THE COMPLETE IDIOT'S GUIDE TO and Design are registered trademarks of Penguin Group (USA) Inc. ISBN: X Library of Congress Catalog. Philosophy continually asks questions about these issues: What exists? To some, philosophy may The Complete Idiot''s Guide to Algebra - Yola. Statistics, statistics everywhere, but not a single word can we understand! .. THE COMPLETE IDIOT'S GUIDE TO and Design are registered.
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Not a numbers person? No problem! - Selection from The Complete Idiot's Guide to Statistics, 2nd Edition [Book]. And, guess what, that is sometimes how statisticians are born! xviii The Complete Idiot's Guide to Statistics, Seco nd Edition How This Book Is Organized The. Robert A. Donnelly Jr., Ph.D., is an associate professor of management at Goldey - Beacom College, a private business institution in Wilmington, Delaware.
This is not possible when any person surfing the Internet can participate in the poll. The lesson here is that we are all consumers of statistics. We are constantly surrounded by information provided by someone who is trying to influence us or gain our support. By having a basic understanding about the field of statistics, we increase the likelihood that we can ward off those evil spirits in their attempts to distort the truth. Seventy-three percent of Asian American households in the United States own a computer.
Households with children under the age of 18 are more likely to have access to the Internet 62 percent than family households with no children 53 percent. Hank Aaron hit career home runs. The average SAT score for incoming freshman at a local college was On a recent poll, 67 percent of Americans had a favorable opinion of the President of the United States. You can find additional sample problems on my website: U The field of statistics evolved from the early work of European mathematicians during the seventeenth century.
U Descriptive statistics focuses on the summary or display of data so we can quickly obtain an overview. U Inferential statistics allows us to make claims or conclusions about a population based on a sample of data from that population. U We are all consumers of statistics and need to be aware of the potential misuses that can occur in this field. The validity of any statistical study hinges on the validity of the data from the beginning of the process.
Many things can come into question, such as the accuracy of the data or the source of the data. Without the proper foundation, your efforts to provide a sound analysis will come tumbling down. The issues surrounding data can be surprisingly complex. What could go wrong? Well, plenty can. Because data can be classified in several ways, we need to recognize the difference between quantitative and qualitative data and how each is used.
The data measurement choice we make at the start of the study will determine what kind of statistical techniques we can apply. I can measure how many times Debbie snores over a minute period. I can measure the length of each snore in seconds. That one sounded like an Alaskan seal calling for its young. And the final attempt measures the event by describing volume using words rather than numbers.
Each of these cases just shows a different way to use data. Data that is used to describe something of interest about a population is called a parameter. However, if the data is describing a sample from that population, we refer to it as a statistic. If we average the number of trips per child, this figure would be considered a parameter because the entire population was measured.
We can consider the average that we observe from her class a statistic if we assume it could be used to estimate all threeyear-olds in the country. Data is the building blocks of all statistical studies. You can hire the most expensive, well-known statisticians and provide them with the latest computer hardware and software available, but if the data you provide them is inaccurate or not relevant to the study, the final results will be worthless.
By definition, data is just the raw facts and figures that pertain to a measurement of interest. Information, on the other hand, is derived from the facts for the purpose of making decisions.
One of the major reasons to use statistics is to transform data into information. For example, the table that follows shows monthly sales data for a small retail store. Data is the value assigned to an observation or a measurement and is the building blocks to statistical analysis. The plural form is data and the singular form is datum, referring to an individual observation or measurement. Data that describes a characteristic about a population is known as a parameter. Data that describes a characteristic about a sample is known as a statistic.
Information is data that is transformed into useful facts that can be used for a specific purpose, such as making a decision. You are doing something very wrong. At this rate, you will be out of business by early next year.
Secondary data is data that somebody else has collected and made available for others to use. The U. The Department of the Interior provides all sorts of data about U. For instance, did you know there are species of squirrels in this country? Primary data is data that you have collected for your own use. Rather than each department in the government being responsible for collecting and disbursing data as in the United States, Canada has a national statistical agency known as Statistics Canada www.
The main drawback of using secondary data is that you have no control over how the data was collected. Primary data, on the other hand, is data collected by the person who eventually uses this data.
