Introduction Previous lessons focused on the relationship between samples and populations, and on...

IntroductionPrevious lessons focused on the relationship between samples and populations, and on using random sampling to select a representative sample and reduce sampling bias. This lesson introduces the idea that simple random sampling is not the only method for selecting representative samples, and that the sampling method used often depends on the goal of the research being conducted as well as practical considerations.

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1.3.3: Other Methods of Random Sampling

Introduction, continuedDifferent sampling methods can be helpful tools for a wide variety of research situations. Furthermore, familiarity with these methods allows you to understand the methods used by other researchers who often need to mix and match methods in order to meet practical challenges without compromising the representative nature of their samples.

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Key Concepts• Additional sampling methods include cluster

sampling, systematic sampling, and stratified sampling.

• All of these methods involve random assignment, although none meet the criteria of simple random sampling.

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Key Concepts, continued• With a cluster sample, naturally occurring groups of

population members are chosen for the sample. This method involves dividing the population into groups by geography or other practical criteria. Some of the groups are randomly selected, while others are not. This method allows each member of the population to have a nearly equal chance of selection. Cluster sampling is usually chosen to eliminate excessive travel or reduce the disruption that a study may cause.

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Key Concepts, continued• A systematic sample is a sample drawn by selecting

people or objects from a list, chart, or grouping at a uniform interval. This method involves using a natural ordering of population members, such as by arrival time, location, or placement on a list. Once the order is established, every nth member (e.g., every fifth member) is chosen. If the starting number is randomly selected, then each member of the population has a nearly equal chance of selection. Systematic sampling is usually chosen when relative position in a list may be related to key variables in a study, or when it is useful to a researcher to space out data gathering.

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Key Concepts, continued• For a stratified sample, the population is divided into

subgroups so that the people or objects within the subgroup share relevant characteristics. This method involves grouping members of the population by characteristics that may be related to parameters of interest. Once the groups are formed, members of each group are randomly selected so that the number of members in the sample with given characteristics is approximately proportional to the number of members in the population with the same characteristics. Stratified sampling has been used for many years to predict the results of state and national elections.

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Key Concepts, continued• A convenience sample is a sample for which members

are chosen in order to minimize time, effort, or expense. Convenience sampling involves gathering data quickly and easily. The advantage of convenience sampling is that, in some cases, preliminary estimates of population parameters can be obtained quickly. The main disadvantage of convenience sampling is that the samples are prone to serious biases. As a result, the estimates obtained are seldom accurate and the statistics are difficult to interpret.

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Key Concepts, continued• While simple random samples provide unbiased

estimates, there are situations in which the goal of the research is better served by other forms of sampling. These include situations in which the goal is to count all members of a population and situations in which the sample provides a comparison group.

• It is unwise to use a sampling method simply because it is the most convenient. Unless the sample is representative of the population of interest, the statistics that are produced may be misleading.

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Key Concepts, continued• A larger sample is not always a better sample.

There is less variability in measures taken from a large sample, but if the large sample is biased, the researcher will likely obtain estimates that are inaccurate.

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Common Errors/Misconceptions• mistakenly believing that a larger sample is always a

better sample

• ignoring bias when making estimates regarding the entire population

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Guided Practice

Example 2Pearce wants to conduct a survey of shoppers at the local mall. He obtains a list of the major stores, restaurants, and other establishments and creates the following table that includes each destination’s name, location (zone), category, and category rank. The category rank represents where the mall destination falls in a list of all the establishments in the same category; for example, Aéropostale is second in the list of clothing stores, so its category rank is 2.

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Use the table (shown on the following slides) and two methods to choose a cluster sample of 5 establishments at which Pearce can interview shoppers.

• Method 1: Give each zone an equal chance of selection.

• Method 2: Give each establishment an equal chance of selection.

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Establishment Zone Category Category rankAbercrombie & Fitch D Clothing 1

Aéropostale D Clothing 2

Amato’s A Food 1

American Eagle B Clothing 3

Arby’s A Food 2

AT&T CTechnology/electronics 1

babyGap D Clothing 4

Banana Republic E Clothing 5

Barton’s Couture D Clothing 6

Bath & Body Works B Bath/beauty 1

The Body Shop D Bath/beauty 2(continued)

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(continued)

Establishment Zone Category Category rank

Build-A-Bear Workshop B Toys/hobbies 1

Bureau of Motor Vehicles D Services 2

Charley’s Subs A Food 3

Chico’s D Clothing 7

The Children’s Place B Clothing 8

Claire’s A Accessories 1

Coach B Accessories 2

Coldwater Creek C Clothing 9

dELiA*s B Clothing 10

Dube Travel A Services 1

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(continued)


Eddie Bauer D Clothing 11

Express D Clothing 12

f.y.e. ATechnology/electronics 2

Foot Locker B Clothing 13

Francesca’s B Clothing 14G.M. Pollack & Sons C Jewelry 4

GameStop A Toys/hobbies 2

Gap D Clothing 15Gloria Jean’s Coffee C Food 4

Go Games C Toys/hobbies 3

Gymboree E Clothing 16

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(continued)


