Examining numerical data

This chapter focuses on the mechanics and construction of summary statistics and graphs. We use statistical software for generating the summaries and graphs presented in this chapter and book. However, since this might be your first exposure to these concepts, we take our time in this chapter to detail how to create them. Mastery of the content presented in this chapter will be crucial for understanding the methods and techniques introduced in rest of the book.

In this section we will explore techniques for summarizing numerical variables. For example, consider the loan_amount variable from the loan50 data set, which represents the loan size for all 50 loans in the data set. This variable is numerical since we can sensibly discuss the numerical difference of the size of two loans. On the other hand, area codes and zip codes are not numerical, but rather they are categorical variables.

Throughout this section and the next, we will apply these methods using the loan50 and county data sets, which were introduced in Section 1.2. If you’d like to review the variables from either data set, see Figures 1.3 and 1.5.

Scatterplots for paired data

A scatterplot provides a case-by-case view of data for two numerical variables. In Figure 1.8 on Section 1.2, a scatterplot was used to examine the homeownership rate against the fraction of housing units that were part of multi-unit properties (e.g. apartments) in the county data set. Another scatterplot is shown in Figure 2.1, comparing the total income of a borrower (total_income) and the amount they borrowed (loan_amount) for the loan50 data set. In any scatterplot, each point represents a single case. Since there are 50 cases in loan50, there are 50 points in Figure 2.1.

Looking at Figure 2.1, we see that there are many borrowers with an income below $100,000 on the left side of the graph, while there are a handful of borrowers with income above $250,000.

Dot plots and the mean

Sometimes two variables are one too many: only one variable may be of interest. In these cases, a dot plot provides the most basic of displays. A dot plot is a one-variable scatterplot; an example using the interest rate of 50 loans is shown in Figure 2.3. A stacked version of this dot plot is shown in Figure 2.4.

The mean, often called the average, is a common way to measure the center of a distribution of data. To compute the mean interest rate, we add up all the interest rates and divide by the number of observations: $$\begin{aligned} \bar{x} = \frac{\text{10.90\%} + \text{9.92\%} + \text{26.30\%} + \cdots + \text{6.08\%}}{50{}} = 11.57{}\% \end{aligned}$$ The sample mean is often labeled $\bar{x}$. The letter $x$ is being used as a generic placeholder for the variable of interest, interest_rate, and the bar over the $x$ communicates we’re looking at the average interest rate, which for these 50 loans was 11.57%. It is useful to think of the mean as the balancing point of the distribution, and it’s shown as a triangle in Figures 2.3 and 2.4.

The loan50 data set represents a sample from a larger population of loans made through Lending Club. We could compute a mean for this population in the same way as the sample mean. However, the population mean has a special label: $\mu$.

The symbol $\mu$ is the Greek letter mu and represents the average of all observations in the population. Sometimes a subscript, such as $_x$, is used to represent which variable the population mean refers to, e.g. $\mu_x$. Often times it is too expensive to measure the population mean precisely, so we often estimate $\mu$ using the sample mean, $\bar{x}$.

Histograms and shape

Dot plots show the exact value for each observation. This is useful for small data sets, but they can become hard to read with larger samples. Rather than showing the value of each observation, we prefer to think of the value as belonging to a bin. For example, in the loan50 data set, we created a table of counts for the number of loans with interest rates between 5.0% and 7.5%, then the number of loans with rates between 7.5% and 10.0%, and so on. Observations that fall on the boundary of a bin (e.g. 10.00%) are allocated to the lower bin. This tabulation is shown in Figure 2.5. These binned counts are plotted as bars in Figure 2.6 into what is called a histogram, which resembles a more heavily binned version of the stacked dot plot shown in Figure 2.4.

Histograms provide a view of the data density. Higher bars represent where the data are relatively more common. For instance, there are many more loans with rates between 5% and 10% than loans with rates between 20% and 25% in the data set. The bars make it easy to see how the density of the data changes relative to the interest rate.

Histograms are especially convenient for understanding the shape of the data distribution. Figure 2.6 suggests that most loans have rates under 15%, while only a handful of loans have rates above 20%. When data trail off to the right in this way and has a longer right tail, the shape is said to be right skewed.⁵

Data sets with the reverse characteristic – a long, thinner tail to the left – are said to be left skewed. We also say that such a distribution has a long left tail. Data sets that show roughly equal trailing off in both directions are called symmetric.

