Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)
Part IV. Picturing and Measuring - Graphs, Measures, Stats, and Sets
Chapter 19. Figuring Your Chances: Statistics and Probability
In This Chapter
Knowing how statistics works with both qualitative and quantitative data
Finding out how to calculate a percentage and the mode of a sample
Calculating the mean and median
Finding the probability of an event
Statistics and probability are two of the most important and widely used applications of math. They're applicable to virtually every aspect of the real world — business, biology, city planning, politics, meteorology, and many more areas of study. Even physics, once thought to be devoid of uncertainty, now relies on probability.
In this chapter, I give you a basic understanding of these two mathematical ideas. First, I introduce you to statistics and the important distinction between qualitative and quantitative data. I show you how to work with both types of data to find meaningful answers. Then I give you the basics of probability. I show you how the probability that an event will occur is always a number from 0 to 1 — that is, usually a fraction, decimal, or percent. After that, I demonstrate how to build this number by counting both favorable outcomes and possible outcomes. Finally, I put these ideas to work by showing you how to calculate the probability of tossing coins.
Gathering Data Mathematically: Basic Statistics
Statistics is the science of gathering and drawing conclusions from data, which is information that's measured objectively in an unbiased, reproducible way.
An individual statistic is a conclusion drawn from this data. Here are some examples:
· The average working person drinks 3.7 cups of coffee every day.
· Only 52% of students who enter law school actually graduate.
· The cat is the most popular pet in the United States.
· In the last year, the cost of a high-definition TV dropped by an average of $575.
Statisticians do their work by identifying a population that they want to study: working people, law students, pet owners, buyers of electronics, whoever. Because most populations are far too large to work with, a statistician collects data from a smaller, randomly selected sample of this population. Much of statistics concerns itself with gathering data that's reliable and accurate. You can read all about this idea in Statistics For Dummies, 2nd Edition, by Deborah J. Rumsey (Wiley).
In this section, I give you a short introduction to the more mathematical aspects of statistics.
Understanding differences between qualitative and quantitative data
Data — the information used in statistics — can be either qualitative or quantitative. Qualitative data divides a data set (the pool of data that you've gathered) into discrete chunks based on a specific attribute. For example, in a class of students, qualitative data can include
· Each child's gender
· His or her favorite color
· Whether he or she owns at least one pet
· How he or she gets to and from school
You can identify qualitative data by noticing that it links an attribute — that is, a quality — to each member of the data set. For example, four attributes of Emma are that she's female, her favorite color is green, she owns a dog, and she walks to school.
On the other hand, quantitative data provides numerical information — that is, information about quantities, or amounts. For example, quantitative data on this same classroom of students can include the following:
· Each child's height in inches
· Each child's weight in pounds
· The number of siblings each child has
· The number of words each child spelled correctly on the most recent spelling test
You can identify quantitative data by noticing that it links a number to each member of the data set. For example, Carlos is 55 inches tall, weighs 68 pounds, has three siblings, and spelled 18 words correctly.
Working with qualitative data
Qualitative data usually divides a sample into discrete chunks. As my sample — which is purely fictional — I use 25 children in Sister Elena's fifth-grade class. For example, suppose all 25 children in Sister Elena's class answer the three yes/no questions in Table 19-1.
Table 19-1 Sister Elena's Fifth-Grade Survey
Question |
Yes |
No |
Are you an only child? |
5 |
20 |
Do you own any pets? |
14 |
11 |
Do you take the bus to school? |
16 |
9 |
The students also answer the question “What is your favorite color?” with the results in Table 19-2.
Even though the information that each child provided is non-numerical, you can handle it numerically by counting how many students made each response and working with these numbers.
Given this information, you can now make informed statements about the students in this class just by reading the charts. For instance,
· Exactly 20 children have at least one brother or sister.
· Nine children don't take the bus to school.
· Only one child's favorite color is yellow.
Playing the percentages
You can make more sophisticated statistical statements about qualitative data by finding out the percentage of the sample that has a specific attribute. Here's how you do so:
1. Write a statement that includes the number of members who share that attribute and the total number in the sample.
Suppose you want to know what percentage of students in Sister Elena's class are only children. The chart tells you that 5 students have no siblings, and you know that 25 kids are in the class. So you can begin to answer this question as follows:
o Five out of 25 children are only children.
2. Rewrite this statement, turning the numbers into a fraction:
In the example, of the children are only children.
3. Turn the fraction into a percent, using the method I show you in Chapter 12.
You find that , so 20% of the children are only children.
