1. What do Samples Tell Us
Variability and Bias

2. Introduction
The process of statistical inference involves using information from a sample to draw conclusions about a wider population.
Population
Sample
Collect data from a representative Sample...
Make an Inference about the Population.

3. Parameters and Statistics
Definition:
A parameter is a number that describes some characteristic of the population.
In statistical practice, the value of a parameter is usually not known because we cannot examine the entire population.
A statistic is a number that describes some characteristic of a sample.
The value of a statistic can be computed directly from the sample data. We often use a statistic to estimate an unknown parameter.

4. ? Sampling Variability Variability and Bias
This basic fact is called sampling variability: the value of a statistic varies in repeated random sampling.
To make sense of sampling variability, we ask, “What would happen if we took many samples?”
Variability and Bias
Population
Sample ?
Sample
Sample
Sample
Sample
Sample
Sample
Sample

5. Describing Sampling Distributions
“How trustworthy is a statistic as an estimator of the parameter?”.
Variability and Bias
Center: Biased and unbiased estimators
Definition:
A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the true value of the parameter being estimated.

6. Variability of a Statistic
Describing Sampling Distributions
Spread: Low variability is better! .
Variability and Bias
n=100
n=1000
Larger samples have a clear advantage over smaller samples. They are much more likely to produce an estimate close to the true value of the parameter.
The variability of a statistic is described by the spread of its sampling distribution. This spread is determined primarily by the size of the random sample. Larger samples give smaller spread.
Variability of a Statistic

7. Describing Sampling Distributions
Bias, variability, and shape
We can think of the true value of the population parameter as the bull’s- eye on a target and of the sample statistic as an arrow fired at the target. Both bias and variability describe what happens when we take many shots at the target.
Variability and Bias
Bias means that our aim is off and we consistently miss the bull’s-eye in the same direction. Our sample values do not center on the population value.
High variability means that repeated shots are widely scattered on the target. Repeated samples do not give very similar results.
The lesson about center and spread is clear: given a choice of statistics to estimate an unknown parameter, choose one with no or low bias and minimum variability.

8. Inference for Sampling
The purpose of a sample is to give us information about a larger population.
The process of drawing conclusions about a population on the basis of sample data is called inference.
Sampling and Surveys
Why should we rely on random sampling?
To eliminate bias in selecting samples from the list of available individuals.
The laws of probability allow trustworthy inference about the population
Results from random samples come with a margin of error that sets bounds on the size of the likely error.
Larger random samples give better information about the population than smaller samples.

9. What do you think Margin of Error tells us?
olitics
What do you think Margin of Error tells us?
General_election_polls
Do you notice a trend in the size of the sample and the margin of error?

10. Sample Surveys: What Can Go Wrong?
Most sample surveys are affected by errors in addition to sampling variability.
Good sampling technique includes the art of reducing all sources of error.
Sampling and Surveys
Definition
Undercoverage occurs when some groups in the population are left out of the process of choosing the sample.
Nonresponse occurs when an individual chosen for the sample can’t be contacted or refuses to participate.
A systematic pattern of incorrect responses in a sample survey leads to response bias.
The wording of questions is the most important influence on the answers given to a sample survey.

11. Bias & Variability

12. What are the types of sampling that are biased?
Review Question:
What are the types of sampling that are biased?
Sampling Bias

13. Bias in Surveys: Nonresponse
People can’t be reached or refuse to respond
30% nonresponse not abnormal(even with call back)
Response bias
Respondents lie or don’t remember clearly
Can be caused by interviewer (race, gender, appearance etc)
Undercoverage
Groups or types of people are missed
Phone surveys miss poor people
1990 census missed 2% of the people
Wording of questions
The question can be worded to slant responses one way or the other
Question can be hard to understand

14. Never trust the results of a sample or survey if you don’t see the exact question asked.
Look at sample design, rate of nonresponse, and date of survey too.