Sampling and Data

Learning Outcomes

  • Apply various types of sampling methods to data collection.


Data may come from a population or from a sample. Small letters like
xx
or
yy
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. 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.

All data that are the result of measuring are quantitative continuous data assuming that we can measure accurately. Measuring angles in radians might result in such numbers as
π6,π3,π2,π,3π4\displaystyle\frac{\pi}{6},\frac{\pi}{3},\frac{\pi}{2},\pi,\frac{3\pi}{4}
, and so on. If you and your friends carry backpacks with books in them to school, the numbers of books in the backpacks are discrete data and the weights of the backpacks are continuous data.

Example

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 (three, four, two, and one) are the quantitative discrete data.

Try It

The data are the number of machines in a gym. You sample five gyms. One gym has
1212
machines, one gym has
1515
machines, one gym has
1010
machines, one gym has
2222
machines, and the other gym has
2020
machines. What type of data is this?





Example

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
6.26.2
,
77
,
6.86.8
,
9.19.1
,
4.34.3
. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data because weights are measured.

try it

The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are
144144
sq. feet,
160160
sq. feet,
190190
sq. feet,
180180
sq. feet, and
210210
sq. feet. What type of data is this?






Omitting Categories and 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.

Frequency Percent
Asian
8,7948,794
36.136.1
%
Black
1,4121,412
5.85.8
%
Filipino
1,2981,298
5.35.3
%
Hispanic
4,1804,180
17.117.1
%
Native American
146146
0.60.6
%
Pacific Islander
236236
1.01.0
%
White
5,9785,978
24.524.5
%
TOTAL
22,04422,044
out of
24,38224,382
90.490.4
% out of
100100
%
Figure 1. Ethnicity of Students

The following graph is the same as the previous graph but the "Other/Unknown" percent (
9.69.6
%) has been included. The "Other/Unknown" category is large compared to some of the other categories (Native American,
0.60.6
%, Pacific Islander
1.01.0
%). This is important to know when we think about what the data are telling us.

This particular bar graph in Figure 2 can be difficult to understand visually. The graph in Figure 3 is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.

Figure 2. Bar Graph with Other/Unknown Category

Figure 3. Pareto Chart with Bars Sorted by Size

Sampling

The following video introduces the different methods that statisticians use collect samples of data.



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
nn
individuals is equally likely to be chosen by any other group of
nn
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, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has
3131
members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all
3131
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 the following table.

ID Name ID Name ID Name
0000
Anselmo
1111
King
2121
Roquero
0101
Bautista
1212
Legeny
2222
Roth
0202
Bayani
1313
Lundquist
2323
Rowell
0303
Cheng
1414
Macierz
2424
Salangsang
0404
Cuarismo
1515
Motogawa
2525
Slade
0505
Cuningham
1616
Okimoto
2626
Stratcher
0606
Fontecha
1717
Patel
2727
Tallai
0707
Hong
1818
Price
2828
Tran
0808
Hoobler
1919
Quizon
2929
Wai
0909
Jiao
2020
Reyes
3030
Wood
1010
Khan
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. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows:

0.943600.94360
;
0.998320.99832
;
0.146690.14669
;
0.514700.51470
;
0.405810.40581
;
0.733810.73381
;
0.043990.04399


Lisa reads two-digit groups until she has chosen three class members (that is, she reads
0.943600.94360
as the groups
9494
,
4343
,
3636
,
6060
). Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers.

The random numbers
0.943600.94360
and
0.998320.99832
do not contain appropriate two digit numbers. However the third random number,
0.146690.14669
, contains
1414
(the fourth random number also contains
1414
), the fifth random number contains
0505
, and the seventh random number contains
0404
. The two-digit number
1414
corresponds to Macierz,
0505
corresponds to Cuningham, and
0404
corresponds to Cuarismo. Besides herself, Lisa's group will consist of Marcierz, Cuningham, and Cuarismo.

USING THE TI-83, 83+, 84, 84+ CALCULATOR

To generating Random Numbers:

  • Press MATH.
  • Arrow over to PRB.
  • Press
    55
    :randInt(. Enter
    00
    ,
    3030
    ).
  • Press ENTER for the first random number.
  • Press ENTER two more times for the other
    22
    random numbers. If there is a repeat press ENTER again.


Note: randInt(
0,30,30, 30, 3
) will generate
33
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 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. 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
nn
th piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains
20,00020,000
residence listings. You must choose
400400
names for the sample. Number the population
11
20,00020,000
and then use a simple random sample to pick a number that represents the first name in the sample. Then choose every fiftieth name thereafter until you have a total of
400400
names (you might have to go back to the beginning of your phone list). Systematic sampling is frequently chosen because it is a simple method.

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.

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 with replacement is very low.

In a college population of
10,00010,000
people, suppose you want to pick a sample of
1,0001,000
randomly for a survey. For any particular sample of
1,0001,000
, if you are sampling with replacement,

  • the chance of picking the first person is
    1,0001,000
    out of
    10,00010,000
    (
    0.10000.1000
    );
  • the chance of picking a different second person for this sample is
    999999
    out of
    10,00010,000
    (
    0.09990.0999
    );
  • the chance of picking the same person again is
    11
    out of
    10,00010,000
    (very low).


If you are sampling without replacement,

  • the chance of picking the first person for any particular sample is
    10001000
    out of
    10,00010,000
    (
    0.10000.1000
    );
  • the chance of picking a different second person is
    999999
    out of
    9,9999,999
    (
    0.09990.0999
    );
  • you do not replace the first person before picking the next person.


Compare the fractions
99910,000\displaystyle\frac{{999}}{{10,000}}
and
9999,999\displaystyle\frac{{999}}{{9,999}}
. For accuracy, carry the decimal answers to four decimal places. To four decimal places, these numbers are equivalent (
0.09990.0999
).

Sampling without replacement instead of sampling with replacement becomes a mathematical issue only when the population is small. For example, if the population is
2525
people, the sample is ten, and you are sampling with replacement for any particular sample, then the chance of picking the first person is ten out of
2525
, and the chance of picking a different second person is nine out of
2525
(you replace the first person).

If you sample without replacement, then the chance of picking the first person is ten out of
2525
, and then the chance of picking the second person (who is different) is nine out of
2424
(you do not replace the first person).

Compare the fractions
925\displaystyle\frac{{9}}{{25}}
and
924\displaystyle\frac{{9}}{{24}}
. To four decimal places,
925\displaystyle\frac{{9}}{{25}}
=
0.36000.3600
and
924\displaystyle\frac{{9}}{{24}}
=
0.37500.3750
. To four 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.

Watch the following video to learn more about sources of sampling bias.



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