Chapter 2. Sadistics

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Chapter 2. Sadistics

• Juggling
• Basic
• Statistical
• Notions
• Important For
• Teachers
• While
• Maintaining
• Sanity

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Remember . . .

• There are three kinds of people in the world.
• Those who are good at math, and
• Those who arent.

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Raw Scores without organization or context,
scores by themselves are meaningless

• George Carlins famous line Heres a partial
  football score just in, Green Bay 6.
• Ohio State 35 Michigan 32
• Annika Sorenstam 35 Lorena Ochoa 32
• John and Charley are taking a trip together.
  Charley has lived for 7 years John has lived for
  27. Who is the older of the two?
• Charley
• John
• Charley is a dog. In human years he is 46 e.g.,
  (7-1) x 5 16 46. John is a man. In human
  years he is 27.

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Topic List

• Descriptive vs. Inferential Statistics
• Concepts of Data and Variables
• Frequency Tables
• Bar Graphs and Histograms
• Measures of Central Tendency
• Measures of Variability
• Shapes of Distributions
• z-scores
• Correlation

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Two Main Areas of StatisticsDescriptive vs.
Inferential

• Descriptive Statistics is used to organize,
  consolidate or summarize data we have in front of
  us. Typically in descriptive statistics we
  describe
• a set of data elements by graphically displaying
  the information or
• its central tendencies and how it is distributed
  in relation to this center or
• the relationship between two data elements.
• Inferential Statistics is a leap into the
  unknown. We use samples (a selected portion of
  the data set) to draw inferences about
  populations (the complete set of data elements).

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Variables

• A good place to begin is with the concept of
  variables. Our students vary with regard to
  many characteristics related to aptitude and
  achievement. We can think of these variable
  characteristics using three levels of generality.
  

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English Lesson When Speaking of Data

• As a teacher, you will see lots of data.
• The dictionary defines data as facts or
  figures. Notice the word data is plural and
  when speaking of data you need to use plural verb
  forms. It is an unfortunate giveaway that you
  might NOT know what you are talking about if you
  use the phrase this data is instead of these
  data are in professional conversation.
• Datum is the singular form for data. No one
  ever uses this term.

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Making and Reading Frequency Tables Part
1Frequency Distributions - with special focus
on bins (also known as intervals, categories and
class intervals)

• Purpose of Creating these Tables To organize
  data in ways to make our inspection of those data
  much more manageable.
• Frequency Distribution
• We construct or read a table of counts per score.
• BUT, when we have many scores, we create
  intervals (I like the term bins) and place the
  individual scores in the bins. When making bins
• Determine your score range
• Determine an appropriate number of bins. Rule of
  thumb no fewer than 5 or more than 20 class
  intervals work best for a frequency table.
• Make sure no overlap exists so that no data fall
  into more than one bin.
• Count each score in its one and only appropriate
  bin.
• Notice that in the resulting table, individual
  scores are lost.

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Making and Reading Frequency Tables Part
2Cumulative Distributions

• Cumulative frequency distribution A distribution
  that indicates cumulative frequency counts (cum
  f) in each bin, and/or percentage of the total
  number of cases at and below the upper limit of
  the associated bin. Sometimes this is referred
  to simply as cumulative distribution or
  cumulative frequency.
• Note Educators are using the description
  statistics of cumulative distributions when
  speaking of students relative standing.
• Percentile The point on the original measurement
  scale at and below which a specified percentage
  of scores falls. Also called a percentile point.
• Percentile rank The percentile rank of a score
  is the point on the percentile scale that gives
  the percentage of scores falling at and below a
  students specified score.

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Frequency Distribution Table
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Tables are Nice, but Pictures are Nicer

• Frequency distributions are often converted into
  graphic form.
• Bar Graph Individual counts. The count bins
  are separated on the horizontal line.
• Histogram Grouped counts. The bins touch each
  other on the horizontal line.
• Pie Graph Either individual or grouped counts.
  The media likes to display data using these
  graphs.
• Explore the CSERD (Computational Science
  Education Reference Desk) Interactive Website.
  This is a Pathways project of the National
  Science Digital Library and funded by the
  National Science Foundation.

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Ideas of Data Centers How Does Data Cluster?.
. . . starting with a concept from Garrison
Keillor.

• Keillors hometown is Lake Wobegon, located near
  the geographic center of Minnesota.
• Keillor reports that in Lake Wobegon "all the
  women are strong, all the men are good looking,
  and all the children are above average."

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Central Tendency

• While graphs and charts are useful to visually
  represent data, they are inconvenient they are
  difficult to display and can not be easily
  remembered apart from the visual. It is
  frequently useful to reduce data to a number
  (sometimes called an index number) that is easy
  to remember, is easy to communicate, yet captures
  the essence of the complete data set it
  represents.
• One such index is called Measures of Central
  Tendency (i.e., how do the raw data tend to
  cluster)
• Mean the arithmetical average
• Median the middle score
• Mode the most occurring score
• So, these are measures of center regarding the
  data, but we are also concerned about how the raw
  data are spread out around the center.

