Probability and Statistics - Analysis of Single-Variable Data

Analysis of Single-Variable Data

Univariate, or single-variable, data is usually given as a list of numbers. It's the type of list you'd take with you to the store if you had to go number shopping and were afraid you would forget to pick up a bag of 3's. 

Be careful: Put your list of numbers in order before trying to analyze them.

• Range

  
  Let's start with something near and dear to everyone's heart: test scores. Suppose ten students took a math test and received the following scores:

  61, 90, 72, 100, 80, 84, 88, 92, 92, 78

  It's difficult to make sense out of an unordered list of numbers, so let's start by ordering the list. We'll order the numbers from least to greatest, so they go the same direction as the number line. Also, so that we end on a high note.

  61, 72, 78, 80, 84, 88, 90, 92, 92, 100

  Since the original list had two copies of the score 92, we keep both copies of 92 in the ordered list also. Each score represents a separate individual, so we don't want to toss someone out just because she tied someone else's score.

  Since the scores on the test go from 61 to 100, the difference between the highest and lowest scores is

  100 – 61 = 39.

  This number, the difference between the highest and lowest values, is called the range of the data.

• Mode

  The phrase "a la mode," used today to mean "with ice cream," literally means "in the fashion." As we know, there's nothing more fashionable than ice cream. In statistics, the mode of a set of data is the most frequently occurring value. In other words, the mode is the most fashionable value. You can wear it around your neck as a wrap, or as a hat. It's incredibly versatile.

  Sample Problem

  The colors of ten students' backpacks are recorded below. Who knows why, but apparently someone thinks it's important.

  black, blue, red, blue, green, blue, black, orange, blue, blue

  What is the most fashionable backpack color?

  Putting your personal preference aside, the most fashionable backpack color is blue, since more students have a blue backpack than any other color. In more mathematical terms, blue is the mode of this set of data. However, by the time autumn rolls around, blue is going to be out and orange is totally going to be in.

  Samples

  In the list of values

  12, 13, 13, 13, 14, 15, 16, 16, 16, 17, 17

  there are two modes: 13 and 16. The numbers 13 and 16 each occur three times, which is more than any of the other numbers. Therefore, 13 and 16 are both fashionable. Explains why the two of them are so cliquish.

  The list of values

  50, 60, 70, 80, 90

  has no mode. Since each number occurs only once, there aren't any "fashionable" numbers in this list. Still, it wouldn't hurt you to be civil to them.

  Be careful: 0 is a perfectly good number, and a perfectly good mode. Having a mode of 0 is different from having no mode at all.

• Mean/Average

  When thinking about averages, we like to think about cookies. Okay, we always like to think about cookies...averages are only a convenient excuse.

  Suppose Paul has 3 cookies and Mary has 5. That distribution doesn't seem quite fair. Let's redistribute so Paul and Mary each receive the same number of cookies. We have 3 + 5 = 8 cookies total, which means to divide the cookies fairly between Paul and Mary, we should give each person 4 cookies. We should do it quickly though, because Mary just caught our eye and we're fairly certain she's onto us.

  Sample Problem

  Anita got 2 cookies, Jonas got 3, and Ella got 2. If we were to redistribute the cookies fairly, how many would each person get?

  We have 2 + 3 + 2 = 7 cookies total. If we divide these cookies evenly between the 3 people, we see that each person should receive:

  

  Hope they're chewy rather than crunchy cookies, or cutting these things will be a nightmare.

  The mean, or average, of a set of values is the size of a "fair" portion. To find the average of a set of values, we add up all the values and divide by the number of portions. "Average'' and ''mean'' refer to the same thing, and you may be asked to find either one. Their synonymousness doesn't extend outside the world of algebra though. Your mother won't care if you tell her a bully at school was being "average" to you.

  Sample Problem

  Louisa got 5 cookies, Danielle got 8 cookies, and Marcus got 2 cookies. What is the average number of cookies each person got?

  The average number of cookies is the number each person would get if we divided the cookies fairly. We add up all the different numbers of cookies:

  5 + 8 + 2 = 15

  Then we divide by the number of portions, which is 3, since there are 3 people:

  15 ÷ 3 = 5

  The average number of cookies received by each person is 5. Again, we'd better take care of this situation with haste. Mary from our earlier problem caught wind of the cookie surplus and is on her way over.

  Another phrase you might hear is the phrase "on average." This means roughly the same thing as "find the average of..." In the example above we could say that, on average, each person got 5 cookies. It's just another way of saying "5 is the average of the number of cookies each person got."

  Word problems involving averages can do some interesting things. They can't bend their legs behind their heads...nothing that interesting, but still. Interesting.

  Sample Problem

  Linda bought four cookies that cost $0.25 each and two cookies that cost $0.50 each. On average, how much did she spend per cookie?

  We want to make a list of all the cookie prices, and find the average of the numbers in that list. Since Linda bought four cookies for $0.25 each, the number $0.25 will show up four times in the list. Similarly, the number $0.50 will show up twice. The prices of the cookies were:

  0.25, 0.25, 0.25, 0.25, 0.50, 0.50

  If we add these numbers and divide by 6 (the total number of cookies purchased), we get

  ,

  or about $0.33 per cookie. Maybe next time Linda will consider buying cookies by the box. Preferably in bulk from Costco.

