Advanced High School Statistics: First Edition

Subsection 2.2.1 Measures of center

In the previous section, we saw that modes can occur anywhere in a data set. Therefore, mode is not a measure of center. We understand the term center intuitively, but quantifying what is the center can be a little more challenging. This is because there are different definitions of center. Here we will focus on the two most common: the mean and median.

The mean, sometimes called the average, is a common way to measure the center of a distribution of data. To find the mean number of characters in the 50 emails, we add up all the character counts and divide by the number of emails. For computational convenience, the number of characters is listed in the thousands and rounded to the first decimal.

The sample mean is often labeled \(\bar{x}\text{.}\) The letter \(x\) is being used as a generic placeholder for the variable of interest, num_char, and the bar on the \(x\) communicates that the average number of characters in the 50 emails was 11,600.

The email50 data set represents a sample from a larger population of emails that were received in January and March. We could compute a mean for this population in the same way as the sample mean, however, the population mean has a special label: \(\mu\text{.}\) The symbol \(\mu\) is the Greek letter mu and represents the average of all observations in the population. Sometimes a subsript, such as \(_x\text{,}\) is used to represent which variable the population mean refers to, e.g. \(\mu_x\text{.}\)

Example 2.2.5 used what is called a weighted mean, which will not be a key topic in this textbook. However, we have provided an online supplement on weighted means for interested readers:

www.openintro.org/stat/down/supp/wtdmean.pdf

The median provides another measure of center. The median splits an ordered data set in half. There are 50 character counts in the email50 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 middle observations: \((6,768+7,012)/2 = 6,890\text{.}\) When there are an odd number of observations, there will be exactly one observation that splits the data into two halves, and in this case that observation is the median (no average needed).

Graphically, we can think of the mean as the balancing point. The median is the value such that 50% of the area is to the left of it and 50% of the area is to the right of it.

Subsection 2.2.2 Standard deviation as a measure of spread

The U.S. Census Bureau reported that in 2012, the median family income was $62,241 and the mean family income was $82,743.⁴www.census.gov/hhes/www/income/

Is a family income of $60,000 far from the mean or somewhat close to the mean? In order to answer this question, it is not enough to know the center of the data set and its range (maximum value - minimum value). We must know about the variability of the data set within that range. Low variability or small spread means that the values tend to be more clustered together. High variability or large spread means that the values tend to be far apart.

Here, we introduce the standard deviation as a measure of spread. Though its formula is a bit tedious to calculate by hand, the standard deviation is very useful in data analysis and roughly describes how far away, on average, the observations are from the mean.

We call the distance of an observation from its mean its deviation. Below are the deviations for the \(1^{st}_{}\text{,}\) \(2^{nd}_{}\text{,}\) \(3^{rd}\text{,}\) and \(50^{th}_{}\) observations in the num_char variable. For computational convenience, the number of characters is listed in the thousands and rounded to the first decimal.

If we square these deviations and then take an average, the result is about equal to the sample variance , denoted by \(s_{}^2\text{:}\)

We divide by \(n-1\text{,}\) rather than dividing by \(n\text{,}\) when computing the variance; you need not worry about this mathematical nuance for the material in this textbook. Notice that squaring the deviations does two things. First, it makes large values much larger, seen by comparing \(10.1^2\text{,}\) \((-4.6)^2\text{,}\) \((-11.0)^2\text{,}\) and \(4.2^2\text{.}\) Second, it gets rid of any negative signs.

The standard deviation is defined as the square root of the variance:

The standard deviation of the number of characters in an email is about 13.13 thousand. A hscript of \(_x\) may be added to the variance and standard deviation, i.e. \(s_x^2\) and \(s_x^{}\text{,}\) as a reminder that these are the variance and standard deviation of the observations represented by \(x_1^{}\text{,}\) \(x_2^{}\text{,}\) ..., \(x_n^{}\text{.}\) The \(_{x}\) subsript is usually omitted when it is clear which data the variance or standard deviation is referencing.