Hannoush Jewelers A Jewelry 1

Hometown Buffet C Food 5

Hot Topic A Clothing 17

Icing by Claire’s A Accessories 3

J.Crew E Clothing 18

J.Jill B Clothing 19

Johnny Rockets A Food 6

Just Puzzles B Toys/hobbies 4

Kamasouptra A Food 7

Kay Jewelers D Jewelry 2La Biotique A Bath/beauty 3

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(continued)


Lane Bryant D Clothing 20

LensCrafters A Services 3

Lids A Accessories 4

LOFT D Clothing 21

LUSH E Bath/beauty 4

MasterCuts A Bath/beauty 5

Mayflower Massage A Services 4

Mrs. Field’s Cookies A Food 8

Olympia Sports D Toys/hobbies 5

On Time A Accessories 5

Origins B Bath/beauty 6

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(continued)


PacSun A Clothing 22

Panda Express A Food 9

The Picture People A Services 5

Piercing Pagoda E Jewelry 3

Pretzel Time/TCBY C Food 10

Pro Vision A Services 6

Qdoba A Food 11

Radio Shack C Technology/electronics 3

Red Mango A Food 12

Regis Salon A Bath/beauty 7

Sarku Japan A Food 13

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Sbarro A Food 14

Sephora E Bath/beauty 8

Starbucks A Food 15

Sunglass Hut B Accessories 6

Super Hearing Aids A Services 7

Swarovski D Jewelry 5

T & C Nails A Bath/beauty 9

T-Mobile B Technology/electronics 4

Teavana D Food 16

Verizon Wireless A Technology/electronics 5

Guided Practice: Example 2, continued

Method 1: Give each zone an equal chance of selection.

1. Number the zones.

The mall is divided into 5 zones, so assign each zone a number 1 through 5.

Let A = 1, B = 2, C = 3, D = 4, and E = 5.

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2. Select a zone of the mall.

Randomly select 1 of the 5 zones using 5 cards from a standard deck or a random number generator.

Suppose that a 4 is chosen. This corresponds to Zone D.

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3. Label the businesses in the chosen zone. There are 16 establishments in Zone D, so label each one with a number from 1 to 16.

1 = Abercrombie & Fitch 9 = Express

2 = Aéropostale 10 = Gap

3 = babyGap 11 = Kay Jewelers

4 = Barton’s Couture 12 = Lane Bryant

5 = The Body Shop 13 = LOFT

6 = Bureau of Motor Vehicles 14 = Olympia Sports

7 = Chico’s 15 = Swarovski

8 = Eddie Bauer 16 = Teavana 22



4. Randomly select 5 of the establishments in the selected zone. Using 16 cards or a random number generator, randomly select 5 establishments from Zone D. Discard repeats.

Results will vary, but suppose the numbers 1, 4, 7, 8, and 12 are randomly chosen.

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Guided Practice: Example 2, continuedThese numbers correspond to the following establishments:

1 = Abercrombie & Fitch

4 = Barton’s Couture

7 = Chico’s

8 = Eddie Bauer

12 = Lane Bryant

The corresponding cluster sample of 5 establishments at which Pearce can interview shoppers consists of Abercrombie & Fitch, Barton’s Couture, Chico’s, Eddie Bauer, and Lane Bryant.

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Guided Practice: Example 2, continuedMethod 2: Give each establishment an equal chance of selection.

1. Label each establishment.There are 75 establishments, so label each of them with a number from 1 to 75.

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2. Randomly select a number from 1 to 75.

Randomly select one of the 75 establishments using 75 cards or a random number generator.

Suppose a 10 is chosen. This corresponds to Barton’s Couture.

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3. Since this is a cluster sample, choose 4 other establishments in the same zone.

Barton’s Couture is in Zone D.

There are 16 establishments in Zone D, so label each one with a number from 1 to 16, as shown on the next slide.

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Guided Practice: Example 2, continued1 = Abercrombie & Fitch 9 = Express

2 = Aéropostale 10 = Gap

3 = babyGap 11 = Kay Jewelers

4 = Barton’s Couture 12 = Lane Bryant

5 = The Body Shop 13 = LOFT

6 = Bureau of Motor Vehicles 14 = Olympia Sports

7 = Chico’s 15 = Swarovski

8 = Eddie Bauer 16 = Teavana

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4. Randomly select 4 other establishments in Zone D.

Using 16 cards or a random number generator, randomly select 4 additional establishments from Zone D. Discard repeats.

Results will vary, but suppose the numbers 1, 9, 13, and 15 are randomly chosen.

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These numbers correspond to the following stores:

1 = Abercrombie & Fitch

9 = Express

13 = LOFT

15 = Swarovski

The corresponding cluster sample of 5 establishments at which Pearce can interview shoppers consists of Barton’s Couture, Abercrombie & Fitch, Express, LOFT, and Swarovski.

