More useless statistics
Over at the ExploringDataBlog, Ron Pearson just wrote a post about the cases when means are useless. In fact, it’s possible to calculate a whole load of stats on your data and still not really understand it. The canonical dataset for demonstrating this (spoiler alert: if you are doing an intro to stats course, you will see this example soon) is the Anscombe quartet.
The data set is available in R as
anscombe, but it requires a little reshaping to be useful.
anscombe2 <- with(anscombe, data.frame( x = c(x1, x2, x3, x4), y = c(y1, y2, y3, y4), group = gl(4, nrow(anscombe)) ))
Note the use of
gl to autogenerate factor levels.
So we have four sets of x-y data, which we can easily calculate summary statistics from using
ddply from the
plyr package. In this case we calculate the mean and standard deviation of y, the correlation between x and y, and run a linear regression.
library(plyr) (stats <- ddply(anscombe2, .(group), summarize, mean = mean(y), std_dev = sd(y), correlation = cor(x, y), lm_intercept = lm(y ~ x)$coefficients, lm_x_effect = lm(y ~ x)$coefficients )) group mean std_dev correlation lm_intercept lm_x_effect 1 1 7.500909 2.031568 0.8164205 3.000091 0.5000909 2 2 7.500909 2.031657 0.8162365 3.000909 0.5000000 3 3 7.500000 2.030424 0.8162867 3.002455 0.4997273 4 4 7.500909 2.030579 0.8165214 3.001727 0.4999091
Each of the statistics is almost identical between the groups, so the data must be almost identical in each case, right? Wrong. Take a look at the visualisation. (I won’t reproduce the plot here and spoil the surprise; but please run the code yourself.)
library(ggplot2) (p <- ggplot(anscombe2, aes(x, y)) + geom_point() + facet_wrap(~ group) )
Each dataset is really different – the statistics we routinely calculate don’t fully describe the data. Which brings me to the second statistics joke.
A physicist, an engineer and a statistician go hunting. 50m away from them they spot a deer. The physicist calculates the trajectory of the bullet in a vacuum, raises his rifle and shoots. The bullet lands 5m short. The engineer adds a term to account for air resistance, lifts his rifle a little higher and shoots. The bullet lands 5m long. The statistician yells “we got him!”.