• Median

  The median of a list of numbers is the "middle" number. To find the median, first we put our list of numbers in order. Then we cross off pairs of numbers (one from the top of the list and one from the bottom of the list, or one from each end of the list, depending on how we've written it out) until we're left with one number. That number is the median. If the median was a contestant on Survivor, he would totally be taking home that $1,000,000 prize.

  Sample Problem

  Find the median of the following list:

  3, 4, 6, 5, 7, 10, 11

  First, we put the numbers in order:

  3, 4, 5, 6, 7, 10, 11

  Then we take off one number from each end of the list:

  4, 5, 6, 7, 10

  We do it again:

  5, 6, 7

  And one more time:

  6

  We're left with the number 6, so that's the middle number in the list. The median is 6. The tribe has spoken.

  It's pretty easy when we've got an odd number of entries. We crossed off one number from each end of the list, and repeated until we were left with one number, which left us with exactly one number. However, if our list has an even number of entries to start with, we get two middle numbers. Keeping with our Survivor analogy, they'd probably need to fight each other to the death to determine a clear winner. But since this is a family example, let's instead find the median between these two middle numbers.

  To find the median, we take the mean (or average) of these two numbers in the middle.

  Sample Problem

  Find the median of the following list:

  2, 3, 4, 8, 9, 10

  Our list is already in order from smallest to biggest, so we cross off the first and last number:

  3, 4, 8, 9

  Then we do it again:

  4, 8

  Now we have only two numbers left. If we cross them both off we'll have no numbers left, which won't be useful for anything, aside from getting rid of all those noisy numbers so we can finally get a few minutes of shut-eye. Instead, we take the average of the two numbers we have left:

  

  The median of the list is 6.

• Quartiles

  
  Quartiles are numbers that break a list of data up into quarters ("quarters,'' "quartiles''...makes sense, right?). We're totally stuffed full of cookies at the moment, so let's turn our attention to, um, brownies.

  To cut a pan of brownies into quarters, we need to make three cuts:

  

  Similarly, to break a list of data into quarters, we need to find three quartiles.

  To cut a pan of brownies into quarters, first we cut the pan down the middle:

  

  Then we cut the left half down the middle:

  

  ...and finally cut the right half down the middle:

  

  It appears we're using an exceptionally long pan. We're amazed that thing even fit in the oven.

  To find quartiles, we do basically the same thing as we did with the brownies. We just don't eat them afterward, unless we want to have a severe bellyache.

  Sample Problem

  Find the quartiles of the list of numbers

  1, 4, 5, 7, 8, 10, 23, 25, 28, 32, 40.

  To find the quartiles, first we cut the list in half down the middle; that is, we find the median:

  1, 4, 5, 7, 8, 10, 23, 25, 28, 32, 40

  Now we look at the left half of the list (not including the median):

  1, 4, 5, 7, 8

  And find its median:

  1, 4, 5, 7, 8

  Finally, we look at the right half of the list (not including the median):

  23, 25, 28, 32, 40

  And find its median:

  23, 25, 28, 32, 40

  The quartiles of this list are:

  5, 10, 28

  The quartiles are often called Q₁, Q₂, and Q₃ (or first quartile, second quartile, and third quartile). If you ever see Q₄, he's an impostor and should be escorted from the premises immediately. Remember, there may be four quarters, but there are only three cuts. In the example above, we had

  

  In summary, to find the quartiles of an ordered list, there are four steps:

  1. Find the median, which is quartile Q₂.
      
  2. Find the median of the "left'' half of the list (not including Q₂), which is quartile Q₁.
      
  3. Find the median of the "right'' half of the list (not including Q₂), which is quartile Q₃.
      
  4. Draw seven new quartiles from the bag and wait for your opponent to take their next turn. Wait a second...

  Sample Problem

  Find the quartiles of the list

  2, 4, 6, 6, 7, 10, 14.

  First we find the median, also known as Q₂:

  2, 4, 6, 6, 7, 10, 14

  We know Q₂ = 6.

  Now we find the median of the left half of the list, not including Q₂. The median of

  2, 4, 6

  is 4, so Q₁ = 4.

  Finally, we find the median of the right half of the list, not including Q₂. The median of

  7, 10, 14

  is 10, so Q₃ = 10.

  A couple of warnings are in order here. First of all, don't venture past the "No Trespassing" sign. That should go without saying. Second, and more to the point, we've said the quartiles are the numbers that divide the data set into four pieces, which is true. However, to make things confusing, "quartile'' can also refer to one of those four pieces. In the list

  1, 2, 3, 4, 5, 6, 7, 8

  we would say the quartiles are

  Q₁ = 2.5, Q₂ = 4.5, Q₃ = 6.5.

  We could also say the quartiles are the four sets into which the data has been divided:

  

  As if that discrepancy wasn't bad enough, there is no global consensus on how to find the quartiles, so some calculators and computer programs may come up with different quartiles than you do. It doesn't mean you're doing something wrong; it just means you're following a different recipe. Your calculator may be making chicken parmesan and you might be making a skirt steak, but don't sweat it. It's all delicious.

  Dr. Math talks here about some of the different ways quartiles can be calculated. We've gone with the most common way, but be sure it's in agreement with your teacher and textbook. We're honored you want to do it the Shmoop way, but we're not the ones grading your pop quizzes.