Data may come from a population or from a sample. Lowercase letters like or generally are used to represent data values. Most data can be put into the following categories:
Qualitative data are the result of categorizing or describing attributes of a population. Qualitative data are also often called categorical data. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O–, or B+. Researchers often prefer to use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type.
Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous.
All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three.
Data that are not only made up of counting numbers, but that may include fractions, decimals, or irrational numbers, are called quantitative continuous data. Continuous data are often the results of measurements like lengths, weights, or times. A list of the lengths in minutes for all the phone calls that you make in a week, with numbers like 2.4, 7.5, or 11.0, would be quantitative continuous data.
Data Sample of Quantitative Discrete Data
The data are the number of books students carry in their backpacks. You sample five students. Two students carry three books, one student carries four books, one student carries two books, and one student carries one book. The numbers of books, 3, 4, 2, and 1, are the quantitative discrete data.
The data are the number of machines in a gym. You sample five gyms. One gym has 12 machines, one gym has 15 machines, one gym has 10 machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this?
Data Sample of Quantitative Continuous Data
The data are the weights of backpacks with books in them. You sample the same five students. The weights, in pounds, of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data.
The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. ft., 160 sq. ft., 190 sq. ft., 180 sq. ft., and 210 sq. ft. What type of data is this?
You go to the supermarket and purchase three cans of soup (19 ounces tomato bisque, 14.1 ounces lentil, and 19 ounces Italian wedding), two packages of nuts (walnuts and peanuts), four different kinds of vegetable (broccoli, cauliflower, spinach, and carrots), and two desserts (16 ounces pistachio ice cream and 32 ounces chocolate chip cookies).
Name data sets that are quantitative discrete, quantitative continuous, and qualitative.
Try to identify additional data sets in this example.
The data are the colors of backpacks. Again, you sample the same five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. The colors red, black, black, green, and gray are qualitative data.
The data are the colors of houses. You sample five houses. The colors of the houses are white, yellow, white, red, and white. What type of data is this?
You may collect data as numbers and report it categorically. For example, the quiz scores for each student are recorded throughout the term. At the end of the term, the quiz scores are reported as A, B, C, D, or F.
Work collaboratively to determine the correct data type: quantitative or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words the number of.
- the number of pairs of shoes you own
- the type of car you drive
- the distance from your home to the nearest grocery store
- the number of classes you take per school year
- the type of calculator you use
- weights of sumo wrestlers
- number of correct answers on a quiz
- IQ scores (This may cause some discussion.)
Determine the correct data type, quantitative or qualitative, for the number of cars in a parking lot. Indicate whether quantitative data are continuous or discrete.
A statistics professor collects information about the classification of her students as freshmen, sophomores, juniors, or seniors. The data she collects are summarized in the pie chart Figure 1.3. What type of data does this graph show?
A large school district keeps data of the number of students who receive test scores on an end of the year standardized exam. The data he collects are summarized in the histogram. The class boundaries are 50 to less than 60, 60 to less than 70, 70 to less than 80, 80 to less than 90, and 90 to less than 100.
Qualitative Data Discussion
Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts, frequencies, and percentages or proportions, relative frequencies. For instance, to calculate the percentage of part time students at De Anza College, divide 9,200/22,496 to get .4089. Round to the nearest thousandth—third decimal place and then multiply by 100 to get the percentage, which is 40.9 percent.
So, the percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.
|De Anza College||Foothill College|
Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data.
Two graphs that are used to display qualitative data are pie charts and bar graphs.
In a pie chart, categories of data are shown by wedges in a circle that represent the percent of individuals/items in each category. We use pie charts when we want to show parts of a whole.
In a bar graph, the length of the bar for each category represents the number or percent of individuals in each category. Bars may be vertical or horizontal. We use bar graphs when we want to compare categories or show changes over tim
A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).
It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the best graph depending on the data and the context. Our choice also depends on what we are using the data for.