When collecting primary data, you want to ensure that the results will not be biased by the manner in which it is collected. You can obtain primary data in many ways, such as direct observation, surveys, and experiments. Random Thoughts The Internet has also become a rich source of data for statistics published by various industries.
If you can muddle your way through the 63, sites that come back from the typical Internet search engine, you might find something useful. I once found a Japanese study on the effect of fluoride on toad embryos www. Before this discovery, I was completely oblivious to the fact that toads even had teeth, much less a cavity problem. Examples of these studies would be observing wild animals stalking their prey in the forest or teenagers at the mall on Friday night or is that the same example?
The advantage of this method is that the subjects will unlikely be influenced by the data collection. Focus groups are a direct observational technique where the subjects are aware that data is being collected. Businesses use focus groups to gather information in a group setting controlled by a moderator. The subjects are usually paid for their time and are asked to comment on specific topics. An example of a treatment could be the use of a new medical drug. Two groups would be established.
The first is the experimental group who receive the new drug, and the second is the control group who think they are getting the new drug but are in fact getting no medication. The reactions from each group are measured and compared to determine whether the new drug was effective.
The claims that the experimental studies are attempting to verify need to be clear and specific. I just recently read about an herb called ginkgo biloba. According to this article, people who make money selling funny-sounding herbs claim ginkgo biloba will keep your mind sharp as you age. Sounds like something everyone would want. As stated, this claim might prove difficult to verify. And then, how do you measure sharpness of mind? These are some of the challenges that statistical experiments face.
The benefit of experiments is that they allow the statistician to control factors that could influence the results, such as gender, age, and education of the participants. The concern about collecting data through experiments is that the response of the subjects might be influenced by the fact that they are participating in a study. The design of experiments for a statistical study is a very complex topic and goes beyond the scope of this book.
The questionnaire needs to be carefully designed to avoid any bias see Chapter 1 or confusion for those participating. Some participants respond in a way they feel the survey would like them to. This is very similar to the manner in which hostages bond with their captors. The survey can be administered by e-mail, snailmail, or telephone.
A question posed in a positive tone will tend to invoke a more positive response and vice versa. A good strategy is to test your questionnaire with a small group of people before releasing it to the general public. Whatever method you employ, your primary concern should always be that the sample is representative of the population in which you are interested.
U Quantitative data uses numerical values to describe something of interest. See page U Qualitative data uses descriptive terms to measure or classify something of inter- est. One example of qualitative data is the name of a respondent in a survey and his or her level of education.
The next section covers qualitative data in more detail. The final way to classify data is by the way it is measured. This distinction is critical because it affects which statistical techniques we can use in our analysis of the data. Measurement classification can be made in several levels.
It has all the properties of nominal data with the added feature that we can rank-order the values from highest to lowest. An example is if you were to have a lawnmower race.
However, we cannot say how much faster. Ordinal data can be either qualitative or quantitative. An example of quantitative data is rating movies with 1, 2, 3, or 4 stars. However, we still may not claim that a 4-star movie is 4 times as good as a 1-star movie. Now we can get to work with the mathematical operations of addition and subtraction when comparing values. For this data, we can measure the difference between the different categories with actual numbers and also provide meaningful information.
Temperature measurement in degrees Fahrenheit is a common example here. For instance, 70 degrees is 5 degrees warmer than 65 degrees. Why not? Simply because we cannot argue that degrees is twice as warm as 50 degrees. This is as good as it gets as far as data is concerned. Now we can perform all four mathematical operations to compare values with absolutely no feelings of guilt.
Examples of this type of data are age, weight, height, and salary. Ratio data has all the features of interval data with the added benefit of a true 0 point. For instance, 0 salary indicates the absence of any salary. Wrong Number Interval data does not have a true 0 point. For example, 0 degrees Fahrenheit does not represent the absence of temperature, even though it may feel like it.
To help explain this, try baking a cake at twice the recommended temperature in half the recommended time.
With a true 0 point, we can use the rules of multiplication and division to compare data values. This allows us to say that a person who is 6 feet in height is twice as tall as a 3-foot person or that a yearold person is half the age of a 40 year old. The distinction between interval and ratio data is a fine line. There are endless examples of ratio data. I can type maybe 20 words a minute on a good day.
