Statistical Reasoning

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Statistical Reasoning
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Statistical Reasoning

• Descriptive Statistics are used to organize and
  summarize data in a meaningful way.
• Frequency distributions Where are the majority
  of the scores?
• Used to organize raw scores, or data, so that
  information makes sense at a glance.
• They take scores and arrange them in order of
  magnitude and the number of times each score
  occurs.

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Multiple Choice
Composite
Essay
Mean9.3 SD2.3
Mean10.2 SD2.0
Mean34.3 SD4.2
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Histograms Frequency Polygons

• These are ways of showing your frequency
  distribution data.
• Histogram graphically represents a frequency
  distribution by making a bar chart using vertical
  bars that touch
• When you have a continuous scale (for example,
  scores on a test go from 0-100, continuously
  getting larger.) the bars touch, because you have
  to have a class for each score to fall into, and
  you cant have any gaps.
• Different than a Bar Graph which is used when you
  have non-continuous classes (example, which
  candidate do you support, Obama or McCain? Youd
  have a bar for each, with gaps in between,
  because you cant fall between two candidates,
  you have to pick one.)
• Frequency Polygon graphically represents a
  frequency distribution by marking each score
  category along a graphs horizontal axis, and
  connecting them with straight lines (line graph)

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Histogram Uses a Bar Graph to show data
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Frequency Polygon Uses a line graph to show data
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Skewed Curves

• Skewed Distribution when more scores pile up on
  one side of the distribution than the other.
  Positively skewed means more people have low
  scores. Negatively skewed means more people have
  high scores.
• Positive Negative refers to the direction of
  the tail of the curve, they do not mean good
  or bad.

A Positive Skew has a tail that goes to the
right.A Negative Skew has a tail that goes to
the left.
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Measures of Central Tendency

• A single number that gives us information about
  the center of a frequency distribution.
  Measures of central tendency 3 types

4, 4, 3, 4, 5

• Modemost common4
• (Reports what there is more of Used in data
  with no
• connection. Cant average men women.)
• 2. Meanarithmetic average20/54
• (has most statistical value but is susceptible to
  the effects of extreme scores )
• 3. Medianmiddle score4
• (1/2 the scores are higher, half are lower. Used
  when there are extreme scores)

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Central Tendency An extremely high or low
price/score can skew the mean. Sometimes the
median is better at showing you the central
tendency.1968 TOPPS Baseball Cards

• Nolan Ryan 1500
• Billy Williams 8
• Luis Aparicio 5
• Harmon Killebrew 5
• Orlando Cepeda 3.50
• Maury Wills 3.50
• Jim Bunning 3
• Tony Conigliaro 3
• Tony Oliva 3
• Lou Pinella 3
• Mickey Lolich 2.50

• Elston Howard 2.25
• Jim Bouton 2
• Rocky Colavito 2
• Boog Powell 2
• Luis Tiant 2
• Tim McCarver 1.75
• Tug McGraw 1.75
• Joe Torre 1.5
• Rusty Staub 1.25
• Curt Flood 1

With Ryan Median2.50 Mean74.14
Without Ryan Median2.38 Mean2.85
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Does the mean accurately portray the central
tendency of incomes? NO!
What measure of central tendency would more
accurately show income distribution? Median the
majority of the incomes surround that number.
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Measures of Variability

• Gives us a single number that presents us with
  information about how spread out scores are in a
  frequency distribution. (See example of why this
  is important).
• Range Difference b/w a high low score
• Take the highest score and subtract the lowest
  score from it. (can be skewed by an extreme
  score)
• Standard Deviation How spread out is your data?
• The larger this number is, the more spread out
  scores are from the mean.
• The smaller this number is, the more consistent
  the scores are to the mean

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Calculating Standard DeviationHow spread out
(consistent) is your data?

• Calculate the mean.
• 2. Take each score and subtract the mean from
  it.
• Square the new scores to make them positive.
• Mean (average) the new scores
• Take the square root of the mean to get back to
  your original measurement.
• 6. The smaller the number the more closely
  packed the data. The larger the number the more
  spread out it is.

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Standard Deviation
Deviation Squared Numbers multiplied by itself
added together
Punt Distance
Deviation from Mean
Standard Deviation
36 38 41 45
36 - 40 -4 38 40 -2 41 40 1 45 40
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16 4 1 25
Mean 160/4 40 yds
Variance 46/4 11.5
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Standard Normal Distribution Curve

• Characteristics of the normal curve
• Bell shaped curve where the mean, median and mode
  are all the same and fall exactly in the middle

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Multiple Choice
Composite
Essay
Mean9.3 SD2.3
Mean10.2 SD2.0
Are these scores consistent? Is there a skew?
Mean34.3 SD4.2
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Skewed Curves
Skewed Distribution when more scores pile up on
one side of the distribution than the other.
Positively skewed means more people have low
scores. Negatively skewed means more people have
high scores.
A Positive Skew has a tail that goes to the
right.A Negative Skew has a tail that goes to
the left.
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Z-ScoresA number expressed in Standard Deviation
Units that shows an Individual scores deviation
from the mean.Basically, it shows how you did
compared to everyone else. Z-score means you
are above the mean, Z-score means you are
below the mean.
Z-Score your score minus the average score
divided by standard deviation.
Which class did you perform better in compared to
your classmates?
Z score in Biology 168-160 8, 8 / 4 2 Z
Score
Z score in Psych 44-38 6, 6/2 3 Z Score
You performed better in Psych compared to your
classmates.
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Correlation

• Correlation shows the relationship between two
  variables.
• The closer to or - one the stronger the
  relationship between the two variables.
• This enables us to predict. However, correlation
  does not mean causation.

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Positive Correlation

• As the value of one variable increases (or
  decreases) so does the value of the other
  variable.
• When A goes UP B goes UP or
• When A goes Down, B goes Down
• A perfect positive correlation is 1.0.
• The closer the correlation is to 1.0, the
  stronger the relationship.

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Negative Correlation

• As the value of one variable increases, the value
  of the other variable decreases.
• When A goes UP B goes Down or
• When A goes Down, B goes Up
• A perfect negative correlation is -1.0.
• The closer the correlation is to -1.0, the
  stronger the relationship.

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Zero Correlation

• There is no relationship whatsoever between the
  two variables.

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Lets Review
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Inferential Statistics

• Techniques that allow a researcher to determine
  whether a studys outcome is more than just
  chance events.
• Usually you would use inferential statistics to
  try to predict things about a population based on
  a sample.
• For example, we surveyed 50 staff members in the
  district about their level of education and are
  trying to use that to predict the average level
  of education for all staff in the district.

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Statistical Significance

• p value likelihood a result is caused by
  chance. In other words, are they statistically
  significant? If the answer is yes, then they can
  be generalized to a larger population
• This is bad to a researcher. They want this
  number to be as small as possible to show that
  any change in their experiment was caused by an
  independent variable and not some outside force.
• Results are considered statistically significant
  if the probability of obtaining it by chance
  alone is less than .05 or a P-Score of 5.
  p .05
• This means the researcher must be 95 certain
  their results are not caused by chance.
• Replication of the experiment will prove the p
  value to be true or not.

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Does the sample represent the population?

• Non-biased sample-good
• Low variability-good
• Larger samples-good
• Population is a complete set of something.
• Sample is a subset of a population.