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Guided Practice: Example 2, continuedNote: Method 1 will probably be more convenient because the smaller zones (Zone C and Zone E) have an equal chance of selection. Since small zones have fewer establishments, the establishments in a small zone will probably be closer together, on average, than the establishments in a large zone, making it easier on Pearce to conduct his survey. Using Method 2 means that the establishments in smaller zones have less chance of being selected.

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Guided Practice

Example 3Kylie wants to estimate the total number of times customers enter different establishments at the same mall described in Example 2. Kylie has 10 electronic devices that can count the number of customers entering a given establishment. Use the tables provided in Example 2 to select a stratified sample (by category) of 10 establishments at which Kylie can install her counting devices.

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1. Construct a table that shows the number of establishments in each category. Refer to the table in Example 2 to determine the number of establishments in each category. Organize the results in a new table (shown on the next slide).

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CategoryNumber of

establishments

Clothing 22

Food 16

Bath/beauty 9

Services 7

Accessories 6

Jewelry 5

Technology/electronics 5

Toys/hobbies 5

Total 75


2. Determine the number of establishments to select from each category. Since Kylie needs to select 10 establishments from only 8 categories, select 2 establishments from the largest 2 categories, and 1 from each remaining category. Two stores each from the Clothing and Food categories will be selected, since these are the largest categories.

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3. Organize the list of establishments by category, then number each item within a category.

Create tables to organize the 8 categories of establishments (shown beginning on the next slide).

Number the stores from 1 to n, where n is the number ranking of a particular establishment in a list of all the members of the same category. For example, babyGap is fourth in the list of clothing stores, so its value for n is 4.

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FoodName n Name n

Amato’s 1 Panda Express 9Arby’s 2 Pretzel Time/TCBY 10Charley’s Subs 3 Qdoba 11Gloria Jean’s Coffee 4 Red Mango 12Hometown Buffet 5 Sarku Japan 13Johnny Rockets 6 Sbarro 14Kamasouptra 7 Starbucks 15Mrs. Field’s Cookies 8 Teavana 16


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ClothingName n Name n

Abercrombie & Fitch 1 Express 12Aéropostale 2 Foot Locker 13American Eagle 3 Francesca’s 14babyGap 4 Gap 15Banana Republic 5 Gymboree 16Barton’s Couture 6 Hot Topic 17Chico’s 7 J.Crew 18The Children’s Place 8 J.Jill 19Coldwater Creek 9 Lane Bryant 20dELiA*s 10 LOFT 21Eddie Bauer 11 PacSun 22


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Bath/beautyName n

Bath & Body Works 1The Body Shop 2La Biotique 3LUSH 4MasterCuts 5Origins 6Regis Salon 7Sephora 8T & C Nails 9

ServicesName n

Dube Travel 1Bureau of Motor Vehicles 2LensCrafters 3Mayflower Massage 4The Picture People 5Pro Vision 6Super Hearing Aids 7


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AccessoriesName n

Claire’s 1Coach 2Icing by Claire’s 3Lids 4On Time 5Sunglass Hut 6

JewelryName n

Hannoush Jewelers 1Kay Jewelers 2Piercing Pagoda 3G.M. Pollack & Sons 4Swarovski 5


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Technology/electronicsName n

AT&T 1f.y.e. 2Radio Shack 3T-Mobile 4Verizon Wireless 5

Toys/hobbiesName n

Build-A-Bear Workshop 1GameStop 2Go Games 3Just Puzzles 4Olympia Sports 5


4. Randomly select the appropriate number of stores in each category. Using cards or a random number generator, randomly select 2 of the 22 clothing stores, 2 of the 16 food stores, 1 of the 9 bath/beauty stores, 1 of the 7 service stores, 1 of the 6 accessories stores, 1 of the 5 jewelry stores, 1 of the 5 technology/electronics stores, and 1 of the 5 toys/hobbies stores.

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Guided Practice: Example 3, continuedResults will vary, but suppose the following numbers were selected:• Clothing: The random integers 12 and 9 were

selected.• Food: The random integers 9 and 16 were selected.• Bath/beauty: The random integer 5 was selected.• Services: The random integer 1 was selected.• Accessories: The random integer 5 was selected.• Jewelry: The random integer 5 was selected.• Technology/electronics: The random integer 5 was

selected.• Toys/hobbies: The random integer 5 was selected.

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5. Match each random number with the establishment that falls in that position in the category list.From our tables, we can use the randomly generated numbers to select a stratified sample.

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Guided Practice: Example 3, continuedThe following stores represent the stratified sample.

• Clothing: 9 = Coldwater Creek and 12 = Express

• Food: 9 = Panda Express and 16 = Teavana

• Bath/beauty: 5 = MasterCuts

• Services: 1 = Dube Travel

• Accessories: 5 = On Time

• Jewelry: 5 = Swarovski

• Technology/electronics: 5 = Verizon Wireless

• Toys/hobbies: 5 = Just Puzzles

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Guided Practice: Example 3, continuedNote: It is possible with a simple random sample that one or more of the categories will be left out if 10 stores are selected using simple random sampling. By using stratified sampling, each category is represented.

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Introduction Previous lessons focused on the relationship between samples and populations, and on...

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