The variance is useful for mathematical reasons, but the standard deviation is easier to interpret because it has the same units as the data set. The units for variance will be the units squared (e.g. meters\(^2\)). Formulas and methods used to compute the variance and standard deviation for a population are similar to those used for a sample.⁵The only difference is that the population variance has a division by \(n\) instead of \(n-1\text{.}\) However, like the mean, the population values have special symbols: \(\sigma_{}^2\) for the variance and \(\sigma\) for standard deviation. The symbol \(\sigma\) is the Greek letter sigma.

The empirical rule tells us that usually about 68% of the data will be within one standard deviation of the mean and about 95% will be within two standard deviations of the mean. However, as seen in Figure 2.2.10 and Figure 2.2.11, these percentages are not strict rules.⁶We will learn where these two numbers come from in Chapter 4 when we study the normal distribution.

When describing any distribution, comment on the three important characteristics of center, spread, and shape. Also note any especially unusual cases.

In this chapter we use standard deviation as a descriptive statistic to describe the variability in a given data set. In Chapter 5 we will use the standard deviation to assess how close a sample mean is to the population mean.

Subsection 2.2.3 Box plots and quartiles

A box plot summarizes a data set using five summary statistics while also plotting unusual observations, called outliers. Figure 2.2.16 provides a box plot of the num_char variable from the email50 data set.

The five summary statistics used in a box plot are known as the five-number summary, which consists of the minimum, the maximum, and the three quartiles (\(Q_1\text{,}\) \(Q_2\text{,}\) \(Q_3\)) of the data set being studied.

\(Q_2\) represents the second quartile, which is equivalent to the 50th percentile (i.e. the median). Previously, we saw that Q\(_2\) (the median) for the email50 data set was the average of the two middle values: \(\frac{\text{ 6,768 } + \text{ 7,012 } }{2} = \text{ 6,890 }\text{.}\)

\(Q_1\) represents the first quartile, which is the 25th percentile, and is the median of the smaller half of the data set. There are 25 values in the lower half of the data set, so \(Q_1\) is the middle value: 2,454 characters. \(Q_3\) represents the third quartile, or 75th percentile, and is the median of the larger half of the data set: 15,829 characters.

We calculate the variability in the data using the range of the middle 50% of the data: \(Q_3 - Q_1 = \text{ 13,375 }\text{.}\) This quantity is called the interquartile range (IQR, for short). It, like the standard deviation, is a measure of variability or spread in data. The more variable the data, the larger the standard deviation and IQR tend to be.

To build a box plot, draw an axis (vertical or horizontal) and draw a scale. Draw a dark line denoting \(Q_2\text{,}\) the median. Next, draw a line at \(Q_1\) and at \(Q_3\text{.}\) Connect the \(Q_1\) and \(Q_3\) lines to form a rectangle. The width of the rectangle corresponds to the IQR and the middle 50% of the data is in this interval.

Extending out from the rectangle, the whiskers attempt to capture all of the data reidxing outside of the box, except outliers. In Figure 2.2.16, the upper whisker does not extend to the last three points, which are beyond \(Q_3 + 1.5\times IQR\) and are outliers, so it extends only to the last point below this limit.⁸You might wonder, isn't the choice of \(1.5 \times IQR\) for defining an outlier arbitrary? It is! In practical data analyses, we tend to avoid a strict definition since what is an unusual observation is highly dependent on the context of the data. The lower whisker stops at the lowest value, 33, since there are no additional data to reach. Outliers are each marked with a dot or asterisk. 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.

Subsection 2.2.4 Calculator: summarizing 1-variable statistics

Calculating the summary statistics will return the following information. It will be necessary to hit the down arrow to see all of the summary statistics.

Enter the following 10 data points into the first list on a calculator: 5, 8, 1, 19, 3, 1, 11, 18, 20, 5. Find the summary statistics and make a box plot of the data.

Solution

The summary statistics should be \({\bar{x}} = 9.1\text{,}\) \({Sx} = 7.475\text{,}\) \({Q1} = 3\text{,}\) etc. The box plot should be as follows.

