Basic Statistical Concepts

1
Basic Statistical Concepts

• Chapter 2

2
Objectives

• Describe the 4 scales of measurement and be able
  to give an example of each
• Define the term variable and describe the
  different types of variables
• Independent, dependent, discrete, continuous
• Distinguish between random sampling and random
  assignment
• Describe and use summation notation

3
Measurement

• Measurement is the assignment of numbers to
  objects
• Ex Score on a test as a measure of knowledge or
  mastery of the material
• The numbers really have no meaning It is
  interpretation of the numbers that has meaning
• Ex IQ score

4
Scales of Measurement

• Characteristics of relationships among numbers
  assigned to objects
• There are 4 types of scales of measurement
• Nominal
• Ordinal
• Interval
• Ratio
• Each subsequent scales contains the properties of
  the scale preceding it

5
Nominal

• Numbers or labels are used only to distinguish
  among objects
• Ex Color of cars in the parking lot (red, blue,
  green, etc.)
• Ex Sex of students in the class (male or female)
• Primary purpose of the nominal scale is
  classification of the data or observations

6
Ordinal

• Numbers used only to place objects, people, or
  events in order along a continuum
• Ex Finishing position in a race Your list of
  10 favorite movies Movie ratings (G, PG, R)
• Gives no information about the differences just
  information about the relative order
• Ex On a 1 to 10 scale, we cannot say that the
  difference between 1 2 is the same as the
  difference between 3 4

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Interval

• Equal intervals between objects represent equal
  differences
• Ex The difference between 80? F and 40? F the
  difference between 120? F and 80? F
• This scale has no true zero point
• Therefore, we cannot says that 80? F is twice as
  warm as 40? F
• Ratios are not meaningful

8
Ratio

• Scale with a true zero point
• Ex A 20lb weight is twice as heavy as a 10lb
  weight
• This type of scale is rare in many areas of the
  social sciences
• Important to note Variables and not the numbers
  define the scale
• Ex Temperature is interval if solely interested
  in temp. differences (62? 64? 72? 74?), but
  if using temp. as an index of comfort then it is
  no longer interval

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Scales of measurement examples

• Student ID number
• Nominal because it is just a label
• Age
• Ratio because it has a true zero point
• Class Rank
• Ordinal because we can only tell position not
  differences
• Temperature in this room
• Interval because differences are meaningful and
  there is no true zero

10
Variables

• Properties of objects that can take on different
  values
• Ex Hair color, gender, number of beers consumed
  on Friday night
• Discrete Variables are variables that can take on
  only a small set of possible values
• Ex Year in school, gender, college major
• Continuous Variables are variables that can take
  on any value
• Ex Time, speed, etc. are variables that can
  theoretically go from zero to eternity

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Variables (cont.)

• Independent Variable
• The manipulated variable
• May be quantitative or categorical (qualitative)
• Ex Test difficulty (easy or hard) gender
• Dependent Variable
• Measured variable
• Usually quantitative
• Ex Test score height

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Variables Example

• A study was conducted to determine if the type of
  music played while studying influences ones
  score on a test of the material being studied
• 2 groups one listened to classical music the
  other group listened to techno music
• After 1 week of studying with the music, a test
  was given to each group
• What are the IV, DV?
• Are they continuous or discrete?

13
Random Sampling Assignment

• Random Sampling means any element of the
  population has an equal chance of being included
  in the sample
• Random Assignment is the allocation or
  assignment of participants to groups by random
  process
• Random Number Table (Appendix E)
• We do this because we need to make sure that any
  group differences reflect true differences and
  not bias

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Take Home Examples

• What scale of measurement are the following
  variables
• Height, shoe size, social security number, bank
  account balance, NCAA football rankings
• A study was conducted to test if the number of
  alcoholic drinks you consume affects the number
  of drinks you spill.
• What is the IV and DV?

15
Tomorrow

• Finish reading Chapter 2
• Get the book and course packet
• Check out class web site
• http//oak.cats.ohiou.edu/kl305290/course.htm

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Answers to take home problems

• Height
• Ratio
• Shoe size
• Interval
• Social security number
• Nominal
• Bank account balance
• Ratio
• NCAA football rankings
• Ordinal

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Answers to take home problems

• A study was conducted to test if the number of
  drinks you consume affects the number of phone
  numbers you get.
• IV
• The number of drinks
• DV
• The number of phone numbers obtained

18
Summation Notation

• Shorthand symbols used to represent mathematical
  operations
• Variables are represented using UPPERCASE
  LETTERS (X)
• One specific value of a variable is denoted by an
  UPPERCASE LETTERsubscript (X1)
• If you want to talk about any value of the
  variable, use Xi

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Summation Notation (cont.)

• ? (Sigma) is the symbol for summation
• Common summation notation
• ?X Sum of the values of X
• ?X2 Sum of the squared values of X
• (?X)2 Squared sum of the values of X
• ?XY Sum of values of X times the values of Y
• ?X ?Y Sum of the values of X times the sum of
  the values of Y
• Constants (C) any value that remains the same

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Examples

• Using the following numbers
• X 2, 3, 4, 1
• Y 5, 3, 1, 1
• Compute
• ?X
• ?Y
• ?X2
• (?X)2
• ?XY
• ?X ?Y

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Summation Rules

• The sum of X minus Y will equal the sum of X
  minus the sum of Y
• ?(X-Y) ?X - ?Y
• The sum of X times C will equal C times the sum
  of X
• ?CX C?X
• The sum of C plus X will equal the sum of X plus
  the number of scores times C
• ?(X C) ?X NC

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Examples

• Using the following numbers
• X 2, 3, 4, 1
• Y 5, 3, 1, 1
• C 5
• Prove that
• ?(X-Y) ?X - ?Y
• ?CX C?X
• ?(X C) ?X NC
• Solve
• ?Y C

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Take Home Examples

• Using the following numbers
• X 5, 7, 4, 3
• Y 2, 6, 4, 6
• Compute
• ?X-Y
• ?Y
• ?Y2
• (?Y)2
• ?X ?Y

24
Monday

• Get the text and course packet
• Read Chapter 3 (you will not be responsible for
  the section on stem and leaf plots)
• We will be discussing how to display data and
  start talking about basic descriptive statistics
• Check out the class web site