Because we can correctly say that John types three times faster than me, typing speed is an example of ratio data. Figure 2. As we explore different statistical techniques later in this book, we will revisit these different measurement scales.
You will discover that specific techniques require certain types of data. If you have no interest in using Excel in this manner, just skip this section. The purpose of this last section is to talk about the use of computers with statistics in general and then to make sure your computer is ready to follow us along. Courtesy of www. It can perform all sorts of mathematical calculations but is far from being user friendly.
During my freshman year in college, I downloadd my first handheld calculator, a Texas Instrument model that could only perform the basic math functions. It was the approximate size of a cash register. Fortunately, we have advanced from the Dark Ages and now have awesome, user-friendly computing power at our fingertips. Parts of this book will demonstrate how to solve some of the statistical techniques using Microsoft Excel.
Choosing to skip these parts will not interfere with your grasp of topics in subsequent chapters. This is simply optional material to expose you to statistical analysis on the computer. I also assume you already have a basic working knowledge of how to use Excel. Open the Excel program and left-click with your mouse on the Tools menu as shown in Figure 2. Notice in the figure that the highlighted item is Data Analysis, which may or may not show up under your Tools menu.
To make all the menu items visible, click on the Expand symbol at the bottom of the list the double-downwardpointing arrows. This dialog box provides a list of available add-ins for you to use. Now click on the Tools menu again; Data Analysis will now appear in the list.
This feature is not currently installed. Would you like to install it now? Click the Yes button and follow any further instructions. Then, the Data Analysis option will become available on the Tools menu.
Your Excel program is now ready to perform all sorts of statistical magic for you as we explore various techniques throughout this book.
At this point, you can click Cancel and close out Excel. Each time you open Excel in the future, the Data Analysis option will be available under the Tools menu. Explain your choice. Average monthly temperature in degrees Fahrenheit for the city of Wilmington throughout the year 2. Average monthly rainfall in inches for the city of Wilmington throughout the year 3. Age of the respondents in the survey 6. The year in which the respondent was born 8. The voting intentions of the respondents in the survey classified as Republican, Democrat, or Undecided 9.
The uniform number of each member on a sports team A list of the graduating high school seniors by class rank Final exam scores for my statistics class on a scale of 0 to U Ordinal data has all the properties of nominal data with the additional capability of arranging the observations in order. U Interval data has all the properties of ordinal data with the additional capability of calculating meaningful differences between the observations. U Ratio data has all the properties of interval data with the additional capability of expressing one observation as a multiple of another.
In its basic form, making sense of the patterns in the data can be very difficult because our human brains are not very efficient at processing long lists of raw numbers. We do a much better job of absorbing data when it is presented in summarized form through tables and graphs.
In the next several sections, we will examine many ways to present data so that it will be more useful to the person performing the analysis. Through these techniques, we are able to get a better overview of what the data is telling us.
And believe me, there is plenty of data out there with some very interesting stories to tell.
Stay tuned. The best way to describe a frequency distribution is to start with an example. Anyway, below is a table of the batting averages of individual Pirates at the end of the season. I have not attached names with these averages in order to protect their identities. A frequency distribution is a table that shows the number of data observations that fall into specific intervals. Transforming this data into the frequency distribution shown here makes this fact more obvious.
Batting Average Number of Players. In this example, the intervals are the batting average ranges in the first column of the table. The intervals in a frequency distribution are officially known as classes, and the number of observations in each class is known as class frequencies.
A very confusing phone bill that requires a Ph. Using this data, I have constructed the following frequency distribution. From classes of equal size.
I chose 3 data values to be in each class for this distribution. An example of a class is 0—2, which includes the number of days when John made 0, 1, or 2 calls. Make classes mutually exclusive, or in other words, prevent classes from overlapping.
Try to have no fewer than 5 classes and no more than 15 classes. Too few or too many classes tend to hide the true characteristics of the frequency distribution. Avoid open-ended classes, if possible for instance, a highest class of 15—over. Classes are considered mutually exclusive when observations can only fall into one class. Include all data values from the original table in a class. In other words, the classes should be exhaustive. Too few or too many classes will obscure patterns in a frequency distribution.
Consider the extreme case where there are so many classes that no class has more than one observation. The other extreme is where there is only one class with all the observations residing in that class. This would be a pretty useless frequency distribution! Rather than display the number of observations in each class, this method calculates the percentage of observations in each class by dividing the frequency of each class by the total number of observations.
