Griffiths D. Head First Statistics

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The Case of the Lost Coffee Sales

The Starbuzz CEO has an idea for a brand-new coffee he wants to

sell in his coffee shops, but he’s not sure how popular it’s going to be

with his customers. He asks his new intern to conduct a survey to help

predict the customers’ opinions. The intern will ask customers to taste

the new brew, and tell him what they think.

The intern is really happy to be given such a great opportunity.

First off, he’s been told that if he does the job well, he stands to

get a bonus at the end of the month. Secondly, he gets to give

out free coffee to friendly Starbuzz customers and hear lots of

positive things. Thirdly, he’s been looking for an excuse to talk to

one particular girl who’s a regular visitor to his local coffee shop,

and this could be just the break he needs.

After the intern conducts his survey, he’s delighted to tell the CEO

that everyone loves the new coffee, and it’s bound to be a huge success.

“That’s great,” says the CEO. “We’ll launch it next season.”

When the new coffee is finally launched, sales are poor, and the CEO

has to cancel the range. What do you think went wrong?

Why didn’t the new coffee sell well?

Five Minute

Mystery

Five Minute

Mystery

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How to design a sample

You use samples to make inferences about the population in general, and to

make sure you get accurate results, you need to choose your sample wisely. Let’s

start off by pinning down what your population really is, so you can get as

representative a sample as possible.

Define your sampling units

Once you’ve defined your target population, you need to decide what sort of object

you’re going to sample. Normally these will be the sorts of things you described when

you defined your target population. As an example, this could be a single gumball or

maybe a packet of gumballs.

Define your target population

The first thing to be clear about is what your target population is so that you know

where you’re collecting your sample from. By target population, we mean the group

that you’re reseraching and want to collect results for. The target population you

choose depends, to a large extent, on the purpose of your study. For example, do you

want to gather data about all the gumballs in the world, one particular brand, or one

particular type?

Try to be as precise as possible, as that way it’s easier to make your sample as

representative of your population as possible.

We need data about Mighty

Gumball’s super-long-lasting

gumballs, so your target

population is all of the gumballs.

The sampling unit in the taste

test is a single Mighty Gumball

super-long-lasting gumball.

designing a sample

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Define your sampling frame

Finally, you need a list of all the sampling units within

your target population, preferably with each sampling

unit either named or numbered. This is called the

sampling frame. It’s basically a list from which you can

choose your sample.

Sometimes it’s not possible to come up with a list that covers

the entire target population. As an example, if you want to

collect the views of people living within a certain area, people

moving in or out of an area can affect who you have on your

list of names. If you’re dealing with similar objects such as

gumballs, it might not be possible or practical to name or

number each one.

This seems like a

waste of time. Do I have to

do all of these things? Can’t

I just sample gumballs?

If you don’t design your sample well,

your sample may not be accurate.

Designing your sample can take a bit of extra

preparation time, but this is much better than

spending time and money on a survey only to find

the results are inaccurate. You will have lost time

and money doing the survey, and what’s more,

someone might make wrong decisions based on it.

A poorly designed sample can introduce bias.

Let’s look at this in more detail.

Naming or numbering

each gumball maybe

isn’t that practical.

Gumball #1897652

Gumball #1897653

Gumball #1897654

Gumball #1897655

Gumball #1897656

Gumball #1897657

Gumball #1897658

Gumball #1897659

Gumball #1897660

Gumball #1897661

Gumball #1897662

Gumball #1897663

Gumbal4

Gumball #1897670

Gumball #1897671

Gumball #1897672

Gumball #1897673

Gumball #1897674

Gumball #1897675

Gumball #1897676

Gumball #1897677

Gumball #1897678

Gumball #1897679

Gumball #1897680

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Sometimes samples can be biased

Not every sample is fair. Unless you’re very careful, some sort of bias can creep

in to the sample, which can distort your results. Bias is a sort of favoritism that

you can unwittingly (or maybe knowingly) introduce into your sample, meaning

that your sample is no longer randomly selected from your population

If a sample is unbiased, then it’s representative of the population. It’s a fair

reflection of what the population is like.

