The Shape of a Quantitative Distribution

When you graph quantitative data, you can often see some kind of shape emerge. Here are some typical block structures that illustrate some possible shapes a display might take. We call the set of values "distributions."

The first thing you need to determine is if there is any symmetry to the graph. If you were to visualize a vertical line going down the center, does each side look like a mirror image of the other? No real-life distribution will be perfectly symmetrical, but if it's close, it's worth mentioning.

(Self-Test): Which three of the above graphs look symmetric?
(Answer): B, D, and E.

The next thing you might notice is that some graphs have peaks whereas others look pretty level. We call the peaks "modes." If there's one peak, we say the graph has a unimodal distribution. If there are two peaks, the graph has a bimodal distribution. 

(Self-Test): Which three of the above graphs look unimodal? Which is bimodal?
(Answer): A, B, and C are unimodal; D is bimodal.

FYI, a graph that's mostly level-looking, like graph E, is called uniform.

Now take a look at distributions A and C. Do you see that each has a "tail" at one end? When a distribution is off-center (compared to graph B), we say it is skewed. The direction of the tail is the direction of the skewness. For example, graph A is skewed left because the tail is on the left side of the graph. graph C is skewed right.

When we describe a distribution we try to describe its shape, center, and spread, plus anything unusual about it, such as outliers. This will be the subject of my next post.

(Self-Test): Which of the distributions pictured above might have outliers?
(Answer): Skewed graphs, like A and C, depending on the length of their tails. The longer the tail, the more likely there are outliers.

Picturing Quantitative Data

Just in case you didn't read the post "Picturing Categorical Data," here's a tiny bit of rehash: The first thing to note that different types of data are pictured in different ways. You might recall from an earlier posting that there are two types of data:

• Quantitative data are numbers for which it makes sense to do arithmetic on them. For example, test scores, dollars, miles, etc. it is meaningful to, say, find their average. Quantitative data usually have labels.
• Categorical data are sub-levels of variables that you do not combine with arithmetic. You usually count them. Examples: eye colors (bllue, brown, green, hazel), education levels (high school, undergrad, graduate, etc.). Some categorical variables are numbers, but not ones that make sense to add; examples are zip codes, numbers on athletic jerseys, phone numbers.

This posting deals with quantitative data only. We deal twith categorical data in an earlier post.

There are many ways to picture quantitative data, but let me introduce two that are most used: the boxplot and the histogram. In both cases, let's use the following set of scores, which are arranged in descending order.

98	86	81
95	86	75
92	86	75
91	85	75
91	82	74
90	82	73
89	81	72

  
Boxplots

(Self-Test): In the earlier post, entitled "Percentiles and Quartiles," we discussed the 5 number summary, consisting of which 5 measures?                                            
(Answer): the minimum, the 1st quartile (Q1), the median, the 3rd quartile (Q3), and the maximum.

A boxplot uses these 5 numbers, arranged on a number line according to their values. As we discussed, each of the four regions marked off by these 5 numbers contain 1/4 of the values in the data set, even if they don't have the same width.

There are 21 numbers shown above. The minimum is 72 and the maximum is 98. Since the number are arranged numerically, we can count to the 11th number to find the median: 85.

We now have 10 numbers on each side of the median. The 1st and 3rd quartiles are the medians of the lower 10 and upper 10 numbers, respectively. Q1 is 75, and Q3 is midway between 90 and 91. Remember: just look between the 5th and 6th number from the bottom and the top in each case.

So, our 5-number summary is: 72 (min), 75 (Q1), 85 (median), 90.5 (Q3), and 98 (max). To make a boxplot, draw a number line that stretches from, say, 70 to 100 and place the 5 numbers in their proper places. Then draw a box around the middle 3 numbers to show the span of the middle 50% of the number set:

A boxplot is helpful not only to see the spread of the data set, but also to see the symmetry of the data. For example, the stretch between the minimum and the median is about the same size as the stretch between the median and the maximum, which tells us that the mean lines up pretty well with the median. (Recall that it doesn't always end up this way!) The unevenness of the outer regions (which we call the "whiskers") are uneven, telling us that the data looks more heavily weighted toward the low side of the number line. If the boxplot looked like the following, the data would be roughly symmetric:

Symmetry in a data set is important to recognize, because we can assume other important traits. More about this in a future post.

Now, if a data set has outliers, they are shown with a symbol; for instance, a star:

In cases like this, a new maximum that lies within the boundary for outliers is determined.  As I said in a previous post, you should look for possible reasons for outliers.

Histograms

A histogram tracks the number of observations (like the values in the above table) that lie within consecutive, same-width intervals on a number line. While most people do not make histograms by hand, we'll go through how it's done so you can understand what a histogram looks like. In statistics histograms are used heavily; usually to see the shape of the data distribution.

