Counting

Representing Data using Counts

• All throughout statistics, there are many important metrics and representations of data that involve counting
• In reference to discrete random variables, the concept of counting data comes up everywhere
• Even in reference to continuous random variables, foundational concepts, such as probability density funcitons, involve the concept of counting data

Binning Data

• Before continuous data can be counted up, they need to be divided into discrete blocks, or binned
• Specifically, we want to understand what type of bins to use
• This is tricky because generally we want the bins to be large and coarse enough that there’s a reasonable degree of representation in each, but not so coarse that we lose all structure
• If we make the bins too small, then the variance in the expected number of points in each bin will get too big
• It’s generally a bad idea to have variable-sized bins, but not always
• It may be a good idea to experiment with different bin-sizes until we find a regime where the bin-size doesn’t have much effect on the analysis
• This unfortunately is quite time-consuming, and therefore may not be the best option

Histograms

• A histogram is simply a plot of how many data-points fall into each bin

• As such, it’s a kind of approximation to the probability density function

  • There are actually quite sophisticated techniques for estimating the pdf from histogram data
• Some people want to use the word histogram only for one-dimensional data
• However, we can always use histograms for multi-dimensional data

Percentiles

• Assuming the data are one-dimensional (so they can be put in a simple order), then we can talk about the percentiles
• The $x^{th}$ percentile value is simply the value equaled or exceeded by only $\frac{x}{100}$ of the datapoints
• That is, $x$ percent of the data are at or above that value (similarly for deciles and quartiles)
• Just like the histogram is an approximation to the pdf, the percentiles contain information about the cumulative distribution function

Median and Mode

• The mode is simply the value with the most data-points
• Assuming one-dimensional data, the median is the value such that half of all points have a higher value and half a lower
• If there is a gap in the data, there may well be a range of values which can claim to be the median
• There is a weak convention, in such cases, to use the mid-point of the range

References

• Probability, Statistics and Stochastic Processes