A Review of the Principles of Statistics

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What is statistics?

Statistics can be thought of as a whole subject or discipline ...

It can be thought of as the methods used to collect, process and/or interpret data ...

It can be thought of as the collections of data gathered by those methods ...

It can also be thought of as a specially calculated figures (e.g. averages) to characterize collection ...

Consider how the word statistics is used in the following paragraph to imply these different meanings of the word:

"A student in a class offered by the PSU Statistics department uses statistics (statistical methods) to interpret statistics (data) about the cost of a 1 bedroom apartment in State College, and he/she may summarize finding by quoting a statistics of  'average price per 10 apartments' in various locations of State College."

Statistics are like a bikini; What is revealed is interesting; What is concealed is crucial.
-
R. Taylor

Statistics is the science and art of making decisions based on quantitative evidence.

  • Statistics uses many mathematical tools, but is not purely mathematical, (see Lesson 1 for a discourse on general approach to data analysis).
  • For a statistician, the numbers are meaningless unless put in a context, and transformed into information; which will ultimately lead to valuable knowledge.

Almost all fields of study collect and interpret the data. In statistics variability is a key concept. Statistics (and statisticians) recognize that not all things/people/units/etc are exactly alike.

Most statisticians (and you) are involved in:

  • Helping phrase the question(s) to be answered such that we could have a reasonable data collection and that is amenable to statistical analysis,
  • Design the experiment, survey, or other way to approach the problem,
  • Gather the data,
  • Summarize and analyze the data, and then
  • Draw the conclusions and communicate the results.

Principles of Statistics

The objective of descriptive statistics methods is to summarize a set of observations.

The objective of inferential statistics methods is to make inferences (predictions, decisions) about population based on information contained in a sample, and to quantify the level of uncertainty in our decisions.

Example

Question: What is the cost of a 1 bedroom apartment in State College?

Approach the problem: At the end of the spring semester you randomly picked 15 people who told you how much rent they pay for a 1 bedroom apartment. You also record the location of the apartment. Is this a survey, experiment, or ….?

Here is the data that was gathered:

\$280
\$320
\$320
\$330
\$340
\$370
\$370
\$375
\$380
\$380
\$380
\$390
\$420
\$420
\$430

Some descriptive statistics:

Mean: \$5505 / 15 = \$367, Standard deviation = 42.07986
Median: \$375, seven values above and below
Mode: \$380, it appears three times

Key Concepts in Statistics

Observations
Population
Sample

Parameter
Statistic
Random variables


Validity, Reliability, Bias & Variability

All important, yet often confused definitions:

A valid measurement actually measures what it claims to measure.

A reliable measurement provides every experiementer the same result in successive trials.

A biased measurement will be wrong in the same direction nearly every time.

Variability is the difference in successive measurements of the same thing.

Natural variability means that we are all different