Both means are

the same.

Unbiased samples

An unbiased sample is representative of

the target population. This means that it has

similar characteristics to the population, and

we can use these to make inferences about

the population itself.

The shape of the distribution of an unbiased

sample is similar to the shape of the

population it comes from. If we know the

shape of the sample distribution, we can

use it to predict that of the population to a

reasonable level of confidence.

Biased samples

A biased sample is not representative of

the target population. We can’t use it to make

inferences about the population because

the sample and population have different

characteristics. If we try to predict the shape

of the population distribution from that of

the sample, we’d end up with the wrong

result.

bias in sampling

Biased Sample

duration

frequency

Unbiased Sample

duration

frequency

Sample mean

Population mean

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Sources of bias

So how does bias creep into samples? Through any of the following and more:

A sampling frame where items have been left off, such that

not everything in the target population is included. If it’s not in your

sampling frame, it won’t be in your sample.

An incorrect sampling unit. Instead of individual gumballs, maybe

the sampling unit should have been boxes of gumballs instead.

Individual sampling units you chose for your sample weren’t

included in your actual sample. As an example, you might send

out a questionnaire that not everybody responds to.

Poorly designed questions in a questionnaire. Design your

questions so that they’re neutral and everyone can answer them. An

example of a biased question is “Mighty Gumball candy is tastier than

any other brand, do you agree?” It would be better to ask the person

being surveyed for the name of their favorite brand of confectionary.

Samples that aren’t random. As an example, if you’re conducting

a survey on the street, you may avoid questioning anyone that looks too

busy to stop, or too aggressive. This means that you exclude aggressive

or busy-looking people from your survey.

This sounds hopeless. How can I be certain I

avoid bias? Where does it come from anyway?

You mean I can’t just

try the pink ones???

As you can see, there are lots of sources of bias, and a lot

of it comes down to how you choose your sample.

We need to take a look at ways in which you can choose your sample to minimize

the chances of introducing bias.

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So is the sampling frame a list of

everything that we’re sampling?

A: The sampling frame lists all the

individual units in the population, and it’s

used as a basis for the sample. It’s not the

sample itself, as we don’t sample everything

on it.

How do I put together the sampling

frame?

A: How you do it and what you use

depends on your target population. As an

example, if your target population is all car

owners, then you can use a list of registered

car owners. If your target population is all

the students attending a particular college,

you can use the college registrar.

How about things like telephone

listings? Can I use those for my sampling

frame?

A: It all depends on your target population.

Telephone listings exclude households

without a telephone, and there may also

be households who have elected not to be

listed. If your target population is households

with a listed telephone number, then using

telephone lists is a good idea. If your target

population is all households with a telephone

or even all telephones, then your sampling

frame won’t be entirely accurate—and that

can introduce bias.

Can I always compile a sampling

frame?

A: Not always. Imagine if you had to

survey all the fish in the sea. It would be

impossible to name and number every

individual fish.

Will I always have to have a target

population?

A: Yes. You need to know what your

target population is so that you can make

sure your sample is representative of it.

Thinking carefully about what your target

population is can help you avoid bias.

If you’re sampling for someone else, get

as much detail as possible about who the

target population should be. Make sure you

know exactly what is included and what is

excluded.

Q: Why is bias so bad?

A: Bias is bad because it can mislead

you into drawing wrong conclusions about

your target population, which in turn can

lead you into making wrong decisions. If, for

example, you only sampled pink gumballs,

your survey results might be accurate for

all pink gumballs, but not for all gumballs in

general. There may be significant differences

between the different color gumballs.

How can the questions in a

questionnaire cause bias?

A: Bias often creeps in through the

phrasing of questions.

First off, if you present a series of statements

and ask respondents to agree or disagree,

it’s more likely that people will agree unless

they have strong negative feelings. This

means that the results of your survey will be

biased towards people agreeing.

