Griffiths D. Head First Statistics

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using statistical sampling

How to choose a simple random sample

There are two main ways of using simple random sampling: by drawing lots or

using random numbers.

Drawing lots

Drawing lots is just like pulling names out of a hat. You write the

name or number of each member of the sampling frame on a piece

of paper or ball, and then place them all into a container. You then

draw out n names or numbers at random so that you have enough

for your sample.

Random number generators

If you have a large sampling frame, drawing lots might not be

practical, so an alternative is to use a random number generator,

or random number tables. For this, you give each member of the

sampling frame a number, generate a set of n random numbers,

and then pick the members of the set whose assigned numbers

correspond to the random numbers that were generated.

It’s important to make sure that each number has an equal chance

of occuring so that there’s no bias.

Simple random sampling isn’t without its problems. What do you think

could go wrong with it?

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There are other types of sampling

Even simple random sampling has its problems.

With simple random sampling, there’s still a chance that your sample will not

represent the target population. For example, you might end up randomly

drawing only yellow gumballs for your sample, and the other colors would be left

out.

So how can we avoid this?

We can use stratified sampling...

An alternative to simple random sampling is stratified sampling. With this type of

sampling, the population is split into similar groups that share similar characteristics.

These charateristics or groups are called strata, and each individual group is called

a stratum. As an example, we could split up the gumballs into the different colors,

yellow, green, red, and pink, so that each color forms a different stratum.

Once you’ve done this, you can perform simple random sampling on each stratum to

ensure that each group is represented in your overall sample. To do this, look at the

proportions of each stratum within the overall population and take a proportionate

number from each. As an example, if 50% of the gumballs that Mighty Gumball

produce are red, half of your sample should consist of red gumballs.

Each color is a

separate stratum.

We sample a proportionate

number from each stratum.

stratified, cluster, and systematic sampling

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using statistical sampling

...or we can use cluster sampling...

Cluster sampling is useful if the population has a number of similar

groups or clusters. As an example, gumballs might be sold in packets,

with each packet containing a similar number of gumballs with similar

colors. Each packet would form a cluster.

With cluster sampling, instead of taking a simple random sample of

units, you draw a simple random sample of clusters, and then

survey everything within each of these clusters. As an example, you

could take a simple random sample of packets of gumballs, and then

taste all the gumballs in these packets.

Cluster sampling works because each cluster is similar to the others,

and an added advantage is that you don’t need a sampling frame of

the whole population in order to achieve it. As an example, if you were

surveying trees and used particular forests as your cluster, you would

only need to know about each tree within only the forests you’d selected.

The problem with cluster sampling is that it might not be entirely

random. As an example, it’s likely that all of the gumballs in a packet

will have been produced by the same factory. If there are differences

between the factories, you may not pick these up.

...or even systematic sampling

With systematic sampling, you list the population in some sort of order,

and then survey every kth item, where k is some number. As an example,

you could choose to sample every 10th gumball.

Systematic sampling is relatively quick and easy, but there’s one key

disadvantage. If there’s some sort of cyclic pattern in the population,

your sample will be biased. As an example, if gumballs are produced

such that every 10th gumball is red, you will end up only sampling red

gumballs, and this could lead to you drawing misleading conclusions

about your population.

Each packet forms a cluster.

You can pick every 10th gumball to get a systematic sample.

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Does using one of these methods

of sampling guarantee that the sample

won’t be biased?

A: They don’t guarantee that the sample

won’t be biased, but they do minimize the

chances of this happening. By really thinking

about your target population and how you

can make your sample representative of it,

you stand a much better chance of coming

up with an unbiased, representative sample.

Do I have to use any of these

methods? Can’t I just choose items at

random.

A: Choosing items at random is simple

random sampling. Yes, this is one approach

you can take, but one thing to be aware of is

that there is a chance your sample will not

be representative of the population at large.

But why? Surely if I choose items

at random, then my sample is bound to

be representative of the target population.

A: Not necessarily. You see, if you choose

sampling units at random, then there’s a

chance that purely at random, you could

choose a sample that doesn’t effectively

represent the target population. As an

example, if you choose customers of the

Statsville Health Club completely at random,

there’s a chance that you might choose only

attendees of one particular class, or of one

particular gender.

There might also be a case where you think

you’re sampling at random, when really

you’re not. As an example, if you conduct a

survey to find out customer satisfaction, but

leave it up to customers whether or not they

respond, you may well end up with a biased

sample as customers have to be sufficiently

motivated to respond. The customers who

are most motivated to take part in the survey

will be those who are either strongly satisfied

or strongly dissatisfied. You are less likely to

hear from those customers without strong

feelings, yet those people may make up the

bulk of the population.

How about if I just increase the size

of my sample? Will that get around bias?

A: The larger your sample, the less

chance there is of your sample being

biased, and this is one way of minimizing the

chances of getting a biased sample using

simple random sampling. The trouble is, the

larger your sample, the more cumbersome

and time-consuming it can be to gather data.

What’s the difference between

stratified sampling and clustered

sampling?