In addition to looking at whether a distribution is skewed or symmetric, histograms can be used to identify modes. A mode is represented by a prominent peak in the distribution. There is only one prominent peak in the histogram of loan_amount.

A definition of mode sometimes taught in math classes is the value with the most occurrences in the data set. However, for many real-world data sets, it is common to have no observations with the same value in a data set, making this definition impractical in data analysis.

Figure 2.7 shows histograms that have one, two, or three prominent peaks. Such distributions are called unimodal, bimodal, and multimodal, respectively. Any distribution with more than 2 prominent peaks is called multimodal. Notice that there was one prominent peak in the unimodal distribution with a second less prominent peak that was not counted since it only differs from its neighboring bins by a few observations.

Looking for modes isn’t about finding a clear and correct answer about the number of modes in a distribution, which is why prominent is not rigorously defined in this book. The most important part of this examination is to better understand your data.

Variance and standard deviation

The mean was introduced as a method to describe the center of a data set, and variability in the data is also important. Here, we introduce two measures of variability: the variance and the standard deviation. Both of these are very useful in data analysis, even though their formulas are a bit tedious to calculate by hand. The standard deviation is the easier of the two to comprehend, and it roughly describes how far away the typical observation is from the mean.

We call the distance of an observation from its mean its deviation. Below are the deviations for the $1^{st}_{}$, $2^{nd}_{}$, $3^{rd}$, and $50^{th}_{}$ observations in the interest_rate variable: $$\begin{aligned} x_1^{}-\bar{x} &= 10.90- 11.57{} = -0.67\text{ } \\ x_2^{}-\bar{x} &= 9.92- 11.57{} = -1.65\\ x_3^{}-\bar{x} &= 26.30- 11.57{} = 14.73\\ &\ \vdots \\ x_{50}^{}-\bar{x} &= 6.08- 11.57{} = -5.49 \end{aligned}$$ If we square these deviations and then take an average, the result is equal to the sample variance, denoted by $s_{}^2$: $$\begin{aligned} s_{}^2 &= \frac{(-0.67)_{}^2 + (-1.65)_{}^2 + (14.73)_{}^2 + \cdots + (-5.49)_{}^2}{50{}-1} \\ &= \frac{0.45 + 2.72 + 216.97 + \cdots + 30.14}{49} \\ &= 25.52{} \end{aligned}$$ We divide by $n - 1$, rather than dividing by $n$, when computing a sample’s variance; there’s some mathematical nuance here, but the end result is that doing this makes this statistic slightly more reliable and useful.

Notice that squaring the deviations does two things. First, it makes large values relatively much larger, seen by comparing $(-0.67)^2$, $(-1.65)^2$, $(14.73)^2$, and $(-5.49)^2$. Second, it gets rid of any negative signs.

The standard deviation is defined as the square root of the variance: $$\begin{aligned} s = \sqrt{25.52{}} = 5.05{} \end{aligned}$$ While often omitted, a subscript of $_x$ may be added to the variance and standard deviation, i.e. $s_x^2$ and $s_x^{}$, if it is useful as a reminder that these are the variance and standard deviation of the observations represented by $x_1^{}$, $x_2^{}$, ..., $x_n^{}$.

Like the mean, the population values for variance and standard deviation have special symbols: $\sigma_{}^2$ for the variance and $\sigma$ for the standard deviation. The symbol $\sigma$ is the Greek letter sigma.

In practice, the variance and standard deviation are sometimes used as a means to an end, where the “end” is being able to accurately estimate the uncertainty associated with a sample statistic. For example, in Chapter 5 the standard deviation is used in calculations that help us understand how much a sample mean varies from one sample to the next.

Box plots, quartiles, and the median

A box plot summarizes a data set using five statistics while also plotting unusual observations. Figure 2.10 provides a vertical dot plot alongside a box plot of the interest_rate variable from the loan50 data set.

The first step in building a box plot is drawing a dark line denoting the median, which splits the data in half. Figure 2.10 shows 50% of the data falling below the median and other 50% falling above the median. There are 50 loans in the data set (an even number) so the data are perfectly split into two groups of 25. We take the median in this case to be the average of the two observations closest to the $50^{th}$ percentile, which happen to be the same value in this data set: $(\text{9.93\%{}} + \text{9.93\%{}}) / 2 = \text{9.93\%{}}$. When there are an odd number of observations, there will be exactly one observation that splits the data into two halves, and in such a case that observation is the median (no average needed).

