Sampling and Data

Student Learning Outcomes

By the end of this chapter, the student should be able to:

•    Recognize and differentiate between key terms.

•    Apply various types of sampling methods to data collection.

•    Create and interpret frequency tables.

Introduction

You are probably asking yourself the question, “When and where will I use statistics?”. If you read any newspaper or watch television, or use the Internet, you will see statistical information. There are statistics about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or watch a news program on television, you are given sample information. With this information, you may make a decision about the correctness of a statement, claim, or “fact.” Statistical methods can help you make the “best educated guess.”

Since you will undoubtedly be given statistical information at some point in your life, you need to know some techniques to analyze the information thoughtfully. Think about buying a house or managing a budget. Think about your chosen profession. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics.

Included in this chapter are the basic ideas and words of probability and statistics. You will soon under- stand that statistics and probability work together. You will also learn how data are gathered and what “good” data are.

 

Statistics

The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives.

Probability

Probability is a mathematical tool used to study randomness.  It deals with the chance (the likelihood) of  an event occurring.  For example,  if you toss a fair coin 4 times,  the outcomes may not be 2 heads and 2   tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is  1   or 0.5.   Even though the outcomes of a    few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction 996 is equal to 0.498 which is very close to 0.5, the expected probability.

The theory of probability began with the study of games of chance such as poker.  Predictions take the form   of probabilities. To predict the likelihood of an earthquake, of rain,  or whether you will get an A in this  course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of  return on a client’s investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data.

Key Terms

In statistics, we generally want to study a population. You can think of a population as an entire collection of persons, things, or objects under study. To study the larger population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.

Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique.  If you wished to compute the overall grade point average at your school,  it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students’ grade point averages. In presidential elections, opinion poll samples of 1,000 to 2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country.  Manufacturers of canned carbonated drinks take samples to determine if a 16 ounce can contains 16 ounces of carbonated drink.

From the sample data, we can calculate a statistic. A statistic is a number that is a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter. A parameter is a number that is a property of the population. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.

One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample. We are interested in both the  sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.

A variable,  notated by capital letters like  X and Y, is a characteristic of interest for each person or thing in a population. Variables may be numerical or categorical. Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category.  If we let  X equal the number of points earned by one math student at the end of a term,  then  X is a numerical variable. If we let Y be a person’s party affiliation, then examples of Y include Republican, Democrat, and Independent. Y is a categorical variable. We could do some math with values of X (calculate the average number of points earned, for example), but it makes no sense to do math with values of Y (calculating an average party affiliation makes no sense).

Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value.

Two words that come up often in statistics are mean and proportion.  If you were to take three exams in  your math classes and obtained scores of 86, 75, and 92, you calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women,  then the proportion of men students is  22  and the proportion of women students is 18 .

NOTE: The words “mean” and “average” are often used interchangeably. The substitution of one word for the other is common practice. The technical term is “arithmetic mean” and “average” is technically a center location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.”

Example 1

Define the key terms from the following study: We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly survey 100 first year students at the college. Three of those students spent $150, $200, and $225, respectively.

Solution 1

The population is all first year students attending ABC College this term.

The sample could be all students enrolled in one section of a beginning statistics course at ABC College (although this sample may not represent the entire population).

The parameter is the average (mean) amount of money spent (excluding books) by first year col- lege students at ABC College this term.

The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample.

The variable could be the amount of money spent (excluding books) by one first year student. Let X = the amount of money spent (excluding books) by one first year student attending ABC College.

The data are the dollar amounts spent by the first year students. Examples of the data are 200, and $225.

1.4   Data may come from a population or from a sample. Small letters like x or y generally are used to represent data values. Most data can be put into the following categories:

•    Qualitative

•    Quantitative

Qualitative data are the result of categorizing or describing attributes of a population. 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 the 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 0, 1, 2, 3, etc.

Qualitative Data Discussion

Below are tables of part-time vs full-time students at De Anza College in Cupertino,  CA and Foothill College in Los Altos, CA for the Spring 2010 quarter.  The tables display counts (frequencies) and percentages or proportions (relative frequencies). 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

 

	Number	Percent
Full-time	9,200	40.9%
Part-time	13,296	59.1%
Total	22,496	100%

Table 1

Foothill College

 

	Number	Percent
Full-time	4,059	28.6%
Part-time	10,124	71.4%
Total	14,183	100%

Table 2

Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. There are no strict rules concerning what graphs to use. Below are pie charts and bar graphs, two graphs that are used to display qualitative data.

