Shavelson Ch' 4 Descriptive Statistics and Scales of Measurement

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Shavelson Ch. 4Descriptive Statistics(and
Scales of Measurement)

• Psyc 3000
• Spring 2006
• By Ryan Redner

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What will we be doing today?

• Practice Frequency distributions!
• Nominal, ordinal, interval, and ratio scales
  (Shavelson Ch. 1)
• Descriptive statistics Measures of central
  tendency and variability (Shavelson Ch. 4)
• Questions you can ask any time
• Tell me if anything is unclear (pretty please!)

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More Frequency Distributions!From Ch.3

• Cumulative Frequency Distribution
• Relative Frequency distribution

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PracticeRelative and cumulative Frequency Chart!

• Data
• 125,121,126,124,125, 124, 127, 122, 125, 127,
  126,123,125,126,124,120,123,122

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Data and frequency
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Relative Frequency
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Cumulative Frequency
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S4.1 Given a data classification system be able
to say whether it is nominal, ordinal, interval,
or ratio Also be able to generate your own
examples. Quantification of Variables (SH Ch. 1)

• Four scales of measurement Nominal, ordinal,
  interval, and ratio
• Numbers can be used to represent levels of an
  attribute (e.g., 1M 2F, heat rate, temperature,
  weight, height, test scores)
• These scales of measurement are used to
  characterize variables

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Scales of Measurement1. Nominal Scale

• Uses numbers to stand for names or categories
• Group as if they were the same in respect to one
  attribute
• Number assignment is arbitrary
• Numbers do not reflect order or size
• Can you think of any examples?
• 1female 2male 88red 65blue 99green

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Scales of Measurement2. Ordinal Scale

• Degrees of an attribute can be identified
• Uses numbers to order persons or objects on a
  continuum
• Does not say how far apart items fall
• Can you think of any examples?
• Size of cars (1 shortest) Honda 1, Ford2,
  Porsche3
• The Student with the highest grade may be
  assigned a number one, and the next highest grade
  a number two etc.

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Scales of Measurement3. Interval Scale

• Differing levels of an attribute can be
  identified AND equal distances between the
  attribute can be identified
• Intervals between units are equal
• Assigns numbers to persons or objects such that
  the numbers of units of measurement is equal to
  number of attribute possessed
• Zero point is arbitrary (zero does not mean none)

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Scales of Measurement3. Interval Scale
(continued)

• Can you think of any examples?
• Temperature (zero does not mean gone)
• Questionnaires (e.g., Likert-scale)
• Ice cream is good for breakfast
• -2 Strongly disagree
• -1 Disagree
• 0 Neither agree or disagree
• 1 Agree
• 2 Strongly agree

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Scales of Measurement4. Ratio Scale

• Equal intervals between numbers
• An actual zero point
• Only differs from interval scale with because of
  the actual zero point
• Can be thought of as an interval scale with zero
  meaning absence

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Scales of Measurement4. Ratio Scale (continued)

• Can you think of any examples?
• Weight
• Heart Rate

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Example Time!!! (1)Nominal, ordinal, interval,
or ratio?

• Blonde
• Brown
• Red
• White
• Gray

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Example Time!!! (2)Nominal, ordinal, interval,
or ratio?

• IQ 0, 102, 114,115

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Example Time!!! (3)Nominal, ordinal, interval,
or ratio?

• Height?

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Example Time!!! (4)Nominal, ordinal, interval,
or ratio?

• Tallest person in class 1, 2nd tallest 2 and
  so on.

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Practice Scales of Measurement

• Come up with any type of measurement.
• Is it Nominal, Ordinal, Interval, or Ratio?

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Now onto descriptive statistics!

• Woooo!

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S4.2 Be able to explain the purpose of
descriptive statistics.

• Summarizes and describes data which occurred in a
  research study. Specifically, these methods
  summarize and describe data for each of the
  variables in a study.
• The other type of statistics are inferential
  statistics. In inferential stats the objective is
  to draw conclusions about the population (as
  opposed to descriptive which just describes).

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S4.3 What information do measures of central
tendency give? What about measures of variability?

• Measures of Central Tendency (mean, median,
  mode) A score that represents the center of a
  distribution of numbers
• Measures of Variability (range,
  semi-interquartile range, standard deviation,
  variance) Describes the spread or range of
  numbers in a distribution

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S4.4 Given a set of numbers be able to calculate
the mean, median, and mode.