Percentages That Add to More (or Less) Than 100 Percent
Sometimes percentages add up to be more than 100 percent (or less than 100 percent). In the graph, the percentages add to more than 100 percent because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100 percent.
|Students studying technical subjects||40.9%|
|Students studying non-technical subjects||48.6%|
|Students who intend to transfer to a four-year educational institutional||61.0%|
Omitting Categories/Missing Data
The table displays Ethnicity of Students but is missing the Other/Unknown category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart.
|TOTAL||22,044 out of 24,382||90.4% out of 100%|
The following graph is the same as the previous graph but the Other/Unknown percent (9.6 percent) has been included. The Other/Unknown category is large compared to some of the other categories (Native American, .6 percent, Pacific Islander 1.0 percent). This is important to know when we think about what the data are telling us.
This particular bar graph in Figure 1.9 can be difficult to understand visually. The graph in Figure 1.10 is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.
Pie Charts: No Missing Data
The following pie charts have the Other/Unknown category included since the percentages must add to 100 percent. The chart in Figure 1.11b is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in Figure 1.11a.
Marginal Distributions in Two-Way Tables
Below is a two-way table, also called a contingency table, showing the favorite sports for 50 adults: 20 women and 30 men.
This is a two-way table because it displays information about two categorical variables, in this case, gender and sports. Data of this type (two variable data) are referred to as bivariate data. Because the data represent a count, or tally, of choices, it is a two-way frequency table. The entries in the total row and the total column represent marginal frequencies or marginal distributions. Note—The term marginal distributions gets its name from the fact that the distributions are found in the margins of frequency distribution tables. Marginal distributions may be given as a fraction or decimal: For example, the total for men could be given as .6 or 3/5 since Marginal distributions require bivariate data and only focus on one of the variables represented in the table. In other words, the reason 20 is a marginal frequency in this two-way table is because it represents the margin or portion of the total population that is women (20/50). The reason 25 is a marginal frequency is because it represents the portion of those sampled who favor football (25/50). Note: The values that make up the body of the table (e.g., 20, 8, 2) are called joint frequencies.
Conditional Distributions in Two-Way Tables
The distinction between a marginal distribution and a conditional distribution is that the focus is on only a particular subset of the population (not the entire population). For example, in the table, if we focused only on the subpopulation of women who prefer football, then we could calculate the conditional distributions as shown in the two-way table below.
To find the first sub-population of women who prefer football, read the value at the intersection of the Women row and Football column which is 5. Then, divide this by the total population of football players which is 25. So, the subpopulation of football players who are women is 5/25 which is .2.
Similarly, to find the subpopulation of women who play football, use the value of 5 which is the number of women who play football. Then, divide this by the total population of women which is 20. So, the subpopulation of women who play football is 5/20 which is .25.
After deciding which graph best represents your data, you may need to present your statistical data to a class or other group in an oral report or multimedia presentation. When giving an oral presentation, you must be prepared to explain exactly how you collected or calculated the data, as well as why you chose the categories, scales, and types of graphs that you are showing. Although you may have made numerous graphs of your data, be sure to use only those that actually demonstrate the stated intentions of your statistical study. While preparing your presentation, be sure that all colors, text, and scales are visible to the entire audience. Finally, make sure to allow time for your audience to ask questions and be prepared to answer them.
Suppose the guidance counselors at De Anza and Foothill need to make an oral presentation of the student data presented in Figures 1.5 and 1.6. Under what context should they choose to display the pie graph? When might they choose the bar graph? For each graph, explain which features they should point out and the potential display problems that might exist.
Suppose you were asked to give an oral presentation of the data graphed in the pie chart in Figure 1.11(b). What features would you point out on the graph? What potential display problems with the graph should you check before giving your presentation?
Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. Each method has pros and cons. In a simple random sample, each group has the same chance of being selected. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 31 members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all 31 names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in Table 1.7.
