1. Chapter 7 of the textbook Pages 215-250
Sampling
Chapter 7 of the textbook
Pages

2. Introduction
Descriptive statistics allow us to describe and summarize our data
Inferential statistics allow us to infer unknown values and/or the probability of events occurring using the data we have (this let’s us test hypotheses)
Probability is at the heart of inferential statistics because we base our results on samples

3. Sampling Error
Sampling error occurs when sample characteristics deviate from population characteristics (i.e., an unrepresentative sample)
Based on random chance
How do we know what the population characteristics are?
How can sampling error be decreased?
“sampling error is not a “mistake”…. All samples deviate from the population in some way; thus, sampling error is always present”

4. Good Points From Textbook
Uncertainty associated with sampling error is the price one pays for using a sample.
“The appeal of statistics is not that it removes uncertainty but rather that it permits inference in the presence of uncertainty.”

5. Sampling Bias
When the selection of the sample favors inclusion of members of a population with certain characteristics
Based on sampling procedures (a.k.a. sampling design or sampling scheme)
How can sampling bias be minimized?

6. Why Sample?
Samples take less time and money to collect – this is particularly true when very detailed measurements are being taken (e.g., in-depth interviews with follow-ups)
A census has error too (contains non-sampling errors)
The population may be infinite
The act of sampling may be destructive
The population may be hypothetical
Populations may change rapidly (i.e., require repeated measurement)

7. Steps in the Sampling Process
Critical pre-sampling step
Make sure there is a need to sample (i.e., see if available data may suit your study)
This can be challenging and can require a lot of effort, but it is usually MUCH easier than collecting a new sample
Rushing out to sample will often come back to haunt you (think before you collect!)

8. Steps in the Sampling Process
Step 1 – Define your population
Who or what is are you interested in (or not)
This is directly connected with the research question(s) you are asking
Example: Geography Students
Does this mean majors or any students in a geography class?
Does this mean graduate and undergraduate students?
Does this include alumni or only current students?

9. Steps in the Sampling Process
Step 2 – Construct a sampling frame
A sampling frame is an exhaustive list of all individuals in a population (i.e., who/what can be sampled)
A sample is only relevant for the sample frame, not appropriate for making inferences beyond the defined sample frame
Differentiate target population and sample populations
Target populations are all the individuals relevant to a study (i.e., who/what we want to include)
The sample population is who/what we actually sample
Goal is for these to be equal
The target and sample populations can differ when some of the target population can’t be sampled
Example: A list of all geography majors

10. Steps in the Sampling Process
Step 3 – Sampling Design
The procedures we use to select members from the sample frames for the sample
Many ways to do this, we’ll discuss several in the coming slides
Example: how we will go about picking a sample of 20 geography majors

11. Steps in the Sampling Process
Step 4 – Specify the information to be collected
Based on pre-testing (a.k.a. pilot testing) the instrument, tools, staff abilities, logistical constraints, etc.
The data collected relate directly to the research questions
This step assesses feasibility and corrects any problems before sampling begins
Example: What questions we will ask the students and about how long it will take to conduct an interview

12. Steps in the Sampling Process
Step 5 – Data Collection
The actual collection of the sample
Data accuracy and quality are determined during this step (i.e., non-sampling error happens here)
Careful measurement and careful recording of the measurements are key

13. Types of Samples
Non-Probability Sample – the likelihood of an individual being sampled is unknown
The sample may be representative or not, but the quality of the sample cannot be determined
Probability Sample – the likelihood of an individual being sampled is known
Since the probabilities are known, probability theory can be applied to make inferences

14. Non-Probability Samples
Judgmental – personal judgment is used to determine which individuals should be included in a sample
Convenience – sample in which only the convenient or accessible individuals are selected
Quota – data obtained from specific subgroups to avoid over or under representation
Volunteer Sample – individuals self select to take part in a study

15. Probability Samples
Random
Systematic
Stratified Random
Clustered

16. Random Samples
For finite populations – each possible sample of size n has an equal probability of being selected
For infinite populations – all observations chosen are statistically independent
Sampling with & without replacement
What would a random sample look like for spatial data when mapped?

17. Random Number Generators
This is one mechanism for actually getting a random sample
Key components
Uniform probability of a number being selected (i.e., 1/10 chance for each digit 0 to 9)
Independence (i.e., first number has no effect on second number)

18. Systematic Samples
Selecting every kth element of a sampling frame (e.g., taking every 10th element)
This is effectively random if
A) the starting point is random
B) there is no natural periodicity (i.e. pattern) to the data
What would a systematic sample look like for spatial data when mapped?

