What is statistics?

• Statistics is the science of dealing with data.

• Data is any type of info packaged in numerical form.

• Common examples: Political polls, Health/medical studies

Some Basic Definitions

• Population: collection of individuals or objects we want to study statistically

• “What is the population to which the statistical statement applies?”

• N-value: how many individuals/objects there are in the population

Example

• Study: What percentage of the M&Ms in the jar are blue?

• Population: all of the M&Ms in the jar

• N-value: 4392

Census• Census: the process of collecting data by going

through every member of the population

• Our example: Count all M&Ms in the jar, count all of the blue ones, find percentage.

• Drawbacks:– Expensive– Too much work– Almost impossible for large populations

Census vs Survey• Census: the process of collecting data by going

through every member of the population

• Survey: process of collecting data only from some members of the population (and use that data to draw conclusions & make inferences about the entire population)

• Poll: data collection done by asking questions

Use samples!

• Sample: a subgroup of the population chosen to provide the data

• Sampling: the act of selecting a sample– Finding a good sample is EXTREMELY DIFFICULT!!!!

• Sampling frame: the actual subset of the population from which the sample will be drawn

Example

• Study: What percentage of our class likes cheeseburgers?

• Population: all members of our class• N-value: 20• Sampling frame: all of the women in our class• A Sample: all of the women in our class who

are present today

Sampling frames make a difference!• CNN/USA Today/ Gallup Poll, Nov 2004: If the election for Congress were being held

today, which party’s candidate would you vote for in your district?

• Asked of 1866 registered voters nationwide: 49% for Dem, 47% for Rep, 4% undecided

• Asked of 1573 likely voters nationwide: 50% for Rep, 46% for Dem, 3% undecided

Representative Samples

• When a population is highly homogeneous, a very small sample may be representative– Ex: blood samples, thoroughly mixed cake batter,

etc

• More heterogeneous populations -> more difficult to find representative samples

Are these samples representative?• Question: What is the average time it takes a UNL

student to walk to class?

• Samples:– All students living in dorms– All students who use city buses– All students in the Union at noon– All students currently taking math classes

1936 Literary Digest Poll• US presidential election: Alfred Landon (R) vs incumbent Franklin D Roosevelt (D)

• Sampling frame included:– Every person listed in a telephone directory

anywhere in the US– Every person on a magazine subscription list– Every person listed on the roster of a club or

professional association– List of 10 million people created to whom mock

ballots were mailed

1936 Literary Digest Poll• Poll predicted Landon with 57% of vote vs

Roosevelt’s 43%

• Reality: 62% for Roosevelt and 38% for Landon

• What went wrong?!– Think about the sample. – Representative?– Biased?

Bias• Selection bias: when the choice of the sample

has a built-in tendency to exclude a particular group or characteristic within the population

• Literary Digest poll only had 24% response rate

• Low response rate -> nonresponse bias (selection bias)

Lots of different kinds of bias• Leading-question bias:– Are you in favor of paying higher taxes to bail the

federal government out of its disastrous economic policies and its mismanagement of the federal budget?

• Question order bias

• Afraid to answer bias:– Have you ever cheated on your income taxes?

Morals

• Bigger samples aren’t necessarily better samples!

• Watch out for different types of bias!

• A representative sample is key!

Lots of Sampling Methods• Convenience sampling: selection of individuals

included in the sample is dictated by what is easiest or cheapest– Notoriously bad!– Ex: Want to know the average score on the last

quiz? Sample: Look at the scores of the people sitting next to you.

– Ex: Want to know how people feel about making the switch to the Big Ten? Sample: Set up a table outside of your house for people to come by and fill out questionnaire

Quota sampling

• Quota sampling: the sample should have so many women, so many men, so many Christians, so many Muslims, so many urban-dwellers, so many rural farmers, etc

• The proportions in each category in the sample should be the same as those in the population

Example of quota sampling• Intro to Stats has 120 students– 40 freshman– 30 sophomores– 30 juniors– 20 seniors

• To fill out questionnaire, prof selects– 24 freshman– 18 sophomores– 18 juniors– 12 seniors

1948 US Presidential Election

• Gallup poll used detailed quota sampling

• Sample size: 3250 people

• Prediction vs reality: – Thomas Dewey: 49.5% / 44.5%– Harry Truman: 44.5% / 49.9%

• What went wrong?

Simple Random Sampling• SRS: all members of the population have an

equal chance at being included in the sample

• How were previous examples not SRS?

• Examples of methods:– Pull names from a hat– Flip a coin– Random number generator

Stratified Sampling• Break the sampling frame into categories

(strata), then randomly choose a sample from these strata

• Those chosen strata are subdivided into substrata, and a random sample taken.

• Subdivide again and take a random sample, etc

• End up with clusters, but usually reliable

Stratified Sampling Example

Now survey these houses!

More Definitions

• Statistic: Numerical information drawn from a sample

• Parameter: unknown measure (numerical info) from the population

• Hopefully, the statistic will be close to the parameter so conclusions made about the sample will be true for the whole population.

Error and Bias

• Sampling error: the difference between the parameter (estimated) and the statistic

• Sampling error attributed to:– Chance error– Sampling variability: different samples give

different results– Sampling bias: bad sample chosen

Sample Size

• Population size = N

• Sample size = n

• Sampling proportion = n/N

• Modern public opinion polls: 1000 ≤ n ≤ 1500

Capture-Recapture

• Used to estimate the N-value

• Steps:– Choose a sample of size , tag the members,

and release.– After some time, capture a new sample of size and take an exact head count of tagged

individuals. Call that number k. – The N-value is approximately

Small fish in a big pond

• A pond of fish!

• Capture = 200 fish. Tag them.

• Capture = 150 fish. Notice that k = 21 of these fish have tags.

• There are approximately N ≈ (200*150)/21 ≈ 1428 fish

Clinical Studies

• Try to study cause and effect, whereas surveys just observe and report

CORRELATION DOES NOT IMPLY CAUSATION!!!!!!!!!!

Alar Scare• Alar: chemical used by apple growers

• 1973: mice exposed to active chemicals in Alar at 8 times greater than the max tolerated dosage– A child would have to eat 200,000 apples per day

to get that dosage

• Alar doesn’t really cause cancer, but no longer used. Washington State apple industry lost $375 million.

Clinical studies

• Concerned with determining whether a single variable or treatment (vaccine, drug, therapy, etc) can cause a certain effect (disease, symptom, cure, etc)

• Confounding variables: all other possible contributing causes that could produce the same effect

• First step: isolate the treatment under investigation from confounding variables

Controlled Study

• Subjects are divided into two different groups:– Treatment group: consists of subjects receiving

the actual treatment– Control group: consists of subjects that are not

receiving any treatment (for comparison only)

• Randomized controlled study: subjects are assigned to the treatment group or control group randomly....hopefully groups are representative samples

Placebos

• Placebo: fake treatment intended to look like the real treatment

• Controlled placebo study: controlled study in which control group is given a placebo

• Placebo effect: just the idea of getting treatment can produce positive results

Don’t tell them about the placebo!

• Blind study: neither the members of the treatment group nor the members of the control group know to which of the two groups they belong

• Double-blind study: the scientists conducting the study don’t know either

Homework

• Read Chapter 13• Answer the questions on the Vocabulary

worksheet• Exercises beginning on page 515: 1-4, 13, 17-25, 30-32, 45-48, 57-60, 70