Lisa can use a table of random numbers (found in many statistics books and mathematical handbooks), a calculator, or a computer to generate random numbers. The most common random number generators are five digit numbers where each digit is a unique number from 0 to 9. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows:
.94360, .99832, .14669, .51470, .40581, .73381, .04399.
Lisa reads two-digit groups until she has chosen three class members (That is, she reads .94360 as the groups 94, 43, 36, 60.) Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers.
The table below shows how Lisa reads two-digit numbers form each random number. Each two-digit number in the table would represent each student in the roster above in Table 1.7.
|Random number||Numbers read by Lisa|
The random numbers .94360 and .99832 do not contain appropriate two digit numbers. However the third random number, .14669, contains 14 (the fourth random number also contains 14), the fifth random number contains 05, and the seventh random number contains 04. The two-digit number 14 corresponds to Lundquist, 05 corresponds to Cuningham, and 04 corresponds to Cuarismo. Besides herself, Lisa’s group will consist of Lundquist, Cuningham, and Cuarismo.
To generate random numbers perform the following steps:
- Press MATH.
- Arrow over to PRB.
- Press 5:randInt(0, 30).
- Press ENTER for the first random number.
- Press ENTER two more times for the other two random numbers. If there is a repeat press ENTER again.
Note—randInt(0, 30, 3) will generate three random numbers.
Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample.
To choose a stratified sample, divide the population into groups called strata and then the sample is selected by picking the same number of values from each strata until the desired sample size is reached. For example, you could stratify (group) your high school student population by year (freshmen, sophomore, juniors, and seniors) and then choose a proportionate simple random sample from each stratum (each year) to get a stratified random sample. To choose a simple random sample from each year, number each student of the first year, number each student of the second year, and do the same for the remaining years. Then use simple random sampling to choose proportionate numbers of students from the first year and do the same for each of the remaining years. Those numbers picked from the first year, picked from the second year, and so on represent the students who make up the stratified sample.
To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four homeroom classes from your student population, the four classes make up the cluster sample. Each class is a cluster. Number each cluster, and then choose four different numbers using random sampling. All the students of the four classes with those numbers are the cluster sample. So, unlike a stratified example, a cluster sample may not contain an equal number of randomly chosen students from each class.
A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others.
Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased. They may favor a certain group. It is better for the person conducting the survey to select the sample respondents.
When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause nonsampling errors. A defective counting device can cause a nonsampling error.
In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error.
In statistics, a sampling bias is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others. Remember, each member of the population should have an equally likely chance of being chosen. When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied. For instance, if a survey of all students is conducted only during noon lunchtime hours is biased. This is because the students who do not have a noon lunchtime would not be included.
We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include the following:
- Problems with samples: —A sample must be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not reliable. Reliability in statistical measures must also be considered when analyzing data. Reliability refers to the consistency of a measure. A measure is reliable when the same results are produced given the same circumstances.
- Self-selected samples—Responses only by people who choose to respond, such as internet surveys, are often unreliable.
- Sample size issues—: Samples that are too small may be unreliable. Larger samples are better, if possible. In some situations, having small samples is unavoidable and can still be used to draw conclusions. Examples include crash testing cars or medical testing for rare conditions.
- Undue influence—: collecting data or asking questions in a way that influences the response.
- Non-response or refusal of subject to participate: —The collected responses may no longer be representative of the population. Often, people with strong positive or negative opinions may answer surveys, which can affect the results.
- Causality: —A relationship between two variables does not mean that one causes the other to occur. They may be related (correlated) because of their relationship through a different variable.
- Self-funded or self-interest studies—: A study performed by a person or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good, but do not automatically assume the study is bad either. Evaluate it on its merits and the work done.
- Misleading use of data—: These can be improperly displayed graphs, incomplete data, or lack of context.
As a class, determine whether or not the following samples are representative. If they are not, discuss the reasons.
- To find the average GPA of all students in a high school, use all honor students at the university as the sample.
- To find out the most popular cereal among young people under the age of 10, stand outside a large supermarket for three hours and speak to every twentieth child under age 10 who enters the supermarket.