Subsection 2.2.5 Outliers and robust statistics

How are the sample statistics of the num_char data set affected by the observation, 64,401? What would have happened if this email wasn't observed? What would happen to these summary statistics if the observation at 64,401 had been even larger, say 150,000? These scenarios are plotted alongside the original data in Figure 2.2.24, and sample statistics are computed under each scenario in Table 2.2.25.

The median and IQR are called robust estimates because extreme observations have little effect on their values. The mean and standard deviation are much more affected by changes in extreme observations.

Subsection 2.2.6 Linear transformations of data

Suppose that a researcher is looking at a list of 500 temperatures recorded in Celsius (C). The mean of the temperatures listed is given as 27°C with a standard deviation of 3 °C. Because she is not familiar with the Celsius scale, she would like to convert these summary statistics into Fahrenheit (F). To convert from Celsius to Fahrenheit, we use the following conversion:

Fortunately, she does not need to convert each of the 500 temperatures to Fahrenheit and then recalculate the mean and the standard deviation. The unit conversion above is a linear transformation of the following form, where \(a=9/5\) and \(b=32\text{:}\)

Using the examples as a guide, we can solve this temperature-conversion problem. The mean was 27°C and the standard deviation was 3 °C. To convert to Fahrenheit, we multiply all of the values by \(9/5\text{,}\) which multiplies both the mean and the standard deviation by \(9/5\text{.}\) Then we add 32 to all of the values which adds 32 to the mean but does not change the standard deviation further.

Subsection 2.2.7 Comparing numerical data across groups

Some of the more interesting investigations can be considered by examining numerical data across groups. The methods required here aren't really new. All that is required is to make a numerical plot for each group. To make a direct comparison between two groups, create a pair of dot plots or a pair of histograms drawn using the same scales. It is also common to use back-to-back stem-and-leaf plots, parallel box plots, and hollow histograms, the three of which are explored here.

We will take a look again at the county data set and compare the median household income for counties that gained population from 2000 to 2010 versus counties that had no gain. While we might like to make a causal connection here, remember that these are observational data and so such an interpretation would be unjustified.

There were 2,041 counties where the population increased from 2000 to 2010, and there were 1,099 counties with no gain (all but one were a loss). A random sample of 100 counties from the first group and 50 from the second group are shown in Table 2.2.34.(a) to give a better sense of some of the raw data, and Figure 2.2.35 shows a back-to-back stem-and-leaf plot.

Population: Gain               Population: No Gain

                    3| 2 |12
                98766| 2 |66899
      444433222211100| 3 |00112234
  9999888777766666655| 3 |56667788888999
 44433332221111110000| 4 |0000001223344
        9988876665555| 4 |666778
            443221100| 5 |0
               977775| 5 |8
                  320| 6 |
                88766| 6 |
                   21| 7 |

 Legend: 4 | 5 = 45,000 median income

The parallel box plot is a traditional tool for comparing across groups. An example is shown in the left panel of Figure 2.2.36, where there are two box plots, one for each group, placed into one plotting window and drawn on the same scale.

Another useful plotting method uses hollow histograms to compare numerical data across groups. These are just the outlines of histograms of each group put on the same plot, as shown in the right panel of Figure 2.2.36.

Looking at an association is different than comparing distributions. When comparing distributions, we are interested in questions such as, “Which distribution has a greater average?” and “How do the shapes of the distribution differ?” The number of elements in each data set need not be the same (e.g. height of women and height of men). When we look at association, we are interested in whether there is a positive, negative, or no association between the variables. This requires two data sets of equal length that are essentially paired (e.g. height and weight of individuals).

Subsection 2.2.8 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 map it using an intensity map, where colors are used to show higher and lower values of a variable. Figure 2.2.40 and Figure 2.2.42 shows intensity maps for federal spending per capita (fed_spend), poverty rate in percent (poverty), homeownership rate in percent (homeownership), and median household income (med_income). The color key indicates which colors correspond to which values. Note that 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.