Relative frequency distributions display the percentage of observations of each class relative to the total number of observations. The total percentage in a relative frequency distribution should be percent or very close within 1 percent, because of rounding errors. Get it? Cousin, relative? This provides you with the percentage of observations that are less than or equal to the class of interest.
The resulting cumulative frequency distribution is shown here. Cumulative frequency distributions indicate the percentage of observations that are less than or equal to the current class.
According to this table, John used his phone 8 times or less on 84 percent of the days in the month. If designed properly, frequency distribution is an excellent way to tease good information out of stubborn data.
A histogram is a bar graph showing the number of observations in each class as the height of each bar. At least the highest class on the graph is the 0 to 2 calls per day. Things could be worse. How nice! The first thing we need to do is open Excel to a blank sheet and enter our data in Column A starting in Cell A1 use the data from the earlier table. Next enter the upper limits to each class in Column B starting in Cell B1.
For example, in the class 0—2, the upper limit would be 2. Random Thoughts For some bizarre reason, Excel refers to classes as bins. Go figure. Go to the Tools menu at the top of the Excel window and select Data Analysis. Select the Histogram option from the list of Analysis Tools see Figure 3. Data Analysis dialog box. In the Histogram dialog box as shown in Figure 3.
Then, click in the Bin Range list box and click in the worksheet to select cells B1 through B6 the upper limits for the 6 classes. Click OK to generate the frequency distribution and histogram see Figure 3. The problem here is that the histogram looks like an elephant sat on it. Click on the chart to select it, and then click on the bottom border to drag the bottom of the chart down lower, expanding the histogram to look like Figure 3. Frequency distributions and histograms are convenient ways to get an accurate picture of what your data is trying to tell you.
The Chart Wizard allows me more control over the final appearance. A statistician named John Tukey originated the idea during the s. The major benefit of this approach is that all the original data points are visible on the display.
Normally, Brian would only report his better scores, but we statisticians must be unbiased and accurate. Because there were 5 scores in the 70s, there are 5 digits to the right of 7. If we choose to, we can break this display down further by adding more stems. The stem and leaf display splits the data values into stems the first digit in the value and leaves the remaining digits in the value. By listing all of the leaves to the right of each stem, we can graphically describe how the data is distributed.
Here, the stem labeled 7 5 stores all the scores between 75 and The stem 8 0 stores all the scores between 80 and Brian usually scores in the low 80s. You can find an excellent source of information about stem and leaf displays at the Statistics Canada website at www. This type of chart is simply a circle divided into portions whose area is equal to the relative frequency distribution.
If you cannot use colors, use patterns and textures to display pie charts. As you can see, the pie chart approach is much easier on the eye when compared to looking at data from a table. This person must be a pretty good statistics teacher! To construct a pie chart by hand, you first need to calculate the center angle for each slice in the pie, which is illustrated in Figure 3. You determine the center angle of each slice by multiplying the relative frequency of the class by which is the number of degrees in a circle.
These results are shown in the following table. To demonstrate this type of chart see Figure 3. An unnamed filing cabinet. The histogram that we visited earlier in the chapter is actually a special type of bar chart that plots frequencies rather than actual data values. Bar charts are more useful when you want to highlight the actual data values. Our current resident teenagers seem to have a costly compulsion to take very long, very hot showers, and sometimes more than once a day.
As I lie awake at night listening to the constant stream of hot water, all I can envision are dollar bills flowing down the drain. So I have tabulated some data, which shows the number of showers the cleanest kids on the block have taken in each of the recent months with the corresponding utility bill. Notice that at these rates we average more than one shower per day. Because the line connecting the data points seems to have an overall upward trend, my suspicions hold true.
It seems the more showers our waterlogged darlings take, the higher the utility bill. Line charts prove very useful when you are interested in exploring patterns between two different types of data.
They are also helpful when you have many data points and want to show all of them on one graph. Now that you have mastered the art of displaying descriptive statistics, you are ready to move on to calculating them in the next chapter.
The following table represents the exam grades from 36 students from a certain class that I might have taught. Construct a frequency distribution with 9 classes ranging from 56 to Construct a histogram using the solution from Problem 1.