Start by examining the data values. They range from 72 to 98, and the range (as defined in statistics) is 98 - 72 or 26. We want to divide 26 into a number of equal-sized intervals, then find out how many of the numbers in the data set lie in each interval. We will represent that number as the height of that interval's bar.

How many intervals do we need? Well, to me, that's a "squishy" question. You want enough to be able to see a shape, but not so many that you hardly have any values in each interval. A rule of thumb that I use is between 6 and 8. Because our range is 26, I'm going to use 7 for intervals, because 7*4 is 28, which is closer to 26 than a multiple of 6 or 8. So we have the following number line. Each interval will be 4 units wide (again, because 7*4 is 28), so I've numbered the line accordingly.

(Self-Test): What's with the // on the number line above?
(Answer): Number lines should always be to scale, but in this case a number line going from 0 to 100 probably wouldn't fit on the page. So, I use the // to indicate a break in the numbering.

Now I need to make a small decision: Shall I count the labelled numbers as the start or the end of the interval? It doesn't matter, just as long as you're consistent. I'm going to count the numbers as the start of each interval. So, my first interval will go from 72 to 75; my second will be from 76 to 79, and so on.

For each interval, you need to count how many values in the data set fall within that interval. For example, there are 6 numbers that are between 72 and 75, inclusive. So the height of my bar for the first interval will be 6 units high.

Our second interval, which goes from 76 to 79, has no values. So we leave a space to show that zero values are in that interval. The next interval goes from 80 to 83, and there are four values that lie in that interval. So, the bar will be 4 units high.

Continuing on for the rest of the intervals, here is our finished product:

This histogram has an interesting shape. If the dataset represents grades, we can assume that there 6 out of 21 "C" students, while twice that many are in the "B" to "A-" range. I'll be saying more about shapes of distributions in a later post. For now, suffice it to say that the shape of a distribution is an important thing to assess when you're looking at data.

Keeping it Simple -- the Area Principle

If you made it through the [rather long] post entitled "Picturing Categorical Data," this next post brings out a fine point about making graphical displays. If you look through articles and newspapers, or perhaps at slides that people make in your office for presentations, you might see a tendency for folks to want to make them fancy. While this is admirable -- trying to show potentially dry information in a more splashy way -- it can end up being misleading. Read on.

Take a look at this pie chart, which graphs quantities that are 10%, 20%, 30%, and 40% of the whole.

Now compare it to its flashy 3-D counterpart:

"What's the difference?" you might ask. However, I would argue that the 3-D version makes the green slice (30%) look larger than the purple (40%) slice. Can you see it? This doesn't always happen with 3-D displays, but 3-D displays are prone to this. It's something that you want to look out for, just in case.

When a smaller pie slice or bar (in a bar chart) looks larger than a slice or bar that represents a larger quantity, we say that the display violates the Area Principle. In the first display, each piece was proportionately sized relative to the others.

Why make this point? When trying to communicate something graphically, the main point is to get the information across with as little potential confusion as possible -- not to impress people with fancy pictures. Statistics can be mind-boggling to many, so why not try to make things as straightforward as possible? It's the old K.I.S.S. principle. Statistics-challenged people will thank you! (As I'm sure you're thanking me for the short post!)

Has the Stock Market Hit Bottom?

Oh, if only there were a nice statistical way to answer that question! In the August 10, 2011 issue of USA Today, the article at http://www.usatoday.com/money/markets/2011-08-09-has-market-hit-bottom_n.htm offered a lot of insights from experts, but the authors were wise enough to know that there’s just no telling whether the stock market will rise or fall, or whether better days are to come yet.

For anyone who’s not aware, the Dow Jones Industrial average (DJIA) plummeted 535 points on Monday, following the bad news that the US credit rating had dropped from AAA to AA+, according to Standard & Poors and probably a multitude of other events that stripped investors of their confidence. The following day, August 9, the DJIA gained back 435 points, bringing it closer to pre-plummet levels. At this writing (Wednesday, August 10, 2011 at 1:35 p.m.), the Dow is down about 357 points.  

The insights offered by experts as to why the market is plummeting range from emotional sell-offs in light of the US credit rating downgrade, to the state of theoverall world economy, to the lack of warm fuzzies in Congress, but no one is making a the mistake of trying to predict the future. Nothing, not even statistics, can do that reliably.

Theoretically, even if somehow we were able to identify *all* of the factors (“variables” we call them) that make for fluctuations in the stock market, and then we found some realistic way to quantify each of the factors, we could come up with a [probably quite complicated] equation involving all of those variables that we could use to predict future behavior, but it would be just that – a prediction. You simply cannot use statistics to foretell the future. The best one can do is make a fact-based, educated guess.