- To find the average annual income of all adults in the United States, sample U.S. congressmen. Create a cluster sample by considering each state as a stratum (group). By using simple random sampling, select states to be part of the cluster. Then survey every U.S. congressman in the cluster.
- To determine the proportion of people taking public transportation to work, survey 20 people in New York City. Conduct the survey by sitting in Central Park on a bench and interviewing every person who sits next to you.
- To determine the average cost of a two-day stay in a hospital in Massachusetts, survey 100 hospitals across the state using simple random sampling.
A study is done to determine the average tuition that private high school students pay per semester. Each student in the following samples is asked how much tuition he or she paid for the fall semester. What is the type of sampling in each case?
- A sample of 100 high school students is taken by organizing the students’ names by classification (freshman, sophomore, junior, or senior) and then selecting 25 students from each.
- A random number generator is used to select a student from the alphabetical listing of all high school students in the fall semester. Starting with that student, every 50th student is chosen until 75 students are included in the sample.
- A completely random method is used to select 75 students. Each high school student in the fall semester has the same probability of being chosen at any stage of the sampling process.
- The freshman, sophomore, junior, and senior years are numbered one, two, three, and four, respectively. A random number generator is used to pick two of those years. All students in those two years are in the sample.
- An administrative assistant is asked to stand in front of the library one Wednesday and to ask the first 100 undergraduate students he encounters what they paid for tuition the fall semester. Those 100 students are the sample.
You are going to use the random number generator to generate different types of samples from the data.
This table displays six sets of quiz scores (each quiz counts 10 points) for an elementary statistics class.
Instructions: Use the Random Number Generator to pick samples.
- Create a stratified sample by column. Pick three quiz scores randomly from each column.
- Number each row one through 10.
- On your calculator, press Math and arrow over to PRB.
- For column 1, Press 5:randInt( and enter 1,10). Press ENTER. Record the number. Press ENTER 2 more times (even the repeats). Record these numbers. Record the three quiz scores in column one that correspond to these three numbers.
- Repeat for columns two through six.
- These 18 quiz scores are a stratified sample.
- Create a cluster sample by picking two of the columns. Use the column numbers: one through six.
- Press MATH and arrow over to the PRB function.
- Press 5:randInt (“and then enter “1,6). Press ENTER.
- Record the number the calculator displays into the first column. Then, press ENTER.
- Record the next number the calculator displays into the second column.
- Repeat steps (c) and (d) nine more times until there are a total of 20 quiz scores for the cluster sample.
- Create a simple random sample of 15 quiz scores.
- Use the numbering one through 60.
- Press MATH. Arrow over to PRB. Press 5:randInt(1, 60).
- Press ENTER 15 times and record the numbers.
- Record the quiz scores that correspond to these numbers.
- These 15 quiz scores are the systematic sample.
- Create a systematic sample of 12 quiz scores.
- Use the numbering one through 60.
- Press MATH. Arrow over to PRB. Press 5:randInt(1, 60).
- Press ENTER. Record the number and the first quiz score. From that number, count ten quiz scores and record that quiz score. Keep counting ten quiz scores and recording the quiz score until you have a sample of 12 quiz scores. You may wrap around (go back to the beginning).
Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).
- A soccer coach selects six players from a group of boys aged eight to ten, seven players from a group of boys aged 11 to 12, and three players from a group of boys aged 13 to 14 to form a recreational soccer team.
- A pollster interviews all human resource personnel in five different high tech companies.
- A high school educational researcher interviews 50 high school female teachers and 50 high school male teachers.
- A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital.
- A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers.
- A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on average.
Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).
A high school principal polls 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors regarding policy changes for after school activities.
If we were to examine two samples representing the same population, even if we used random sampling methods for the samples, they would not be exactly the same. Just as there is variation in data, there is variation in samples. As you become accustomed to sampling, the variability will begin to seem natural.