 

In a pie chart, categories  of  data  are  represented  by  wedges  in  the  circle  and  are  proportional  in  size to the percent of individuals in each category.

 

In a bar graph, the length of the bar for each category is proportional to the number or percent  of  individuals in each category. Bars may be vertical or horizontal.

 

A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).

 

Look at the graphs  and  determine  which  graph  (pie  or  bar)  you  think  displays  the  comparisons  bet-  ter. This is a matter of preference.

 

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.

Table 3

Table 4

Percentages That Add to More (or Less) Than 100%

Sometimes percentages add up to be more than 100% (or less than 100%).  In the graph,  the percentages  add to more than 100% 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%.

De Anza College Spring 2010

Characteristic/Category	Percent
Full-time Students	40.9%
Students who intend to transfer to a 4-year educational institution	48.6%
Students under age 25	61.0%
TOTAL	150.5%

Table 5

Omitting Categories/Missing Data

Table 6

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. Create a bar graph and not a pie chart.

Missing Data: Ethnicity of Students De Anza College Fall Term 2007 (Census Day)

 

	Frequency	Percent
Asian	8,794	36.1%
Black	1,412	5.8%
Filipino	1,298	5.3%
Hispanic	4,180	17.1%
Native American	146	0.6%
Pacific Islander	236	1.0%
White	5,978	24.5%
TOTAL	22,044 out of 24,382	90.4% out of 100%

Table 7

Bar graph Without Other/Unknown Category

Table 8

The following graph is the same as the previous graph but the “Other/Unknown” percent (9.6%) has been added back in. The “Other/Unknown” category is large compared to some of the other categories (Native American, 0.6%, Pacific Islander 1.0% particularly). This is important to know when we think about what the data are telling us.

This particular  bar  graph  can  be  hard  to  understand  visually.  The  graph  below  it  is  a  Pareto  chart.  The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.

Bar Graph With Other/Unknown Category

Table 9

Pareto Chart With Bars Sorted By Size

Table 10

Pie Charts: No Missing Data

 

The following pie charts have the “Other/Unknown” category added back in (since the percentages must add to 100%). The chart on the right is organized having the wedges by size and makes for a more visually informative graph than the unsorted, alphabetical graph on the left.

Table 11

Sampling

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. Any group of n individuals is equally likely to be chosen by any other group of n individuals if the simple random sampling technique is used. In other words, each sample of the same size has an equal chance of being selected. For example, sup- pose 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 3 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 3 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 shown below.

Class Roster

 

ID	Name
00	Anselmo
01	Bautista
02	Bayani
03	Cheng
04	Cuarismo
05	Cuningham
06	Fontecha
07	Hong
08	Hoobler
09	Jiao
10	Khan
11	King
12	Legeny
13	Lundquist
14	Macierz
15	Motogawa
16	Okimoto
17	Patel
18	Price
19	Quizon
20	Reyes
21	Roquero
22	Roth
23	Rowell
24	Salangsang
25	Slade
26	Stracher
27	Tallai
28	Tran
29	Wai
30	Wood

Table 12

Lisa can either use a table of random numbers (found in many statistics books as well as mathematical handbooks) or a calculator or computer to generate random numbers. For this example, suppose  Lisa chooses to generate random numbers from a calculator. The numbers generated are:

.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 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 Macierz, 05 corresponds to Cunningham, and 04 corresponds to Cuarismo.  Besides herself, Lisa’s group will consist   of Marcierz, and Cunningham, and Cuarismo.

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 take a proportionate number from each stratum. For example, you could stratify (group) your college population by department and then choose a proportionate simple random sample from each stratum (each department) to get a stratified random sample. To  choose a simple random sample from each department,  number each member of  the first department, number each member of the second department and do the same for the remaining departments. Then use simple random sampling to choose proportionate numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department and so on represent the members 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 departments from your college population, the four departments make up the cluster sample. For example, divide your college faculty by department. The departments are the clusters. Number each department and then choose four different numbers using simple random sampling. All members of the four departments with those numbers are the cluster sample.