Similarly, you cannot use the past to predict the future. Most of fall into this trap pretty easily. For example, a lot of people moved their investments to Treasury-related securities, because Treasury issues have shown a steadily upward trend without the volatility of riskier investments, like stocks. In fact, in the past 10 years, inflation-protected securities (TIPS) have out-performed the Dow, the Nasdaq, and the S&P 500 significantly…at about half the risk (as rated by Vanguard).  So are you rushing to your favorite investment site to move your money? If you are, then you’re using past performance to predict the future! Sometimes it works, but just as often it doesn’t.

Has the stock market hit bottom? Read the article, and you be the judge. Just trying to set your expectations about what statistics can and cannot do!

Picturing Categorical Data

They say a picture is worth a thousand words, and that's certainly true with statistical data! The first thing to note that different types of data are pictured in different ways. You might recall from an earlier posting that there are two types of data:

• Categorical data are sub-levels of variables that you do not combine with arithmetic. You usually count them. Examples: eye colors (bllue, brown, green, hazel, gray), education levels (high school, undergrad, graduate, etc.). Some categorical variables are numbers, but not ones that make sense to add; examples are zip codes, numbers on athletic jerseys, and phone numbers.
• Quantitative data are numbers for which it makes sense to do arithmetic on them. For example, with test scores, dollars, miles, etc., it is meaningful to, say, find their average. Quantitative data usually have labels.

In this post, we'll picture categorical data only; I'll cover quantitative data in another post.

There are two ways to "graphically" picture categorical data: bar charts and pie charts. You see these all the time in articles. Let's use the following base data that shows eye colors in a group of 115 people:

Bar Charts
The most straightforward type of display is a bar chart. Bar charts can portray the actual numbers...

...or they can portray percents of the whole (115)...

Notice that the bars can be vertical or horizontal. In either case, the bars are arranged in either increasing or decreasing size.

If you have a number of very small bars, you can put them together as a larger combined bar and label it "Other." Such bars don't necessarily need to be placed in order; they usually appear as the last bar. Suppose, for example, you want to put green, hazel, and gray together as an "Other" bar. Then your graph would look like this:

Sometimes you will see a single bar with sections representing each category in proportion. This is called a segmented (or stacked) bar chart. They can appear with the actual counts (height of the single bar is equal to the sum of the counts), or as percentages (height of the single bar represents 100%). Here's how a segmented bar chart with percents would look. Notice that it's good to arrange the bars in decreasing order from bottom to top:

Segmented bar charts are especially helpful in comparing the same categories in two or more different groups. Suppose we had a second group of people whose segmented bar chart of eye colors looked slightly different. We could put the bars side by side in a single display, with the eye colors in the same order.

It is easy to see that there are fewer brown-eyed people in Group 2, but more blue-eyed people, about the same number of green and gray eyed-people, but fewer with hazel eyes.

One caution with two or more segmented bars: Unless each group is exactly the same size, you should use percents rather than counts. Otherwise it would be nearly impossible to compare the bars.

Pie Charts
A pie chart shows each category as a proportional-sized pie slice. Pie charts always use percents. So, a pie chart for our eye color data would look like this:

Notice that the pieces are arranged in order of size as you go around the pie.

(Self-Test): Suppose there's another group -- this time consisting of 140 people -- whose eye colors are as follows. Make a "Group 3" segmented bar chart to show the differences.

(Answer): See below.

The Interquartile Range and Outliers

In the previous post, I introduced percentiles and quartiles and said that the Interquartile Range (IQR) is found by subtracting Q3 (the third quartile) minus Q1 (the first quartile). The IQR is significant because it tells us where the middle 50% of the numbers in the data set lie. This is especially helpful to know if there are extreme values at the low and/or high end of the data set.

Take the very short data set 40, 80, 86. 88, 90. Notice that the 40 is extremely far away from the other numbers; it is considered an  extreme value. It would make the average (mean) unreasonably small with respect to the majority of the other numbers. We call 40 in this set an outlier. Outliers can occur by coincidence but just as often there is some reason behind them. Statisticians should at least try to see if there's an explanation when they run into an outlier. Then they can analyze the data set with that in mind. Sometimes it's good to do two analyses: one with the outlier, and one without it.

How do we tell if a number is extreme enough to be an outlier? Well, believe it or not, there is a mathematical way and it's pretty straightforward. To test for outliers, we need to know what the 1st and 3rd quartiles are, so we can compute the IQR. Here are the steps:

1. Find the IQR.
2. Multiply the IQR by 1.5.
3. Add the resulting number to Q3 to get an upper boundary for outliers.
4. Subtract the same resulting number (from #2) from Q1 to get a lower boundary for outliers.
5. If a number in the data set lies beyond either boundary, it is considered an outlier.