• Mean The weight of the scores below this number
  are the same as above (i.e., average). Later
  known as
• Median Divides the distribution in half
• Mode Most frequently occurring number

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Calculate the Mode

• The most frequently occurring number (Since you
  will be calculating all of the Measures of
  central tendency it will be beneficial to put
  these in order)
• Example 1 1,5,5,4,4,5,6,7,8,7,5,5,4,4,6,5
• Answer 1,4,4,4,4,5,5,5,5,5,5,6,6,7,7,8 5
  (Unimodal)
• Example 2 12,12,55,43,54,55,97,65
• Answer 12,12,43,54,55,55,65,97 12 and 55
  (Bi-modal)
• Example 3 100,122,200,166,177,145,199
• 100,122,145,166,177,199,200 No mode

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Calculate the Mean

• Add up all the numbers than divide by the number
  of numbers.
• Example 1 12, 14, 11, 17, 16
• Answer14 1214111716
• 5
• Example 2 1,10, 9, 5,3,4,5,8,6,7
• Answer 5.8 11095345867
• 10

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Calculate the Median

• Put the numbers in order. Find the number that is
  in the middle.
• Example 1 6,6,3,2,8,9,7
• In order 2,3,6, 6 ,7,8,9 Middle6
• Example 2 100,150,124,123,188,166,172
• 100,123,124,150,166,172,188 Middle150
• Example 3 10,15,12,13
• In order 10,12,13,15 Middle(1213)/2 12.5

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Calculate the mean, median, and mode!

• Data 20,40,10,30,100,55,60,80,60,50
• Put them in order.
• 10,20,30,40,50,55,60,60,80,100
• Mode 60 (Most frequently occurring)
• Median (5055)/2 52.5
• Mean 50.5 (average)

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S4.5 Be able to explain where the mean, median,
and mode would fall on a positively skewed,
negatively skewed, and normal distribution

• Where would the mean, median, and mode fall on
• A) normal curve?
• B) positively skewed?
• C) negatively skewed?
• D) Which give is the best descriptor of the
  center of the data on each?

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Mean, median, mode on aNormal Distribution

• Normal Distribution
• Momode
• Mdmedian
• Xbarmean

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Mean, median, mode on aPositively Skewed
Distribution

• Positive skew
• Momode
• Mdmedian
• Xbarmean
• A few really high
• scores pull the distribution

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Mean, median, mode on aNegatively Skewed
Distribution

• Negative Skew
• Momode
• Mdmedian
• Xbarmean
• A few really low
• scores pull the distribution

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PracticeCalculate the mean, median, and mode

• Take out a piece of paper, dont put your name on
  it
• Work individually to calculate the mean, median,
  and mode of this distribution
• 1,8,4,6,7,8,4,9,5,2

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Measures of Variability!Range, Variance, and
Standard Deviation

• The last section of this class! Almost done!

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Measures of Variability

• The range, standard deviation, semi-interquartile
  range (not discussed), and variance

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S4.6 Explain what range is. Be able to calculate
the range.

• Range is the highest score in the distribution
  minus the lowest.
• Quick and dirty estimate of the amount of
  variability in a distribution of scores
• Range Highest score-lowest score
• Example 66,55,77,88,99,11,22
• Range (9911) 88
• Range of this distribution 88

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S4.7 Be able to define standard deviation and be
able to describe what information it provides.

• Definition of Standard Deviation (s) the average
  deviation from the mean.
• Most commonly used measure of variability
• The greater the spread of the scores the higher
  the variation
• It is the spread of the scores about the mean of
  the distribution

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S4.8 Be able to define variance and be able to
describe what information it provides.

• Definition of Variance Average squared deviation
  from the mean
• The only difference between this and standard
  deviation is that variance is squared
• To calculate you can just square (e.g., 52)
  standard deviation (s2)
• Used in more advanced statistical tests such as
  analysis of variance

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S4.9 Given the formula for standard deviation
(the information below) and variance be able to
calculate each.

• The test question will provide this information
  and chart
• x x x-x (x-x)2 S(x-x)2
  variance
• n-1
  (the average squared dev from mean)
• 
  S(x-x)2 sdev
• 
  n-1 (the average deviation from mean)
• S(x-x)2

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Calculate standard deviation and variance

• Data 5,3,9,7,6,6 (N6)
• Book puts it in lowest to highest order.
• S(x-x)2 (910019)20
• N-1 (6-1) 5
• S(x-x)2 (20/5) 4 Variance
• n-1
• 4 2 standard deviation
• s 2 (standard dev)
• s 2 4 (variance)

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A most interesting note The normal curve and
standard deviation

• For all normal curves this relationship holds
  true.
• What percentile are you in if you are two
  standard deviations above the mean?
• What percentage of scores are under the curve
  between -1SD and 1SD?

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Any questions?

• Any questions on
• Measures of variability (range, SD, variance)?
• Measures of central tendency (mean, median,
  mode)?
• Nominal, ordinal, interval, or ratio scales?

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Time for you to grade me!

• Be Honest
• All feedback is appreciated! The written is more
  important than the quantitative measures!

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The End!

• Thanks for coming!