To choose a systematic sample, randomly select a starting point and take every nth piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains 20,000 residence listings.  You  must choose 400 names for the sample.  Number the population 1 – 20,000  and then use a simple random sample to pick a number that represents the first name of the sample. Then choose every 50th name thereafter until you have a total of 400 names (you might have to go back to the of your phone list). Systematic sampling is frequently chosen because it is a simple method.

A type of sampling that is nonrandom 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 (favors 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 (for example, they may favor a certain group). It is better for the person conducting the survey to select the sample respondents.

True random sampling is done with replacement. That is, once  a  member  is  picked  that  member  goes back into the population and thus may be chosen more than once. However for practical reasons, in most populations, simple random sampling is done without replacement. Surveys are typically done without replacement. That is, a member of the population may be chosen only once. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Since this is the case,sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same individual more than once using with replacement is very low.

For example, in a college population of 10,000 people, suppose you want to randomly pick a sample of 1000 for a survey. For any particular sample of 1000, if you are sampling with replacement,

•    the chance of picking the first person is 1000 out of 10,000 (0.1000);

•    the chance of picking a different second person for this sample is 999 out of 10,000 (0.0999);

•    the chance of picking the same person again is 1 out of 10,000 (very low).

If you are sampling without replacement,

•    the chance of picking the first person for any particular sample is 1000 out of 10,000 (0.1000);

•    the chance of picking a different second person is 999 out of 9,999 (0.0999);

•    you do not replace the first person before picking the next person.

Compare the fractions 999/10,000 and 999/9,999. For accuracy, carry the decimal answers to 4 place decimals. To 4 decimal places, these numbers are equivalent (0.0999).

Sampling without replacement instead of sampling with replacement only becomes a mathematics issue when the population is small which is not that common. For example, if the population is 25 people, the sample is 10 and you are sampling with replacement for any particular sample, the chance of picking the first person is 10 out of 25 and a different second person is 9 out of 25 (you replace the first person).

If you sample without replacement, the chance of picking the first person is 10 out of 25 and then the second person (which is different) is  9 out of 24 (you do not replace the first person).

Compare the fractions 9/25 and 9/24. To 4 decimal places, 9/25 = 0.3600 and 9/24 = 0.3750. To 4 decimal places, these numbers are not equivalent.

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.

Example 6

Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).

1.    A soccer coach selects 6 players from a group of boys aged 8 to 10, 7 players from a group of boys aged 11 to 12, and 3 players from a group of boys aged 13 to 14 to form a recreational soccer team.

2.    A pollster interviews all human resource personnel in five different high tech companies.

3.  A high school educational researcher interviews 50 high school female teachers and 50 high school male teachers.

4. A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital.

5.  A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers.

6. A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on the average.

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 seem natural.

Example 7

Suppose ABC College has 10,000 part-time students (the population). We are interested in the average amount of money a part-time student spends on books in the fall term. Asking all 10,000 students is an almost impossible task.

Suppose we take two different samples.

First, we use convenience sampling and survey 10 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 is as follows:

$128; $87; $173; $116; $130; $204; $147; $189; $93; $153

The second sample is taken by using a list from the P.E. department of senior citizens who take

P.E. classes and taking every 5th senior citizen on the list, for a total of 10 senior citizens. They spend:

$50; $40; $36; $15; $50; $100; $40; $53; $22; $22

Problem 1

Do you think that either of these samples is representative of (or is characteristic of) the entire 10,000 part-time student population?

Solution

No. The first sample probably consists of science-oriented students. Besides the chemistry course, some of them are taking first-term calculus.   Books for these classes tend to be expensive.   Most of these students are, more than likely, paying more than the average part-time student for their books.  The second sample is a group of senior citizens who are,  more than likely,  taking courses  for health and interest. The amount of money they spend on books is probably much less than the average part-time student. Both samples are biased. Also, in both cases, not all students have a chance to be in either sample.

Problem 2

Since these samples are not representative of the entire population, is it wise to use the results to describe the entire population?

Solution

No. For these samples, each member of the population did not have an equally likely chance of being chosen.