In the example above (40, 80, 86, 88, 100), Q1 is 80 and Q3 is 88. Going through the steps...

1. Q3 - Q1 = 88 - 80 = 8 (The IQR)
2. IQR * 1.5 = 8 * 1.5 = 12
3. Q3 + 12 = 88 + 12 = 100. This is the upper boundary for outliers. Since our maximum is right at 100, we have no high outliers.
4. Q1 - 12 = 80 - 12 = 68. This is the lower boundary for outliers. Since our minimum number, 40, is less than 68, our data set has one low outlier: 40.

Note that there can be any number of outliers: one on each side, two high ones and no low ones, and so forth. If there are outliers, we note them as such, try to explain them if possible, and restate and minimum and maximum at the boundaries we found.

Moral of the Story: How do we try to explain outliers? The best way is to examine the data: where it came from, what it represents, and where and how it was collected, In other words, look at the context. An outlier could turn out to be a simple mis-keying error, or might indicate something more serious. Don't make something up! But do have a look to see if there's something obvious going on.

Connection or Causation?

I was reading an article the other day about a possible connection between mental illness and nose jobs, and thought, "Aha! A perfect possible example of crazy statistics!" But I was not only disappointed on that count, but also pleasantly surprised that the article showed valid statistical practices. What I can focus on in this post are two things:

1. What valid statistical practices were used?
2. How to avoid common pitfalls when reading an article that involves statistics.

First, you might want to quickly read the article:
http://well.blogs.nytimes.com/2011/07/27/some-nose-job-patients-may-have-mental-illness/
Go ahead; I'll wait.

There are enough bad statistical articles out there, which I'm sure I'll happen upon and grab for another post, but as I said, this article is not one of them. First, the author was careful to qualify the difference between people who have valid medical reasons to seek plastic surgery on their noses and those who appear to be negatively obsessed with a perfectly normal-looking (from a plastic surgeon's standpoint) nose. The author was careful to separate the two.

This was a "controlled" study, as statisticians say. First, there was a rather large sample (266) of patients seeking nose jobs in Belgium, who filled out a diagnostic questionnaire that was intended to uncover the condition (called Body Dysmorpphic Disorder (BDD)). The author included all of the vital information -- number of people, location of the study, duration of the study, and a link to the full journal article so that the study could be reproduced if desired. Rather than worrying that a duplicate study will refute the findings of the original study, most statisticians welcome "do-overs" of the study, because it can either strengthen their findings or point out something they might have overlooked. After all, from an ethical standpoint, most researchers are after the truth, so they try to make the specifics as transparent as possible.

Separating those patients who had a "valid" complaint about their nose (for instance, a breathing problem) from those who showed signs of BDD was an excellent example of controlling the study. For example, if they hadn't done this and went on conducting the study, they couldn't have separated the "valid" patients from the BDD patients later. By controlling the study, they were able to determine that only 2% of the "valid" patients showed evidence of BDD while 43% of the "invalid" patients did. If everyone had been lumped together, the results might have been diluted. Researchers need to think of possible variables ahead of time that could muddy the waters later. The variable of "valid" versus "invalid" was one of these. We call these variables lurking variables, because they can work under the surface, making it look like one thing (seeking a nose job) is causing the other (BDD)... or vice versa.

Another interesting and admirable thing the article did was volunteer extra information that the reader might have had upon reading the article; that is, they singled out nose jobs from other plastic surgery as being notable with respect to this connection. They seem to have thought things through very well.

Now, on to you as a reader of such articles. My only caution in this case is that you don't read any type of cause and effect situation into the article. Having BDD doesn't necessarily cause people to seek a nose job, although there seems to be a very strong connection between the two. Further study and repetition of the study would be needed to prove cause and effect. To its credit, the author of this article was careful not to lead you in the causation direction.

Simply put: Just because two variables appear to be connected doesn't prove that one variable causes another. For those readers who recall the "old days" when people smoked cigarettes and didn't know about the dangers of lung cancer, remember how it took years to get even the mildest cautionary note placed on cigarette packs? The first such caution cited a connection between cigarette smoking and lung cancer. After years of further controlled study, researchers were finally able to put stronger cautions on cigarette packs: that cigarette smoking causes lung cancer.

Moral of the Story: This was an example of an article in which the author was very careful not to imply cause and effect. Other articles aren't so clear. As you read articles that involve numbers and statistics, be aware of this and don't jump to causation conclusions!

Self-Test: Name a lurking variable in this statement: "The more firefighters sent to a fire, the more damage is done to the structure."

Answer: Size of the fire is the lurking variable. It influences both variables: the amount of damage done, and the number of firefighters sent to the scene.

(Credit: Stats: Modeling the World, Bock, Velleman, deVeaux. Thanks!