for these data. Which display do you prefer, the boxplot or the histogram? Why? Although boxplots are a nice visual of the five-number summary, they can sometimes miss interesting features in a data set. In particular, shape can be more difficult to judge in a boxplot. Another application of the inter-quartile range is as a way to measure whether an observation is far from the bulk of the distribution. Definition: A value is an outlier according to the 1.5IQR criterion if the value is larger than the upper quartile + 1.5 × box length or smaller than the lower quartile 1.5 × box length. Note: The box length = upper quartile lower quartile, is called the interquartile range or IQR. A modified boxplot will display such outliers separately and then extend the whiskers to the most extreme non-outlier observation. (f) Use the following Technology Detour to create a “modified” boxplot for these data. A re there any outliers? Technology Detour Modified Boxplots In R > boxplot(responsetime, ylab="time until reaction" m Adds labels + horizontal=TRUE) m Makes horizontal OR > iscamboxplot(responsetime) m Uses quartiles In Minitab x Choose Graph > Boxplot . x Specify One Y, Simple , press OK . x Enter the response time variable in the Graph variables box. x Press Scale and check the Transpose value and category scales box. m Turns horizontal x Press OK twice.

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Chance/Rossman, 2015 ISCAM III Investigation 2.2 145 Modelling non-normal data These data are not well modelled by a normal distribution. So can we still make predictions? There are a couple of strategies. One would be to consider whether a rescaling or transformation of the data might create a more normal-looking distribution, allowing us to use the methods from Investigation 2.1. In this case, we need a transformation that will downsize the large values more than the small values. Log transformations are often very helpful in this regard. Definition: A data transformation applies a mathematical function to each value to re-express the data on an alternative scale. Data transformations can also make the data more closely modeled with a normal distribution, which could then satisfy the conditions the Central Limit Theorem and inference procedures based on the t- distribution. (g) Create a new variable which is log(responsetime). (You can use either natural log or log base 10, but so we all do the same thing, let’s use natural log here, which is the default in most software when you say “log.”) x In R > lnresponsetime = log(responsetime) x In Minitab Choose Calc > Calculator , name: lnresponsetime , expression: log('responsetime') Create a histogram of these data and a normal probability plot. Does log(responsetime) approximately follow a normal distribution? What are the mean and standard deviation of this distribution?
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