Now, suppose we take a third sample. We choose ten different part-time students from the dis- ciplines 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/she has a corresponding number. The students spend:

$180; $50; $150; $85; $260; $75; $180; $200; $200; $150

Problem 3

Is the sample biased?

Solution

The sample is unbiased,  but a larger sample would be recommended to increase the likelihood  that the sample will be close to representative of the population. However, for a biased sampling technique, even a large sample runs the risk of not being representative of the population.

 

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.

Critical Evaluation

We need to critically evaluate the statistical studies we read about and analyze before accepting the results of the study. Common problems to be aware of include

 

1. Problems with Samples: A sample should 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 valid.

2. Self-Selected Samples: Responses only by people who choose to respond, such as call-in surveys are often unreliable.

3. Sample Size Issues: Samples that are too small may be unreliable. Larger samples are better if possible. In some situations, small samples are unavoidable and can still be used to draw conclusions, even though larger samples are better. Examples: Crash testing cars, medical testing for rare conditions.

4. Undue influence: Collecting data or asking questions in a way that influences the response.

5. 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.

6. Causality:  A relationship between two variables does not mean that one causes the other to occur.  They may both be related (correlated) because of their relationship through a different variable.

7. Self-Funded or Self-Interest Studies: A study performed by a person or organization in order to sup- port 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.

8. Misleading Use of Data: Improperly displayed graphs, incomplete data, lack of context. Confounding: When the effects of multiple factors on a response cannot be separated. Confounding makes it difficult or impossible to draw valid conclusions about the effect of each factor.

Answers and Rounding Off

A simple way to round off answers is to carry your final answer one more decimal place than was present in the original data. Round only the final answer. Do not round any intermediate results, if possible. If it becomes necessary to round intermediate results, carry them to at least twice as many decimal places as the final answer. For example, the average of the three quiz scores 4, 6, 9 is 6.3, rounded to the nearest tenth, because the data are whole numbers. Most answers will be rounded in this manner.

It is not necessary to reduce most fractions in this course. Especially in Probability Topics (Section 3.1), the chapter on probability, it is more helpful to leave an answer as an unreduced fraction.

 

Frequency

Twenty students were asked how many hours they worked per day. Their responses, in hours, are listed below:

5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3

Below is a frequency table listing the different data values in ascending order and their frequencies.

Frequency Table of Student Work Hours

Table 15

A frequency is the number of times a given datum occurs in a data set.   According to the table above, there are three students who work 2 hours, five students who work 3 hours, etc. The total of the frequency column, 20, represents the total number of students included in the sample.

A relative frequency is the fraction or proportion of times an answer occurs. To find the relative frequencies, divide each frequency by the total number of students in the sample – in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.

Frequency Table of Student Work Hours w/ Relative Frequency

DATA VALUE	FREQUENCY	RELATIVE FREQUENCY
2	3	3/20 or 0.15
3	5	5/20 or 0.25
4	3	3/20 or 0.15
5	6	6/20 or 0.30
6	2	2/20 or 0.10
7	1	1/20 or 0.05

 

Table 16

The sum of the relative frequency column is 20 , or 1.

Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row.

Frequency Table of Student Work Hours w/ Relative and Cumulative Relative Frequency

 

Table 17

The last entry of the cumulative relative frequency column is one,  indicating that one hundred percent of    the data has been accumulated.

NOTE: Because of rounding, the relative frequency column may not always sum to one and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.

The following table represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.

 

Frequency Table of Soccer Player Height

 

HEIGHTS (INCHES)	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
59.95 – 61.95	5	5 = 0.05 100	0.05
61.95 – 63.95	3	3 = 0.03 100	0.05 + 0.03 = 0.08
63.95 – 65.95	15	15 = 0.15 100	0.08 + 0.15 = 0.23
65.95 – 67.95	40	40 = 0.40 100	0.23 + 0.40 = 0.63
67.95 – 69.95	17	17 = 0.17 100	0.63 + 0.17 = 0.80
69.95 – 71.95	12	12 = 0.12 100	0.80 + 0.12 = 0.92
71.95 – 73.95	7	7 = 0.07 100	0.92 + 0.07 = 0.99
73.95 – 75.95	1	1 = 0.01 100	0.99 + 0.01 = 1.00
	Total = 100	Total = 1.00