Bias can also occur if you give a set of

possible answers that don’t cover all

eventualities. As an example, imagine you

need to ask people how often they exercise

in a typical week. You would introduce bias

if you give answers such as “more than 5

times a week,” “3–5 times a week,” “1–2

times a week,” and “I don’t value my health,

so I don’t exercise.” Someone may not

exercise, but disagree with the statement

that they don’t value their health. This would

mean that they wouldn’t be able to answer

the question.

there are no dumb questions

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Look at the following scenarios. What would you choose as a

target population? What’s the sampling unit? How would you

develop a sampling frame? What other things might you need to

consider when forming your sample?

1. Choc-O-Holic Inc. manufactures chocolates, and they have just finished a limited-

edition run of chocolates for the holiday season. They want to check the quality of

those chocolates.

2. The Statsville Health Club wants to conduct a survey to see what their customers

think of their facilities.

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Look at the following scenarios. What would you choose as a

target population? What’s the sampling unit? How would you

develop a sampling frame? What other things might you need to

consider when forming your sample?

1. Choc-O-Holic Inc. manufactures chocolates, and they have just finished a limited-

edition run of chocolates for the holiday season. They want to check the quality of

those chocolates.

2. The Statsville Health Club wants to conduct a survey to see what their customers

think of their facilities.

The target population is all the chocolates in the limited edition run.

The sampling unit is one chocolate.

The sampling frame needs to cover all of the chocolates; as it’s a limited-edition run, it’s possible that

Choc-O-Holic has records of how many chocolates are in the run, including numbers of each type of

chocolate.

When forming the sample, you need to make sure that it is representative of the target population and

unbiased. If there are different types of chocolate in the run, you’d need to make sure that you included

each sort of chocolate.

The target population is all the customers of the Statsville Health Club.

The sampling unit is one customer.

The sampling frame needs to cover all of the customers. It’s likely that the health club has a list of

registered customers, so you could use this as the sampling frame.

As before, you need to make sure that your sample is representative of the population and unbiased. You’d

need to make sure that each of the classes is fairly represented by customer gender, customer age group,

and so on.

sharpen your pencil solution

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Solved: The Case of the Lost Coffee Sales

Why didn’t the coffee sell well?

We don’t know for certain, but there’s a very good chance that the

sample of people surveyed by the intern wasn’t

representative of the target population.

First of all, the intern was looking forward to giving

away free coffee to friendly Starbuzz customers and

hearing positive things. Does this mean he only spoke

to customers who looked friendly to him? Did he get their

real opinions about the coffee, or did he only ask them whether they

agreed it tasted nice?

The intern also hoped to use the job as an opportunity to speak to a girl

at his local coffee shop. Did he spend most of his time in this particular

coffee shop? Did the girl influence his sample choice?

Finally, the CEO launched the new coffee in a different season from the

one in which the survey took place, and this may have affected sales too.

Any or all of these factors could have lead to the sample being

misleading, which in turn led to the wrong decision being made.

Five Minute

Mystery

Solved

Five Minute

Mystery

Solved

430 Chapter 10

How to choose your sample

We’ve looked at how to design your sample and explored types of bias that need

to be avoided. Now we need to select our actual sample from the sample frame.

But how should we go about this?

Simple random sampling

One option is to choose the sample at random. Imagine you have a

population of N sampling units, and you need to pick a sample of n

sampling units. Simple random sampling is where you choose a

sample of n using some random process, and all possible samples of size

n are equally likely to be selected.

With simple random sampling, you have two options. You can either

sample with replacement or without replacement.

Sampling without replacement

Sampling without replacement means that the sampling unit isn’t

replaced back into the population. An example of this is the gumball

taste test; you wouldn’t want to put gumballs that have been tasted back

into the population.

Sampling with replacement

Sampling with replacement means that when you’ve selected each

unit and recorded relevant information about it, you put it back into the

population. By doing this, there’s a chance that a sampling unit might

be chosen more than once. You’d be sampling with replacement if you

decided to question people on the street at random without checking

if you had already questioned them before. If you stop a person for

questioning and then let them go once you’ve finished asking them

questions, you are in effect releasing them back into the population. It

means that you may question them more than once.

You wouldn’t want to replace

gumballs once you’ve tasted them,

so this would be simple random

samping without replacement.

simple random sampling