Transcript Sampling error • Error that occurs in data due to the errors inherent in sampling from a population – Population: the group of.

Sampling error
• Error that occurs in data due to the errors inherent in
sampling from a population
– Population: the group of interest (e.g., all students at UIC)
– Sample: a subset of the population that is studied (i.e.,
people in this class)
• Symbols for making the distinction:
–  = mu = population mean; M = sample mean
–  = sigma = population standard deviation; s = SD = sample SD
– 2 population variance; s2 = sample variance
• Note: The problem of sampling error seems to worry
psychologists more than the other two problems that
we have discussed.
Example
• Let’s begin with an example:
– 21 cards, each with a different score
– shuffle cards
– draw 5 cards at random
– write down the five numbers
– pass all 21 cards to the next person
– find the average of the 5 scores that you selected
Things to notice about the distribution of
sample means from our example
1. The average of these sample means is very close to
the population mean
2. There is variation in the sample means
3. The distribution of sample means is normal.
Sampling Error
• Sampling error: the difference between the
population value of interest (e.g., mean) and the
sample value.
• Our sample value is often referred to as an estimate
of our population value
• If the sample is randomly drawn from the population,
then sampling error will be random and will be
distributed normally
Why does sampling error occur?
• Here is the problem: Different samples drawn from the same
population can have different properties
• When you take a sample from a population, you only have a
subset of the population--a piece of what you’re trying to
understand
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The problems of sampling error
•
Sampling error causes two major problems in psychological
research
1. Because of sampling error, our sample values might not be equal to the
population values.
Election example
2. Because of this obfuscation, we can run into a number of difficulties
testing in testing scientific hypotheses.
ESP experiment example
•
One of the objectives of inferential statistics is to find ways
to quantify the amount of sampling error associated with
various statistics
How Can We Quantify Sampling Error?
• We begin by noting that, like any kind of “error” we’ve
discussed up to this point, we can define it as the
difference between two things.
• In this example, we’re working with the difference
between the sample mean and the population mean
• ( - M) = amount of sampling error for any one
sample
Quantifying Sampling Error
• How much sampling error should we expect in any
one situation?
• One way to answer this question is to study the
distribution of sample means that would be observed
under known population values and known
sampling conditions.
• Forward inference: Making an inference about what
sample statistics will be observed based on certain
facts about the sampling process and the population.
Quantifying Sampling Error
• “Known population values”
–  = 10
–  = 6.05
• “Known sampling conditions”
– Random sampling (every case in the population has an
equal opportunity for being selected for the sample)
– Sample size of 5
• Example: During the last demonstration, we studied the sample
means observed when we took random samples of size 5 from
a population in which the mean was 10.
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Population of
scores
 = 10.00 and
 = 6.05
Sample of 5 scores
drawn randomly
from the population
M = 11.6 and
SD = 6.78
Add cards to deck
and sample again
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Take the mean of each
sample and set it aside
11.6
11
9.2
12.4
11.8
6.8
12
10.2
13.2
9.4
The distribution of
these sample means
can be used to
quantify sampling
error
Sampling Distributions
• By simulating the process of drawing random
samples of size N from a population with a specific
mean and variance, we can learn
– (a) how much error we can expect on average and
– (b) how much variation there will be on average in the errors
observed
• Sampling distribution: the distribution of a sample
statistic (e.g., a mean) when sampled under known
sampling conditions from a known population.
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n=2
n=5
n = 15
mean of sample
means = 10
mean of sample
means = 10
mean of sample
means = 10
SD of sample means =
4.16
SD of sample means =
2.41
SD of sample means =
0.87