Table 18

The data in this table has been grouped into the following intervals:

•    59.95 – 61.95 inches

•    61.95 – 63.95 inches

•    63.95 – 65.95 inches

•    65.95 – 67.95 inches

•    67.95 – 69.95 inches

•    69.95 – 71.95 inches

•    71.95 – 73.95 inches

•    73.95 – 75.95 inches

In this sample, there are 5 players whose heights are between 59.95 – 61.95 inches, 3 players whose heights fall within the interval 61.95 – 63.95 inches, 15 players whose heights fall within the interval 63.95 – 65.95 inches, 40 players whose heights fall within the interval 65.95 – 67.95 inches, 17 players whose heights fall within the interval 67.95 – 69.95 inches, 12 players whose heights fall within the interval 69.95 – 71.95, 7 players whose height falls within the interval 71.95 – 73.95, and 1 player whose height falls within the interval 73.95 – 75.95. All heights fall between the endpoints of an interval and not at the endpoints.

Example 8

From the table, find the percentage of heights that are less than 65.95 inches.

Solution

If you look at the first, second, and third rows, the heights are all less than 65.95 inches.  There are  5 + 3 + 15 = 23 males whose heights are less than 65.95 inches. The percentage of heights less than 65.95 inches is then  23   or 23%. This percentage is the cumulative relative frequency entry in the third row.

Example 9

From the table, find the percentage of heights that fall between 61.95 and 65.95 inches.

Solution

Add the relative frequencies in the second and third rows: 0.03 + 0.15 = 0.18 or 18%.

Example 10

Use the table of heights of the 100 male semiprofessional soccer players. Fill in the blanks and check your answers.

1.    The percentage of heights that are from 67.95 to 71.95 inches is:

2.    The percentage of heights that are from 67.95 to 73.95 inches is:

3.    The percentage of heights that are more than 65.95 inches is:

4.    The number of players in the sample who are between 61.95 and 71.95 inches tall is:

5.    What kind of data are the heights?

6.    Describe how you could gather this data (the heights) so that the data are characteristic of all male semiprofessional soccer players.

Remember, you count frequencies. To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.

Example 11

Nineteen people were asked how many miles, to the nearest mile they commute to work each day. The data are as follows:

2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10

The following table was produced:

Frequency of Commuting Distances

 

DATA	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
3	3	3 19	0.1579
4	1	1 19	0.2105
5	3	3 19	0.1579
7	2	2 19	0.2632
10	3	4 19	0.4737
12	2	2 19	0.7895
13	1	1 19	0.8421
15	1	1 19	0.8948
18	1	1 19	0.9474
20	1	1 19	1.0000

Table 19

Problem                                                               

1.    Is the table correct? If it is not correct, what is wrong?

2.    True or False: Three percent of the people surveyed commute 3 miles. If the statement is not correct, what should it be? If the table is incorrect, make the corrections.

3.    What fraction of the people surveyed commute 5 or 7 miles?

4.    What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between 5 and 13 miles (does not include 5 and 13 miles)?

Summary

Statistics

•    Deals with the collection, analysis, interpretation, and presentation of data

Probability

•    Mathematical tool used to study randomness

Key Terms

•    Population

•    Parameter

•    Sample

•    Statistic

•    Variable

•    Data

Types of Data

•    Quantitative Data (a number)

·    Discrete (You count it.)

·    Continuous (You measure it.)

•    Qualitative Data (a category, words)

Sampling

•    With Replacement: A member of the population may be chosen more than once

•    Without Replacement: A member of the population may be chosen only once

Random Sampling

•    Each member of the population has an equal chance of being selected

Sampling Methods

•    Random

·    Simple random sample

·    Stratified sample

·    Cluster sample

·    Systematic sample

•    Not Random

·    Convenience sample

Frequency (freq. or f)

•    The number of times an answer occurs

Relative Frequency (rel. freq. or RF)

•    The proportion of times an answer occurs

•    Can be interpreted as a fraction, decimal, or percent

Cumulative Relative Frequencies (cum. rel. freq. or cum RF)

An accumulation of the previous relative frequencies

10This content is available online at <http://cnx.org/content/m16023/1.10/>.