Similarly, suppose you want to find out what percentage of children take the bus to school. This time, the chart tells you that 16 children take the bus, so you can write this statement:
· Sixteen out of 25 children take the bus to school.
Now rewrite the statement as follows:
Finally, turn this fraction into a percent: 16 ÷ 25 = 0.64, which equals 64%, so
· 64% of the children take the bus to school.
Getting into the mode
The mode tells you the most popular answer to a statistical question. For example, in the poll of Sister Elena's class (see Tables 19-1 and 19-2), the mode groups are children who
· Have at least one brother or sister (20 students)
· Own at least one pet (14 students)
· Take the bus to school (16 students)
· Chose blue as their favorite color (8 students)
When a question divides a data set into two parts (as with all yes/no questions), the mode group represents more than half of the data set. But when a question divides a data set into more than two parts, the mode doesn't necessarily represent more than half of the data set.
For example, 14 children own at least one pet, and the other 11 children don't own one. So the mode group — children who own a pet — is more than half the class. But 8 of the 25 children chose blue as their favorite color. So even though this is the mode group, fewer than half the class chose this color.
With a small sample, you may have more than one mode — for example, perhaps the number of students who like red is equal to the number who like blue. However, getting multiple modes isn't usually an issue with a larger sample because it becomes less likely that exactly the same number of people will have the same preference.
Working with quantitative data
Quantitative data assigns a numerical value to each member of the sample. As my sample — again, fictional — I use five members of Sister Elena's basketball team. Suppose that the information in Table 19-3 has been gathered about each team member's height and most recent spelling test.
Table 19-3 Height and Spelling Test Scores
Student |
Height in Inches |
Number of Words Spelled Correctly |
Carlos |
55 |
18 |
Dwight |
60 |
20 |
Patrick |
59 |
14 |
Tyler |
58 |
17 |
William |
63 |
18 |
In this section, I show you how to use this information to find the mean and median for both sets of data. Both terms refer to ways to calculate the average value in a quantitative data set. An average gives you a general idea of where most individuals in a data set fall so you know what kinds of results are standard. For example, the average height of Sister Elena's fifth-grade class is probably less than the average height of the Los Angeles Lakers. As I show you in the sections that follow, an average can be misleading in some cases, so knowing when to use the mean versus the median is important.
Finding the mean
The mean is the most commonly used average. In fact, when most people use the word average, they're referring to the mean. Here's how you find the mean of a set of data:
1. Add up all the numbers in that set.
For example, to find the average height of the five team members, first add up all their heights:
o 55 + 60 + 59 + 58 + 63 = 295
2. Divide this result by the total number of members in that set.
Divide 295 by 5 (that is, by the total number of boys on the team):
So the mean height of the boys on Sister Elena's team is 59 inches.
This procedure is summed up (so to speak) in simple formula:
You can use this formula to find the mean number of words that the boys spelled correctly. To do this, plug the number of words that each boy spelled correctly into the top part of the formula, and then plug the number of boys in the group into the bottom part:
Now simplify to find the result:
As you can see, when you divide, you end up with a decimal in your answer. If you round to the nearest whole word, the mean number of words that the five boys spelled correctly is about 17 words. (For more information about rounding, see Chapter 2.)
The mean can be misleading when you have a strong skew in data — that is, when the data has many low values and a few very high ones, or vice versa.
For example, suppose that the president of a company tells you, “The average salary in my company is $200,000 a year!” But on your first day at work, you find out that the president's salary is $19,010,000 and each of his 99 employees earns $10,000. To find the mean, first plug the total salaries ($19,010,000 for the president plus $10,000 for each of 99 employees) into the top of the formula. Next, plug the number of employees (100) into the bottom:
Now calculate:
So the president didn't lie. However, the skew in salaries resulted in a misleading mean.
Finding the median
When data values are skewed (when a few very high or very low numbers differ significantly from the rest of the data), the median can give you a more accurate picture of what's standard. Here's how to find the median of a set of data:
1. Arrange the set from lowest to highest.
To find the median height of the boys in Table 19-3, arrange their five heights in order from lowest to highest.
o
2. Choose the middle number.
The middle value, 59 inches, is the median average height.
To find the median number of words that the boys spelled correctly (refer to Table 19-3), arrange their scores in order from lowest to highest:
·
This time, the middle value is 18, so 18 is the median score.
If you have an even number of values in the data set, put the numbers in order and find the mean of the two middle numbers in the list (see the preceding section for details on the mean). For instance, consider the following:
·
The two center numbers are 5 and 7. Add them together to get 12, and then divide by 2 to get their mean. The median in this list is 6.