A: With stratified sampling, you divide

the population into different groups or strata,

where all the units within a stratum are as

similar to each other as possible. In other

words, you take some characteristic or

property such as gender, and use this as the

basis for the strata. Once you’ve split the

population into strata, you perform simple

random sampling on each stratum.

With clustered sampling, your aim is to

divide the population into clusters, trying to

make the clusters as alike as possible. You

then use simple random sampling to choose

clusters, and then sample everything in

those clusters.

I see. So with stratified sampling,

you make each stratum as different as

possible, and with clustered sampling,

you make each cluster as similar as

possible.

A: Exactly.

So what about systematic

sampling?

A: With systematic sampling, you choose

a number, k, and then choose every kth

item for your sample. This way of sampling

is fairly quick and easy, but it doesn’t mean

that your sample will be representative of

the population. In fact, this sort of sampling

can only be used effectively if there are no

repetitive patterns or organization in the

sampling frame

Drawing lots sounds antiquated.

Do people still do that?

A: It’s not as common as it used to be,

but it’s still a way of sampling.

there are no dumb questions

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1. How could you apply simple random sampling to this problem?

2. How could you apply stratified sampling?

3. What about cluster sampling?

You’ve been given 10 boxes of chocolates and been asked to sample the chocolates in them.

There are whilte, milk, and dark chocolates in the boxes. Your target population is all of the

chocolates, and the sampling unit is one chocolate.

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1. How could you apply simple random sampling to this problem?

2. How could you apply stratified sampling?

3. What about cluster sampling?

You’ve been given 10 boxes of chocolates and been asked to sample the chocolates in them.

There are whilte, milk, and dark chocolates in the boxes. Your target population is all of the

chocolates, and the sampling unit is one chocolate.

You could apply simple random sampling by choosing chocolates at random, either through drawing

lots or using random numbers. That way, each chocolate stands an equal chance of being sampled.

For stratified smpling, you divide the chocolates into strata and apply simple random sampling to

each one. Each strata comprises of a group of chocolates with similar characteristics, so you could

use the different types of chocolate. One stratum could be white chocolates, another one could be

milk chocolates, and the final one could be dark chocolates.

For cluster sampling, you divide the chocolates into groups, but this time each group needs to be

similar. Assuming each box of chocolates is similar, you could take one of the boxes, and sample all of

the chocolates in it.

exercise solution

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How would you go about conducting a sample survey of Mighty Gumball’s super-long-lasting

gumballs? The gumballs come in four different colors, and they’re all made in the same factory.

Assume you have to start your sample from scratch.

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How would you go about conducting a sample survey of Mighty Gumball’s super-long-lasting

gumballs? The gumballs come in four different colors, and they’re all made in the same factory.

Assume you have to start your sample from scratch.

A population is the entire collection

of things you are studying.

A sample is a relatively small

selection taken from the population

that you can use to draw conclusions

about the population itself.

To take a sample, start off by defining

your target population, the population

you want to study. Then decide on

your sampling units, the sorts of

things you need to sample. Once

you’ve done that, draw up a sampling

frame, a list of all the sampling units

in your target population.

A sample is biased if it isn’t

representative of your target

population.

Simple random sampling is where

you choose sampling units at random

to form your sample. This can be



with or without replacement. You can

perform simple random sampling by

drawing lots or using random number

generators.

Stratified sampling is where you

divide the population into groups of

similar units or strata. Each stratum

is as different from the others as

possible. Once you’ve done this, you

perform simple random sampling

within each stratum.

Cluster sampling is where you

divide the population into clusters

where each cluster is as similar

to the others as possible. You use

simple random sampling to choose

a selection of clusters. You then

sample every unit in these clusters.

Systematic sampling is where you

choose a number, k, and sample

every kth unit.



The target population is all of Mighty Gumball’s super-long-lasting gumballs, and the sampling unit is an

individual gumball. For the sampling frame, we ideally need some sort of numbered list of the gumballs, but

it’s likely that this isn’t practical. Instead, we’ll settle for a list showing how many gumballs there are in the

population for each color.

The type of sampling you use is subjective, but we’d choose to use stratified sampling, as this may be the best

way of coming up with an unbiased sample. We’d divide the gumballs into their different colors and then use

simple random sampling to choose a proportionate number of each of the four colors. We would then use these

for our sample.

Don’t worry if you got a different answer. The key thing is to think through how you can best make your

survey representative of the population, and you may have different ideas.

exercise solution

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using statistical sampling

Mighty Gumball has a sample

With your help, Mighty Gumball has gathered a sample of their

super-long-lasting gumballs. This means that rather than perform

taste tests on the entire gumball population, they can use their

sample instead.

That’s great! It

means we’ll save time,

money, and teeth.

So what’s next?

We’ve looked at how we can put together a representative sample,

but what we haven’t looked at is how we can use it. We know that

an unbiased sample shares the same characteristics as its parent

population, but what’s the best way of analyzing this?

Keep reading, and we’ll show you how in the next chapter.