Suppose ABC high school has 10,000 upperclassman (junior and senior level) students (the population). We are interested in the average amount of money a upperclassmen spends on books in the fall term. Asking all 10,000 upperclassmen is an almost impossible task.
Suppose we take two different samples.
First, we use convenience sampling and survey ten upperclassman students from a first term organic chemistry class. Many of these students are taking first term calculus in addition to the organic chemistry class. The amount of money they spend on books is as follows:
$128, $87, $173, $116, $130, $204, $147, $189, $93, $153.
The second sample is taken using a list of seniors who take P.E. classes and taking every fifth seniors on the list, for a total of ten seniors. They spend the following:
$50, $40, $36, $15, $50, $100, $40, $53, $22, $22.
It is unlikely that any student is in both samples.
a. Do you think that either of these samples is representative of (or is characteristic of) the entire 10,000 part-time student population?
b. Since these samples are not representative of the entire population, is it wise to use the results to describe the entire population?
Now, suppose we take a third sample. We choose ten different part-time students from the disciplines of chemistry, math, English, psychology, sociology, history, nursing, physical education, art, and early childhood development. We assume that these are the only disciplines in which part-time students at ABC College are enrolled and that an equal number of part-time students are enrolled in each of the disciplines. Each student is chosen using simple random sampling. Using a calculator, random numbers are generated and a student from a particular discipline is selected if he or she has a corresponding number. The students spend the following amounts:
$180, $50, $150, $85, $260, $75, $180, $200, $200, $150.
c. Is the sample biased?
Students often ask if it is good enough to take a sample, instead of surveying the entire population. If the survey is done well, the answer is yes.
A local radio station has a fan base of 20,000 listeners. The station wants to know if its audience would prefer more music or more talk shows. Asking all 20,000 listeners is an almost impossible task.
The station uses convenience sampling and surveys the first 200 people they meet at one of the station’s music concert events. Twenty-four people said they’d prefer more talk shows, and 176 people said they’d prefer more music.
Do you think that this sample is representative of (or is characteristic of) the entire 20,000 listener population?
Variation in Data
Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage:
15.8, 16.1, 15.2, 14.8, 15.8, 15.9, 16.0, 15.5.
Measurements of the amount of beverage in a 16-ounce can may vary because different people make the measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range.
Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This is completely natural. However, if two or more of you are taking the same data and get very different results, it is time for you and the others to reevaluate your data-taking methods and your accuracy.
Variation in Samples
It was mentioned previously that two or more samples from the same population, taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their high school sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however.
Think about what contributes to making Doreen’s and Jung’s samples different.
If Doreen and Jung took larger samples, that is, the number of data values is increased, their sample results (the average amount of time a student sleeps) might be closer to the actual population average. But still, their samples would be, in all likelihood, different from each other. This is called sampling variability. In other words, it refers to how much a statistic varies from sample to sample within a population. The larger the sample size, the smaller the variability between samples will be. So, the large sample size makes for a better, more reliable statistic.
Size of a Sample
The size of a sample (often called the number of observations) is important. The examples you have seen in this book so far have been small. Samples of only a few hundred observations, or even smaller, are sufficient for many purposes. In polling, samples that are from 1,200–1,500 observations are considered large enough and good enough if the survey is random and is well done. You will learn why when you study confidence intervals.
Be aware that many large samples are biased. For example, internet surveys are invariably biased, because people choose to respond or not.
Divide into groups of two, three, or four. Your instructor will give each group one six-sided die. Try this experiment twice. Roll one fair die (six-sided) 20 times. Record the number of ones, twos, threes, fours, fives, and sixes you get in Table 1.10 and Table 1.11 (frequency is the number of times a particular face of the die occurs)
|Face on Die||Frequency|
|Face on Die||Frequency|
Did the two experiments have the same results? Probably not. If you did the experiment a third time, do you expect the results to be identical to the first or second experiment? Why or why not?
Which experiment had the correct results? They both did. The job of the statistician is to see through the variability and draw appropriate conclusions.