The second step in building a box plot is drawing a rectangle to represent the middle 50% of the data. The total length of the box, shown vertically in Figure 2.10, is called the interquartile range (IQR, for short). It, like the standard deviation, is a measure of variability in data. The more variable the data, the larger the standard deviation and IQR tend to be. The two boundaries of the box are called the first quartile (the $25^{th}$ percentile, i.e. 25% of the data fall below this value) and the third quartile (the $75^{th}$ percentile), and these are often labeled $Q_1$ and $Q_3$, respectively.

Extending out from the box, the whiskers attempt to capture the data outside of the box. However, their reach is never allowed to be more than $1.5\times IQR$. They capture everything within this reach. In Figure 2.10, the upper whisker does not extend to the last two points, which is beyond $Q_3 + 1.5\times IQR$, and so it extends only to the last point below this limit. The lower whisker stops at the lowest value, 5.31%, since there is no additional data to reach; the lower whisker’s limit is not shown in the figure because the plot does not extend down to $Q_1 - 1.5\times IQR$. In a sense, the box is like the body of the box plot and the whiskers are like its arms trying to reach the rest of the data.

Any observation lying beyond the whiskers is labeled with a dot. The purpose of labeling these points – instead of extending the whiskers to the minimum and maximum observed values – is to help identify any observations that appear to be unusually distant from the rest of the data. Unusually distant observations are called outliers. In this case, it would be reasonable to classify the interest rates of 24.85% and 26.30% as outliers since they are numerically distant from most of the data.

Robust statistics

How are the sample statistics of the interest_rate data set affected by the observation, 26.3%? What would have happened if this loan had instead been only 15%? What would happen to these summary statistics if the observation at 26.3% had been even larger, say 35%? These scenarios are plotted alongside the original data in Figure 2.11, and sample statistics are computed under each scenario in Figure 2.12.

The median and IQR are called robust statistics because extreme observations have little effect on their values: moving the most extreme value generally has little influence on these statistics. On the other hand, the mean and standard deviation are more heavily influenced by changes in extreme observations, which can be important in some situations.

Transforming data (special topic)

data are very strongly skewed, we sometimes transform them so they are easier to model.

There are some standard transformations that may be useful for strongly right skewed data where much of the data is positive but clustered near zero. A transformation is a rescaling of the data using a function. For instance, a plot of the logarithm (base 10) of county populations results in the new histogram in Figure 2.13(b). This data is symmetric, and any potential outliers appear much less extreme than in the original data set. By reigning in the outliers and extreme skew, transformations like this often make it easier to build statistical models against the data.

Transformations can also be applied to one or both variables in a scatterplot. A scatterplot of the population change from 2010 to 2017 against the population in 2010 is shown in Figure 2.14(a). In this first scatterplot, it’s hard to decipher any interesting patterns because the population variable is so strongly skewed. However, if we apply a log$_{10}$ transformation to the population variable, as shown in Figure 2.14(b), a positive association between the variables is revealed. In fact, we may be interested in fitting a trend line to the data when we explore methods around fitting regression lines in Chapter 8.

Transformations other than the logarithm can be useful, too. For instance, the square root ($\sqrt{\text{original observation}}$) and inverse ($\frac{1}{\text{original observation}}$) are commonly used by data scientists. Common goals in transforming data are to see the data structure differently, reduce skew, assist in modeling, or straighten a nonlinear relationship in a scatterplot.

Mapping data (special topic)

The county data set offers many numerical variables that we could plot using dot plots, scatterplots, or box plots, but these miss the true nature of the data. Rather, when we encounter geographic data, we should create an intensity map, where colors are used to show higher and lower values of a variable. Figures 2.15(b) and 2.16(b) shows intensity maps for poverty rate in percent (poverty), unemployment rate (unemployment_rate), homeownership rate in percent (homeownership), and median household income (median_hh_income). The color key indicates which colors correspond to which values. The intensity maps are not generally very helpful for getting precise values in any given county, but they are very helpful for seeing geographic trends and generating interesting research questions or hypotheses.

Exercises