Construct a relative and a cumulative frequency distribution from the data in Problem 1. Construct a pie chart from the solution to Problem 1. Construct a stem and leaf diagram from the data in Problem 1 using one stem for the scores in the 50s, 60s, 70s, 80s, and 90s. Construct a stem and leaf diagram from the data in Problem 1 using two stems for the scores in the 50s, 60s, 70s, 80s, and 90s. U Histograms provide a graphical overview of data from frequency distributions.
U Pie, bar, and line charts are effective ways to present data in different graphical forms. With that task behind us, we can now proceed to the next step— summarizing our data numerically. If these are not calculated with loving care, our final analysis could be misleading.
And as everybody knows, statisticians never want to be misleading. So this chapter focuses on how to calculate descriptive statistics manually and, if you choose, how to verify these results with our good friend Excel. This is the first chapter that uses mathematical formulas that have all those funnylooking Greek symbols that can make you break out into a cold sweat.
But have no fear. We will slay these demons one by one through careful explanation and, in the end, victory will be ours. The first, measures of central tendency, describes the center point of our data set with a single value. The second category, measures of dispersion, is the topic of Chapter 5. Measures of central tendency describe the center point of a data set with a single value.
Measures of dispersion describe how far individual data values have strayed from the mean. The formula for the sample mean is as follows: As in many teenage households, video games are a common form of entertainment in our family room. Turns out the delay was really between my brain and my fingers. Anyway, the following data set represents the number of hours each week that video games are played in our household.
A weighted mean refers to a mean that needs to go on a diet. Just kidding; I was checking to see whether you were paying attention. A weighted mean allows you to assign more weight to certain values and less weight to others.
Type Score Weight Percent Exam Project Homework 94 89 83 50 35 15 We can calculate your final grade using the following formula for a weighted average. Note that here we are dividing by the sum of the weights rather than by the number of data values.
You earned an A! The weights in a weighted average do not need to add to 1 as in the previous example. I would calculate this by: You can actually calculate the mean of grouped data from a frequency distribution.
After we have determined the midpoint for each class, we can calculate the mean of the frequency distribution using the following equation—which is basically a weighted average formula: Wrong Number The mean of a frequency distribution where data is grouped into classes is only an approximation to the mean of the original data set from which it was derived.
This is true because we make the assumption that the original data values are at the midpoint of each class, which is not necessarily the case. The true mean of the 30 original data values in the cell phone example is only 4. What is the average daily demand during the past 65 days? The median is the value in the data set for which half the observations are higher and half the observations are lower.
We find the median by arranging the data values in ascending order and identifying the halfway point. In this case, that will be the values 5 and 6, resulting in a median of 5. Notice that there are four data values to the left 3, 4, 4, and 4 of these center points and four data values to the right 7, 7, 9, and The median is a measure of central tendency that represents the value in the data set for which half the observations are higher and half the observations are lower.
When there is an even number of data points, the median will be the average of the two center points. Therefore, the median for this data set is 5 hours of video games per week. Again, there are four data values to the left and right of this center point. That wraps up all the different ways to measure central tendency of our data set. However, one question is screaming to be answered, and that is … 9 17 Random Thoughts There can be more than one mode of a data set if more than one value occurs the most frequent number of times.
If you think all the data in your data set is relevant, then the mean is your best choice. This measurement is affected by both the number and magnitude of your values. However, very small or very large values can have a significant impact on the mean, especially if the size of the sample is small. If this is a concern, perhaps you should consider using the median.
The median is not as sensitive to a very large or small value. Consider the following data set from the original video game example: The mean of this sample was 6.
If you think 17 is not a typical value that you would expect in this data set, the median would be your best choice for central tendency. The poor lonely mode has limited applications. It is primarily used to describe data at the nominal scale—that is, data that is grouped in descriptive categories such as gender. If 60 percent of our survey respondents were male, then the mode of our data would be male. To begin, open a blank Excel worksheet and enter the video game data Figure 4.
Click on the Tools menu at the top of the spreadsheet between Format and Data and select Data Analysis. After selecting Data Analysis, you should see the dialog box shown in Figure 4. Select Descriptive Statistics and click OK. The following dialog box will appear Figure 4.
