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Statistics and Data Science: A Modeling Approach

3.6 The Five-Number Summary

So far we have used histograms as our main tool for examining distributions. But histograms aren’t the only tool we have available. In this section we will introduce a few more tools for examining distributions of quantitative variables. In the following section we will introduce some tools for examining the distributions of categorical variables.

Sorting Revisited, and the Min/Max/Median

In the previous chapter we introduced the simple idea of sorting a quantitative variable in order. Before sorting the numbers, it was hard to see a pattern. You could read the numbers, but it was hard to draw any conclusions about the distribution itself.

As soon as we sort the numbers in order, we can see things that are true of the distribution. For example, when we sort even a long list of numbers we can see what the smallest number is and what the largest number is. This shows us something about the distribution that we couldn’t have seen just by looking at a jumbled list of numbers.

We can demonstrate this by looking at the Wt variable in the MindsetMatters data frame. Write some code to sort the housekeepers by weight from lowest to highest, and then see what the minimum and maximum weights are.

require(tidyverse) require(mosaic) require(Lock5withR) require(supernova) # Write code to sort Wt from lowest to highest # Solution 1 arrange(MindsetMatters, Wt) # Solution 2 sort(MindsetMatters$Wt) 3 ex() %>% check_or( check_function(., "arrange", not_called_msg = "If you were trying to use `sort`, that's acceptable but your code must have resulted in an error.") %>% check_result() %>% check_equal(), check_function(., "sort") %>% check_result() %>% check_equal() )
You could use either sort() or arrange() to sort Wt from lowest to highest
DataCamp: ch3-17

 [1]  90 105 107 109 115 115 115 117 117 118 120 123 123 124 125 126 127
[18] 128 130 130 131 133 134 134 135 136 137 137 137 140 140 141 142 142
[35] 142 143 144 145 145 148 150 150 154 154 155 155 156 156 156 157 158
[52] 159 160 161 161 161 162 163 163 164 166 167 167 168 170 172 173 173
[69] 178 182 183 184 187 189 196

Note that you can also use the function arrange() to sort, but that will sort the entire data frame. We just wanted to be able to see the sorted weights in a row, so we just used sort() on the vector (MindsetMatters$Wt).

Now that we have sorted the weights, we can see that the minimum weight is 90 pounds and the maximum weight is 196 pounds. In addition to knowing the minimum and maximum weight, it would be helpful to know what the number is that is right in the middle of this distribution. If there are 75 housekeepers, we are looking for the 38th housekeeper’s weight, because there are 37 weights that are smaller than this number and 37 that are bigger than this number. This middle number is called the median.

Numbers such as the minimum (often abbreviated as min), median, and maximum (abbreviated as max) are helpful for understanding a distribution. These three numbers can be thought of as a three-number summary of the distribution. We’ll build up to the five-number summary in a bit.

There is a function called favstats() (for favorite statistics) that will quickly summarize these values for us. Here is how to get the favstats for Wt from MindsetMatters.

favstats(~ Wt, data = MindsetMatters)
 min  Q1 median    Q3 max     mean       sd  n missing
  90 130    145 161.5 196 146.1333 22.46459 75       0

There are a lot of other numbers that are generated by the favstats() function, but let’s take a look at Min, Median, and Max for now. By looking at the median weight in relation to the minimum and maximum weight, you can tell a little bit about the shape of the distribution.

Try writing code to get the favstats() for the variable Population of countries in the data frame HappyPlanetIndex.

require(tidyverse) require(mosaic) require(Lock5withR) require(supernova) HappyPlanetIndex$Region <- recode( HappyPlanetIndex$Region, '1'="Latin America", '2'="Western Nations", '3'="Middle East and North Africa", '4'="Sub-Saharan Africa", '5'="South Asia", '6'="East Asia", '7'="Former Communist Countries" ) # Modify the code to get favstats for Population of countries in HappyPlanetIndex favstats() favstats(~ Population, data = HappyPlanetIndex) ex() %>% check_function("favstats") %>% check_result() %>% check_equal()
Did you use ~Population and set data = HappyPlanetIndex?
DataCamp: ch3-18

  min    Q1 median     Q3    max     mean       sd   n missing
 0.29 4.455  10.48 31.225 1304.5 44.14545 145.4893 143       0

Create a histogram of Population to see if your intuition about the shape of this distribution from looking at the min/median/max is correct.

require(tidyverse) require(mosaic) require(Lock5withR) require(supernova) HappyPlanetIndex$Region <- recode( HappyPlanetIndex$Region, '1'="Latin America", '2'="Western Nations", '3'="Middle East and North Africa", '4'="Sub-Saharan Africa", '5'="South Asia", '6'="East Asia", '7'="Former Communist Countries" ) # Make a histogram of Population from HappyPlanetIndex using gf_histogram gf_histogram(~ Population, data = HappyPlanetIndex) ex() %>% check_function("gf_histogram") %>% check_result() %>% check_equal()
Use the gf_histogram() function
DataCamp: ch3-19

Histogram of population with results grouped to left

Quartiles and the Five-Number Summary

Another way to think about what we’ve been doing is this. Imagine all the values of a variable are sorted and lined up along the thick blue line below.

