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Two Main Applications

• 1. Descriptive - summarizes large groups of data using numerical and/or graphical techniques
• 2. Inferential - use small sets of data to describe or find out information about larger sets of data

The Basics

• 1. Population - entire set of data we are interested in studying
• 2. Sample - a subset of the population we have collected information from
• 3. Experimental Unit - object we measure to get a single piece of data
• 4. Variables - a characteristic of an experimental unit that can vary from one unit to the next

Two Types of Variables

• 1. Quantitative - NUMERICAL
• Histogram - group similar values together and then draw bars
• Stem-and-Leaf Display - data values are broken into stems and leaves
• 2. Qualitative - CATEGORICAL
• Bar Graph - bar length will depend on the (relative) frequency of the category
• Pie Chart - "slices" of the pie depend on the (relative) frequencies

Reliability

• Inferential statistics provides a measure of reliability for the inferences we make

Example 1)

• I want to estimate the average parking time of all USF students. I ask 300 USF students to estimate their parking times and find that the average was 12 minutes.
• Population: All USF student parkers
• Sample: 300 Student parkers we collected information from
• Experimental Units: A single USF parker
• Variable: Parking time (Quantitative)
• Descriptive: The average of the 300 parking times is 12 minutes.
• Inferential: We are 95% confident that the average parking time of ALL USF students falls between 10.5 and 13.5 mins.

Sources of Data

• 1. Published Source - Data collected by others
• Ex) STOCK PRICES
• 2. Designed Experiment - Researcher controls variables in a study to collect data
• Ex) SCIENCE FAIR
• 3. Observational Study - Observe experimental units in their natural environment
• Ex) SEATBELT STUDY

Methods of Random Sampling

• 1. Physical Mixing - represent every experimental unit in some common matter (pieces of paper will work). Mix together and select "N".
• 2. Random Number Table - calculators/computers online

Numerical Descriptions of Data

• A. Measures of Center
• B. Measure of Variation
• C. Measure of Relative Standing

Measures of Center

• 1. MEAN - Average of data
• 2. MEDIAN - Middle of ordered data
• 3. MODE - Value(s) that occur(s) the most
• Which measure is "best" ?
• Symmetric - Mean is best in symmetric distributions
• Skewed - Median is best in skewed distributions

Measures of Variation

• 1. Range - Largest - Smallest Observation
• 2. Variance - Sum of the squared deviations of the measurements from their mean, divided by 1.
• 3. Standard Deviation - Square root of the variance (which is interpretable)

Chebyshev's Theorem

• - Works for ANY data set
• 1. One standard deviation - At least 0%
• Two standard deviations - At least 75%
• Three standard deviations- At least 89% (Most observations are here)

Empirical Rule

• - Only works for mound shaped symmetric data
• 1. One standard deviation - Approx 68%
• Two standard deviations - Approx 95% (Most of data falls here)
• Three standard deviations - Approx 100%

Modified Empirical Rule

• - Works for SKEWED DATA
• 1. One standard deviation - Approx 60-90%
• Two standard deviations - Approx 90-100% (MOST DATA FALLS HERE)
• Three standard deviations - Approx 100%

Measures of Relative Standing

• 1. Percentile - gives the percent of the data in the distribution that fall below a particular value
• 25th percentile- LOWER QUARTILE (QL)
• 75th percentile- UPPER QUARTILE (QU)
• 50th percentile- MEDIAN
• 2. Z-score - tells us the standard deviations an observation falls from the mean, and in which direction (+ OR -)
• Ex) -3 -2 -1 0 1 2 3

Outliers

• - Measurements that fall outside where we expect them to in a distribution
• Symmetric Data or Skewed Data - Outside 2 standard deviations
• Chebbyshev's Data - Outside 3 standard deviations
• 1. Observation is miscoded
• 2. The observation comes from a different population than one specified
• 3. Observation is the result of "rare" chance

Methods of Detecting Outliers

• 1. Z-SCORE - Use the Z-Score for an observation and the shape of the distribution to determine if its an outlier
• 2. BOX PLOT METHOD - Graphical display that uses the quartiles of the data as the basis for identifying outliers
• Example of Quartiles in Box Plot

Probability Distribution

• - A table, graph, or formula that gives you the probability of observing each item of the random variable
• Rules to follow:
• 1.
• 2.

Expected Value of Discrete Random Variable

• - The average SAT score of USF students is 1200
• -The expected number of children in U.S. households is 1.8
• Expected value:
• These expected values should be interpreted as long-run averages, not as outcomes we expect to observe on a single attempt

Binomial Probability Distribution

• Criteria:
• 1. The experiment consists of "n" identical trials
• 2. There are 2 outcomes, success & failure, possible on each trial
• 3. p(success)= P; p(failure)= 1-P=Q
• 4. Trials are Independent

The Binomial Random Variable

• X = the number of success in the N trials
• Mean:
• Variance
• Standard Deviation:

Cumulative Binomial Table

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