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Consider the two graphs below. These graphs
represent the scores on two quizzes. The mean
score for each quiz is 7.0. Despite the equality
of means, you can see that the distributions are
quite different. Specifically, the scores on Quiz
1 (top graph) are more densely packed while those
on Quiz 2 (bottom graph) are more spread out. The
differences among students was much greater on
Quiz 2 than on Quiz 1.
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Variability

• Our second index is called Measures of
  Variability (i.e., how do the raw data tend to
  spread out or scatter)
• Range list the lowest and highest scores, then
  take the difference (aka subtract) between them
• Standard Deviation (S, SD, s) this is an
  interesting concept it is akin to finding the
  average distance that scores are from the center
• Variance (SD2) mathematically the standard
  deviation squared we more often use the standard
  deviation in educational assessment.

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Shape of Normal Distributions

• The frequency histograms for test score data
  often approximate what is called the normal
  distribution (aka bell curve, normal curve).
• The normal curve has three characteristics
• unimodal one hump
• asymptotic tails never touch the base
• symmetrical mirror image about the center axis

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Normal Curve
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Shape of Other Distributions

• Kurtosis
• platykurtic looks more flat
• leptokurtic looks more peaked
• Skewness
• positive skew means that the tail is to the right
• negative skew means that the tail is to the left.
• --------------------------------------------------
  ------------------
• Back to the normal distribution, lets look at
  transforming a data score to a score that will
  tell us where that score is in relationship to
  the mean. This score is called a z-score.

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z-scores

• Formula
• z X - M
• SD
• Definition A measure of how many standard
  deviations a raw score is from the mean.
• If the z score is negative, we say the score is
  below the mean
• If the z score is positive, we say the score is
  above the mean

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z-scores in normal curve

• This Graph Leads In To Percentile Rank

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Comparing Two Variables

• So far we have only dealt with one variable (aka
  univariate statistics). Sometimes (I would say
  many times) we are curious as to the relationship
  between two variables (aka bivariate statistics).
  We call this curiosity an interest in
  co-relationships or correlation.

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Some History . . . Francis Galton
(1822-1911)and Co-relations

• Cousin of Charles Darwin
• Interested in the mathematical treatment of
  heredity
• Used statistical analysis to study human
  variation
• noted that arranging measures of a physical trait
  in a population (height, e.g.) displays a
  bell-shaped distribution
• Coined term "eugenics"science of improving the
  stock
• variations (deviations) viewed as flaws as well
  as assets
• artificial and natural selection will shift
  median of distribution

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The Eugenics Movement

• Scientific evidence was used to argue that
  social ills like feeble-mindedness, alcoholism,
  pauperism and criminal behavior are hereditary
  traits.
• Aim - "to give the more suitable races or strains
  of blood a better chance of prevailing speedily
  over the less suitable"
• Can no longer rely on natural selection
• unfit survive to childbearing years due to
• advances in medicine
• comforts of civilization
• social welfare
• unfit reproduce at higher rate than fit,
• Must design society by controlling human
  reproduction encourage fit to have children
• prohibit unfit from having children

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Scattergram Can you eye ball the one line you
could draw through the data points that best
describes the graphic display?

• .

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Correlation Coefficient the calculated number
that best describes the relationship between two
variables

• Correlation coefficient symbol is r
  linear relationships
• Range -1.00 through .00 to 1.00
• Sign indicates direction
• indicates that as one variable increases, the
  other variable increases
• - indicates that as one variable increases,
  the other variable decreases
• Number indicates strength
• Although the following table is somewhat
  arbitrary, the following thinking might be useful
  in interpretation
• -1.0 to -0.7 strong converse association.
• -0.7 to -0.3 weak converse association.
• -0.3 to 0.3 little or no association.
• 0.3 to 0.7 weak direct association.
• 0.7 to 1.0 strong direct association.

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Important Notes about r

• Not a percentage (decimal makes it look like one)
• Linear assumption, not curvilinear
• Equal scatter assumption no bunching
• Variability affects r
• Greater the variability, greater the r
• Less the variability, lower the r
• r does not imply causation

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Depth Chart

• During your YSU field work, you will be asked to
  organize data through the creation of frequency
  tables and histograms. Thus, we discussed
  constructing them as well as understanding them.
• Throughout your professional practice, you will
  be asked to utilize measures of central tendency
  and variability. Thus, we emphasized
  understanding them, basic computations, and their
  relationship to z-scores. These concepts are key
  to understanding standard scores.
• In professional publications you will see
  correlation coefficients. We discussed (and you
  were asked to compute) correlation. Correlation
  is a key tool in exploring our next topic
  reliability (and later, validity) .
• Hopefully you will see value in computing
  measures based on your own classroom data. It is
  actually fun to learn to do these basic
  descriptive stats with a software package.
  Commonly used packages include SPSS, SAS,
  Minitab, and SYSTAT. Any system would be OK.
  Start simple.

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Terms/Concepts to Review andStudy on Your Own (1)

• bivariate distribution
• measures of central tendency (mean, median, mode)
• construct
• correlation coefficient
• descriptive statistics
• frequency distribution
• histogram
• inferential statistics

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Terms Concepts to Review andStudy on Your Own (2)

• kurtosis
• normal curve
• range
• scattergram
• skewness
• standard deviation
• variability
• variable
• z-score