19. Stratified Random Samples
The sample frame is first split into fairly homogeneous classes, from which random samples are taken
Why would we do this?
What are the drawbacks of this approach?
What might a stratified random sample look like for spatial data?
Are there other options?

20. Cluster Samples
Data are grouped into heterogeneous clusters and a census is taken of randomly chosen clusters
Why might we use this approach, particularly for spatial data?
Why can this approach be problematic?
What would cluster samples look like on a map?

21. How do we decide which design to use?
Above all the sample should be as representative as possible
Choices made to increase efficiency, decrease cost or time, etc. should be made carefully
Many sampling designs are hybrids

22. Sampling Distributions
Recall from chapter 2 that there are descriptive statistics (e.g., the mean and the standard deviation) for both samples and populations
Recall from chapter 6 that a random variable (X) can be any value (x1 … xn) from a population, each with an associated probability
Therefore, random variables can be defined as functions (f) or probability distributions (e.g., a histogram or a curve) based on the values they can take on

23. Sampling Distributions
Now extend this concept so that our sample is a set of random variables from a population
What is the mean of the sample?
Just like you’d expect, the mean is the sum of the random variables divided by n
BUT, since each random variable is random, the mean itself is also a random variable
Therefore the sample mean can also be defined as a function or distribution

24. Sampling Distributions
Think of this as a new distribution (curve, function, etc.) where the graphed values relate to the mean values of the random variables (X) noted as
Conceptually this is the histogram that you would produce using the mean values from many independent samples from the same population
The sample statistic is the random variable (in this case the mean) based on a sample of random variables
Since the sample statistic is a random variable, it has a distribution, which is known as the sampling distribution

25. Example Let the population = {10,12,13,16,19,20}
The population mean (μ) = 90/6 = 15
The population standard deviation (σ) = 3.65
Let the sample size (n) = 4
All possible samples (15 total combinations):

26. Example Continued

27. Example Continued
In this case, because we have all possible samples (n=4) from our population the
Why are the standard deviation values different?
The standard deviation of a sample distribution is known as its standard error

28. Central Limits Theorem
For a large n (n > 30)
The distribution of will be approximately normal
The peak of the distribution is then the “mean of the means”, and since the distribution is normal we can estimate the actual population mean (μ) with some degree of confidence
The standard deviation of is:
As n increases the distribution of X becomes more peaked (i.e., the variability of decreases and more closely approximates μ)

29. Central Limits Theorem
How is this theorem useful to us?
“it provides a way of deducing the results of a sample based only on a knowledge of population mean and standard deviation”… and “determine the probability that a sample mean statistics is >, <, or within a given interval”
The key to making this possible is the approximate normal distribution, for which we can easily apply z-values

30. Example Height of middle school kids μ = 60 inches σ = 10 inches
What is the probability of having a class (n = 30) with a mean height of 70 inches?
Remember that

31. Geographic Sampling Why might we want to collect a geographic sample?
Many data are distributed spatially, but you could argue that space can be coded aspatially
However:
Recording space in addition to characteristics allows other (independent) variables to be derived as needed
Space can act as a place holder for things we don’t fully understand
What can we do with geographic samples?
Welcome to the field of spatial analysis….

32. Geographic Sampling
The sampling frame (i.e., all possible samples) is typically done using Cartesian coordinates (X and Y values)
Sample sites are then selected by choosing X,Y pairs using some sampling procedure (e.g., random)
Common Geographic Sampling Methods (i.e., the geographic unit/object being sampled)
Quadrats
Transects (traverses)
Point sampling

33. Quadrats A square, areal sample (i.e., a polygon)
Quadrat size depends on feature of interest & research question – picking the “right” size can be challenging
Arrangement of quadrats on the landscape can be any of the types previously mentioned
Random
Systematic
Stratified Random
Clustered
Quadrat orientation can also vary

34. Transects Transects sample a geographic area along lines
Placing the sample lines is how randomness etc. is included in the sample
Options discussed in textbook
Random
Systematic
Stratified random
Stratified systematic
The n value can be the total length (L) or the number of transects (typically of uniform length)

35. Point Samples Sampling at point locations
Locating the points can be done similarly to locating quadrats
Options discussed in textbook
Random
Systematic
Stratified random
Clustered
Stratified, systematic, unaligned sample

36. Example Carolina Vegetation Survey (CVS) Paper is on blackboard (.pdf)
This is one example of a real sampling approach
In biogeography, ecology, etc. we often sample using nested quadrats

37. Summary Think before you sample
Many sampling approaches (spatial or aspatial) exist, choosing the “right” one takes experience and some knowledge of the population
The Central Limits Theorem is related to the distribution of certain sample statistics (the mean in particular) and is important for inference