Thick line with Min, Median, and Max points drawn

The min is the lowest value, the median is the middle value, and the max is the highest value. We have divided the distribution into two equal parts in order to find the median. The two parts can be thought of as halves.

We can divide each half into two parts again to divide the distribution into four parts called quartiles. The quartiles are four equal groups of values. It is as if the long vector of values have been cut into four equal sized pieces.

Thick line with Min, Median, and Max points drawn with two points drawn in between

If we divide the lower half of the distribution in two parts, the bottom part (the lowest .25 of values) is called the first quartile. The upper part (the next lowest .25 of values) is called the second quartile. If we divide the higher half of the distribution in two parts, the middle of that half is called the third quartile. Everyone above that point is in the top quartile (fourth quartile) of the distribution.

It is important to note that what is equal about the four quartiles is the number of data points included in each. Each quartile contains one-fourth of the observations, regardless of what their exact scores are on the variable.

In order to demarcate where a quartile begins and ends, statisticians have given these cut points (the orange lines) boring names: Q0, Q1, Q2, Q3, and Q4.

Thick line with Q0 through Q4 points in five equal sections

When statisticians refer to the “five-number summary” they are referring to these five numbers: the minimum, Q1, the median, Q3, and the maximum. So let’s look again at the favstats() for Wt.

favstats(~ Wt, data = MindsetMatters)
 min  Q1 median    Q3 max     mean       sd  n missing
  90 130    145 161.5 196 146.1333 22.46459 75       0

Now you can see that the favstats() function actually gives you the five-number summary (min, Q1, median, Q3, max), then the mean, standard deviation, n (number of values), and how many cases (in this case, housekeepers) are missing a value for weight. We will delve into the mean and standard deviation in later chapters.

Range and Inter-Quartile Range

The distance between the max and min gives us range, a quick measure of how spread-out the values are in a distribution. Based on the numbers from the favstats() results above, use R as a calculator to find the range of Wt.

require(tidyverse) require(mosaic) require(Lock5withR) require(supernova) # Based on the numbers from the favstats results above, use R as a calculator to find the range of Wt in MindsetMatters ex() %>% check_output_expr("196 - 90")
Subtract the minimum from the maximum to find the range
DataCamp: ch3-20

[1] 106

In distributions like the Population of countries, the range can be very deceptive.

favstats(~ Population, data = HappyPlanetIndex)
  min    Q1 median     Q3    max     mean       sd   n missing
 0.29 4.455  10.48 31.225 1304.5 44.14545 145.4893 143       0

The range looks like it is about 1,304.2 million. But we saw in the histogram that this is due to one or two very populous countries! There was a lot of empty space in that distribution. In cases like this, it might be useful to get the range for just the middle .50 of values. This is called the interquartile range (IQR).

Use the five-number summary of Population to find the IQR. You can use R as a calculator.

require(tidyverse) require(mosaic) require(Lock5withR) require(supernova) HappyPlanetIndex$Region <- recode( HappyPlanetIndex$Region, '1'="Latin America", '2'="Western Nations", '3'="Middle East and North Africa", '4'="Sub-Saharan Africa", '5'="South Asia", '6'="East Asia", '7'="Former Communist Countries" ) # Use R as a calculator to find the IQR of Population from the HappyPlanetIndex data set ex() %>% check_output_expr("31.225 - 4.455")
Subtract Q1 from Q3 to find the IQR
DataCamp: ch3-21

[1] 26.77

Interquartile range ends up being a handy ruler for figuring out whether a data point should be considered an outlier. Outliers present the researcher with a hard decision: should the score be excluded from analysis because it will have such a large effect on the conclusion, or should it be included because, after all, it’s a real data point?

For example, China is a very populous country and is the very extreme outlier in the HappyPlanetIndex, with a population of more than 1,300 million people (another way of saying that is 1.3 billion). If it weren’t there, we would have a very different view of the distribution of population across countries. Should we exclude it as an outlier?

Well, it depends on what we are trying to do. If we wanted to understand the total population of this planet, it would be foolish to exclude China because that’s a lot of people who live on earth! But if we are trying to get a sense of how many people live in a typical country, then perhaps it would make more sense to exclude China.

But then, what about the second-most populous country—India? Should we exclude it too? What about the third-most populous country—the US? Or the fourth—Indonesia? How do we decide what an outlier is? That process seems fraught with subjectivity.

There is no one right way to do it. After all, deciding on what an “outlier” is really depends on what you are trying to do with your data. However, the statistics community has agreed on a rule of thumb to help people figure out what an outlier might be. Any data point bigger than the \(Q3 + 1.5*IQR\) is considered a very large outlier. Anything smaller than the \(Q1 - 1.5*IQR\) is considered a small outlier.