§ Terminology, Clinical Studies, Graphical Representations of Data

Terminology  A statistic is a piece of numerical information taken from a sample.  A parameter is a piece of numerical information about the population being studied.  In other words, a statistic is an estimate for a parameter.

Terminology  Sampling error is the difference between a parameter and the statistic used to estimate it. The causes of this error are: 1. Error due to chance or sampling variability. 2. A poorly chosen sample--sample bias.  If we have a sample of size n from a population of size N then the sampling rate is the ratio n/N.

The Capture-Recapture Method  Step 1: Capture (choose) a sample of size n 1 and tag a certain number of the animals/objects/people.  Step 2: After some amount of time, capture a new sample of size n 2 and take a count of the tagged individuals. Call this number k.  If the second sample is representative then the size of the population is N  (n 1 )(n 2 )/k

Example: The N - value of the Monarch Butterfly  Suppose 150 monarchs are caught, tagged and released.  A few days later 200 more monarchs are caught, of which only 2 are found to be tagged.  Estimate the N - value of the local monarch population.

Clinical Studies  Clinical studies are concerned with determining whether a single variable is causes a certain effect.  The goal is to limit confounding variables--other possible causes.  In a controlled study the subjects are divided into two groups: the treatment group and the control group.  If the subjects are assigned to the two groups randomly then the study is a randomized controlled study.

Clinical Studies  If the control group is given a placebo then the study is a controlled placebo study.  If neither group of subjects knows whether they are receiving treatment or a placebo then the study is said to be blind.  If neither the subjects nor the scientists know who is receiving treatment and who is receiving a placebo then the study is referred to as double-blind.

Graphical Representations of Data A data set is a collection of individual data points. Below is a data set consisting of test scores:

Frequency Table Frequency Score  One way we might summarize the data is in the form of a Frequency Table.  The number below each exam score is the number of students getting that score.

Bar Graphs  Another convenient way to summarize the test scores is in the form of a bar graph:

§ 14.2 Variables

Variables: Quantitative v. Qualitative  A variable is any value or characteristic that varies with members of a population.  In the previous example, test scores would be considered a variable.  A variable is said to be quantitative if it represents a measurable quantity.  A variable that cannot be measured is called qualitative.

Variables: Continuous v. Discrete  If the possible values of a variable are ‘countable’--or if there is some smallest increment we can use- -the variable is said to be discrete.  If the difference between values of a variable can be arbitrarily small, then the variable is called continuous.

Blood Types Example: Blood Types Forty people recently donated blood and their types are listed below: ABOOAOAAAOO AOOAABA AA AAOOAOOBOB OAAAOABAOO

Blood Types Example: Blood Types While this data is qualitative, it is still possible to make both a frequency table and a bar graph to represent it:

Blood Types Example: Blood Types Another way to present the information is in the form of a pie chart.  What differentiates this from the previous tables and graphs is that it shows the percentage, or relative frequency of each blood type in the sample.

 Let’s return for a moment to our test score example...  Suppose the instructor decided to allocate grades as follows: A B C D class intervals   This is an example of using what are called class intervals   When there are too many different values or categories to display our data nicely, we will use these kinds of intervals to simplify the situation.

 The test scores, when sorted into class intervals (in this case the letter grades), can be graphed like this:

Histograms  You may have noticed that in all the cases where we have given a chart or graph that the variable used was discrete.  How can we graphically display continuous variables?  We can use a variation on the bar graph called a histogram.

Example: Age at first marriage. Based on a survey, the frequency table below was obtained for the age of groom at first marriage in the state of Wisconsin Using class intervals of length 10 (years) draw a histogram for the given data , , , # of Grooms Age Interval*

Example: Age at first marriage. Based on a survey, the frequency table below was obtained for the age of groom at first marriage in the state of Wisconsin Using class intervals of length 10 (years) draw a histogram for the given data , , , # of Grooms Age Interval*

Example: Age at first marriage. Now draw a histogram with intervals which are five years in length , , , # of Grooms Age Interval*

Example: Age at first marriage. Now draw a histogram with intervals which are five years in length , , , # of Grooms Age Interval*