Descriptive Statistics dialog box. Then choose the Summary statistics check box and click OK. After you expand columns C and D slightly to see all the figures, your spreadsheet should look like Figure 4.
As you can see, the mean is 6. Piece of cake! Calculate the mean, median, and mode for the following data set: A company counted the number of their employees in each of the following age classes. According to this distribution, what is the average age of the employees in the company? Age Range Number of Employees 20—24 8 25—29 37 30—34 25 35—39 48 40—44 27 45—49 10 6. Calculate the weighted mean of the following values with the corresponding weights. Value Weight 3 2 1 7.
A company counted the number of employees at each level of years of service in the following table. What is the average number of years of service in this company? U The median of a data set is the midpoint of the set if the values are arranged in ascending or descending order. U The median is the single center value from the data set if there is an odd number of values in the set. The median is the average of the two center values if the number of values in the set is even. U The mode of a data set is the value that appears most often in the set.
There can be more than one mode in a data set. But in doing so, we lost information that could be useful. For the video game example, if the only information I provided you was that the mean of my sample was 6. As you will see later, this distinction can be very important. To address this issue, we rely on the second major category of descriptive statistics, measures of dispersion, which describes how far the individual data values have strayed from the mean.
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Our interactive player makes it easy to find solutions to The Complete Idiot's Guide to Statistics, 2nd Edition problems you're working on - just go to the chapter for your book. In other words, here comes your life preserver. Grab on! When a new product is recommended by 4 out of 5 doctors, do we question the validity of the claim? For instance, were the doctors paid for their endorsement? Getting a handle on this widely used tool is a good thing for all of us. In simpler terms, I view statistics as a way to convert numbers into useful information so that good decisions can be made.
These decisions can affect our lives in many ways. For instance, countless medical studies have been performed to determine the effectiveness of new drugs. Statistics form the basis of making an objective decision as to whether this new drug is actually an improvement over current treatments.
The results of statistical studies and the manner in which these results are presented often dictate government policies. Wrong Number Not interpreting statistical information properly can lead to disaster. Coca-Cola performed a major consumer study in and, based on the results, decided to reformulate Coke, its flagship drink. After a huge public outcry, Coca-Cola had to backtrack and bring the original formulation back to market. What a mess! In the s, Marriott conducted an extensive survey with potential customers on their attitudes about current hotel offerings.
After analyzing the data, the company launched Courtyard by Marriott, which has been a huge success. The federal government heavily relies on the national census that is conducted every 10 years to determine funding levels for all the various parts of the country.
The statistical analysis performed on this census data has far-reaching implications for many ongoing programs at the state and federal levels. The entire sports industry is completely dependent on the field of statistics. Can you even imagine baseball, football, or basketball without all the statistical analysis that surrounds them?
You would never know who the top players are, who is currently hot, and who is in a slump. But then, without statistics, how could the players negotiate those outrageous salaries? Statistics is a useful, and sometimes even critical, tool in our everyday life. Population surveys appear to be the primary motivation for the historical development of statistics as we know it today. In fact, according to the Bible, Moses conducted a census more than 3, years ago.
In , Sir William Petty provided the first accounts of the number of deaths in London on a weekly basis. You can thank Mr.
Bernoulli for providing you with a way to solve this type of problem. Later, during the s, English mathematician Thomas Bayes developed probability concepts that have also been very useful to the field of statistics.
Bayes used the probability of known events of the past to predict probabilities of the future. This concept of inference is widely used in statistical techniques today. The term inference refers to a key concept in statistics in which we draw a conclusion from available evidence.
We will raise our glasses to Mr. Gossett as we investigate his efforts in Chapter Edwards Deming has been credited with merging the science of statistics with the field of quality control in manufacturing environments.
Deming spent considerable time in Japan during the s and s promoting the concept of statistical quality for businesses. This technique relies on control charts to monitor a process and the use of statistics to determine whether the process is operating satisfactorily. During the s, the Japanese auto industry gained major market share in this country due mainly to superior quality. Random Thoughts Dr. This list has proven to be invaluable for organizations seeking ways to use statistics to make their processes more efficient.
Through Dr. The purpose of descriptive statistics is to summarize or display data so we can quickly obtain an overview. Inferential statistics allows us to make claims or conclusions about a population based on a sample of data from that population.