Overview
Objectives
Readings
Lecture
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Overview

Objectives

By the end of this page, students should be able to:

Define common types of descriptive statistic;
Describe the difference between inferential and predictive statistics.

Readings

Lecture

Statistics is the oldest form of data science. It really started to hit its stride in the 17th century, but has never been quite as sexy as it is now.

In many ways, it’s difficult to define statistics separate from our earlier definition of data science. Most of the concepts are translateable. At its simplest, statistics is: all the parts of data science that don’t include machine learning. Theoretical statistics far, far predates computers, but in the modern day it’s difficult to do sophisticated stats without considerable computational resources.

Distributions: a Primer

The basic unit of statistics is the distribution. Statistics looks at the variation in the world and claims that this variation is patterned and informative. For example, if you know something about the distribution of human heights, you could make a good guess about how likely a random person is to be a certain height. Or at least, so statistics claims.

We’ll talk a lot more about distributions next lecture, but for now just get used to the idea of thinking about every data point as a single sample from a varied background.

Descriptive Statisics

The most basic set of explorations you can conduct on a dataset is what are called descriptive statistics– this is simply understanding what patterns there are in each individual variable of your dataset. Things like, “How many observations are there?” “What is the average value of this variable?” “What are the minimum and maximum values?”, etc. Running the summary() command on a dataset in R is a great way to get descriptive statistics for each variable. Let’s try running summary() on the Davis dataset, which looks at true versus reported height and weight in adults:

summary(Davis)

##  sex         weight          height          repwt            repht      
##  F:112   Min.   : 39.0   Min.   : 57.0   Min.   : 41.00   Min.   :148.0  
##  M: 88   1st Qu.: 55.0   1st Qu.:164.0   1st Qu.: 55.00   1st Qu.:160.5  
##          Median : 63.0   Median :169.5   Median : 63.00   Median :168.0  
##          Mean   : 65.8   Mean   :170.0   Mean   : 65.62   Mean   :168.5  
##          3rd Qu.: 74.0   3rd Qu.:177.2   3rd Qu.: 73.50   3rd Qu.:175.0  
##          Max.   :166.0   Max.   :197.0   Max.   :124.00   Max.   :200.0  
##                                          NA's   :17       NA's   :17

These descriptive statistics tell us the following:

112 study participants were female, 88 were male.
People’s reported weight ranged from 41 to 124 kilos, with an average (mean) value of 65.62.
17 people didn’t report their weight (hence, the NA's: null values). These are probably the same 17 people who didn’t report their height.

Let’s quickly review the other rows of this summary dataset. There are two types of “average” reported here: the mean, which is what we usually mean when we say “average”, and the median, which is the physical midpoint of the data. To understand the difference, consider this array of how many pets people own. Seven individuals were surveyed, and we sorted the results in ascending order:

## [1] 1 1 1 1 2 2 5

You calculate the mean by adding all these values together and dividing by the number of samples. So here, the mean would be $(1 + 1 + 1 + 1 + 2 + 2 + 5) / 5 = 1.86$ . The median, on the other hand, is whatever value is in the middle of this sorted list– here, it’s 1.

Knowing a median value is useful because 50% of your samples will be greater than (or equal to) it, and the other 50% will be less than (or equal to) it. If your data have many outliers-- unusually small or large values-- the median will give you a better measure of the "middle" of the distribution than a mean would. For example, average home values are usually reported as a median, because otherwise the homes of the ultra-rich would artificially inflate the "average" home price.

Another word for the median is the 50th percentile of the data. You can take other percentiles of the data, too. It’s common to split the data into 4 parts and describe the 25th percentile, also known as the first quartile; the median; and the 75th percentile, or third quartile. So in the Davis dataset above, the 1st Qu row of repwt tells us that the slimmest 25% of the sample had weights at or below 55 kilos, while the 3rd Qu row says that the heaviest 25% of people had weigths at or above 73.5 kilos. What does the median tell you?

The data visualizations we discussed in Module 1 were all founded on either the raw data itself or descriptive statistics like these. It’s open for debate whether or not that counts as “data science”.

Inference vs. Prediction

Descriptive statistics, while useful, isn’t usually considered a statistical “analysis”. Things like regressions or other models are more of the meat-and-potatoes of statistical methods, and these analyses are usually performed for one of two distinct purposes. In both cases, you run some model to establish a relationship between two or more variables.

The first case, known as inference, asks the question “What does the relationships between my variables mean?” For example, if you run an analysis testing how education, access to health care, and income affect mortality, inferential statistics would be interested in knowing which of those three variables had the biggest impact on mortality, whether or not that impact was statistically significant, and what the values and directions were of those relationships.

In the second case, known as prediction, you don’t really care what the values of the relationship are in the model, so long as they allow you to accurately predict outcomes for new data points. So, using the above example of the model for mortality rates, a predictive analysis would be interested in knowing what mortality outcome the model would expect for a person with a given education level, degree of access to health care, and income.

These two types of analysis are of course not mutually exclusive, but give very different content and motivation to statistical analyses.

Now that I’ve waved my hands a bit about what “statistics” mean, let’s actually start learning some real stats: on to distributions!

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