Now recall the company president who makes $19,010,000 a year and his 99 employees who each earn $10,000. Here's how this data looks:
·
As you can see, if you wrote out all 100 salaries, the center numbers would obviously both be 10,000. The median salary is $10,000, and this result is much more reflective of what you'd probably earn if you worked at this company.
Looking at Likelihoods: Basic Probability
Probability is the mathematics of deciding how likely an event is to occur. For example,
· What's the likelihood that the lottery ticket I bought will win?
· What's the likelihood that my new car will need repairs before the warranty runs out?
· What's the likelihood that more than 100 inches of snow will fall in Manchester, New Hampshire, this winter?
Probability has a wide variety of applications in insurance, weather prediction, biological sciences, and even physics.
The study of probability started hundreds of years ago when a group of French noblemen began to suspect that math could help them turn a profit, or at least not lose so heavily, in the gambling halls they frequented.
You can read all about the details of probability in Probability For Dummies, by Deborah J. Rumsey (Wiley). In this section, I give you a little taste of this fascinating subject.
The Silver standard
Probability can be a powerful tool for predicting weather patterns, sports events, and election results. In his bestselling book The Signal and the Noise, Nate Silver discusses how statistical modeling, when done correctly, can permit mathematicians to peer into the future with spooky accuracy. He also discusses why a lot of apparently scientific predictions go wrong. Silver's work is cutting edge, and he does a good job of explaining what statisticians do without too much jargon or complicated equations. Check him out!
Figuring the probability
The probability that an event will occur is a fraction whose numerator (top number) and denominator (bottom number) are as follows (for more on fractions, flip to Chapter 9):
In this case, the number of target outcomes (or successes) is simply the number of outcomes in which the event you're examining does happen. In contrast, the number of total outcomes (or sample space) is the number of outcomes that can happen.
For example, suppose you want to know the probability that a tossed coin will land heads up. Notice that there are two total outcomes (heads or tails), but only one of these outcomes is the target — the outcome in which heads comes up. To find the probability of this event, make a fraction as follows:
So the probability that the coin will land heads up is .
So what's the probability that, when you roll a die, the number 3 will land face up? To figure this one out, notice that there are six total outcomes (1, 2, 3, 4, 5, and 6), but in only one of these does 3 land face up. To find the probability of this outcome, make a fraction as follows:
So the probability that the number 3 will land face up is .
And what's the probability that, if you pick a card at random from a deck, it'll be an ace? To figure this out, notice that there are 52 total outcomes (one for each card in the deck), but in only 4 of these do you pick an ace. So
So the probability that you'll pick an ace is , which reduces to (see Chapter 9 for more on reducing fractions).
Probability is always a number from 0 to 1. When the probability of an outcome is 0, the outcome is impossible. When the probability of an outcome is 1, the outcome is certain.
Oh, the possibilities! Counting outcomes with multiple coins
Although the basic probability formula isn't difficult, sometimes finding the numbers to plug into it can be tricky. One source of confusion is in counting the number of outcomes, both target and total. In this section, I focus on tossing coins.
When you flip a coin, you can generally get two total outcomes: heads or tails. When you flip two coins at the same time — say, a penny and a nickel — you can get four total outcomes:
Outcome |
Penny |
Nickel |
#1 |
Heads |
Heads |
#2 |
Heads |
Tails |
#3 |
Tails |
Heads |
#4 |
Tails |
Tails |
When you flip three coins at the same time — say, a penny, a nickel, and a dime — eight outcomes are possible:
Notice the pattern: Every time you add a coin, the number of total outcomes doubles. So if you flip six coins, here's how many total outcomes you have:
The number of total outcomes equals the number of outcomes per coin (2) raised to the number of coins (6): Mathematically, you have 2^{6} = 64.
Here's a handy formula for calculating the number of outcomes when you're flipping, shaking, or rolling multiple coins, dice, or other objects at the same time:
Suppose you want to find the probability that six tossed coins will all fall heads up. To do this, you want to build a fraction, and you already know that the denominator — the number of total outcomes — is 64. Only one outcome is the target outcome, so the numerator is 1:
So the probability that six tossed coins will all fall heads up is .
Here's a more subtle question: What's the probability that exactly five out of six tossed coins will all fall heads up? Again, you're building a fraction, and you already know that the denominator is 64. To find the numerator (target outcomes), think about it this way: If the first coin falls tails up, then all the rest must fall heads up. If the second coin falls tails up, then again all the rest must fall heads up. This is true of all six coins, so you have six target outcomes:
Therefore, the probability that exactly five out of six coins will fall heads up is , which reduces to (see Chapter 9 for more on reducing fractions).