Descriptive Statistics

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Descriptive Statistics

• Measures of Central Tendency

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Why? What? And How

• Remember, data reduction is key
• Are the scores generally high or generally low?
• Where the center of the distribution tends to be
  located
• Three measures of central tendency
• Mode
• Median
• Mean
• Which one you report is related to the scale of
  measurement and the shape of the distribution

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Mode

• The most frequently occurring score
• Look at the simple frequency of each score
• Unimodal or bimodal
• Report mode when using nominal scale, the most
  frequently occurring category
• If you have a rectangular distribution do not
  report the mode

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Median

• Score at the 50th percentile (Mdn)
• If normal distribution the mdn is the same as the
  mode
• Arrange scores from lowest to highest, if odd
  number of scores the mdn is the one in the
  middle, if even number of scores then average the
  two scores in the middle
• Used when have ordinal scale and when the
  distribution is skewed

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Mean

• Score at the exact mathematical center of
  distribution (average)
• M ?X/N
• Used with interval and ratio scales, and when
  have a symmetrical and unimodal distribution
• Not accurate when distribution is skewed because
  it is pulled towards the tail

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Deviations around the Mean

• The score minus the mean
• Include plus or minus sign
• Sum of deviations of the mean always equals zero
• ?(X-M)

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Uses of the Mean

• Describes scores
• Deviation of mean gives us the error of our
  estimate of the score, with total error equal to
  zero
• Predict scores
• Describe a scores location
• Describe the population mean (?) which is a
  parameter
• Typically estimate ?

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Summarizing Results

• Used in all research methods including
  observational, survey, correlational, and
  experimental
• Compute the mean of the dependent variable for
  each of the conditions or levels of the
  independent variable
• Mean dependent score changes as function of
  changes in the IV
• Graphing the results using line or bar graphs

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Measures of Variability

• Extent to which the scores differ from each other
  or how spread out the scores are
• Tells us how accurately the measure of central
  tendency describes the distribution
• Shape of the distribution

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Why do we care about variability?
Where would you rather vacation, Gulfside
Bungalows, where the mean temperature is 70
degrees, or Kalahari Condos where the mean
temperature is 70 degrees?
Gulfside temperature range day 72 night
68   Kalahari temperature range day
110 night 30 Also variability in terms of
the range of temperature at each of these places
over the years that temperature has been
documented
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Range

• Can report the lowest and highest value
• Or report the maximum difference between the
  lowest and highest
• Semi-interquartile range used with the median
  one half the distance between the scores at the
  25th and 75th percentile

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Variance and Standard Deviation

• Definitional and computational formulas (remember
  order of operations)
• Again, most psychological research uses interval
  and ratio scales of measurement and assume a
  normal distribution
• Goal is to assess the average or typical amount
  the scores differ from the mean
• Biased estimates of the population variance

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Sample Variance

• Uses the deviation from the mean
• Remember, the sum of the deviations always equals
  zero, so you have to square each of the
  deviations
• S2X sum of squared deviations divided by the
  number of scores (p. 107 and 108)
• Provides information about the relative
  variability

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Some Limits

• It isnt the average deviation
• Interpretation doesnt make sense because
• Number is too large
• And it is a squared value

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So, Standard Deviation

• Take the square root of the variance
• P.109 and 110
• SX
• Uses the same units of measurement as the raw
  scores
• How much scores deviate below and above the mean

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The standard deviation
What is a standard deviation (in English)?
the mean of deviations from the mean (sort of)
What is
s
(lowercase sigma) is the population standard
deviation.
S
the sample standard deviation
(s-hat) is the sample estimate of s
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The deviation (definitional) formula for the
population standard deviation

• The larger the standard deviation the more
  variability there is in the scores
• The standard deviation is somewhat less
  sensitive to extreme outliers than the range (as
  N increases)

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The deviation (definitional) formula for the
sample standard deviation
Whats the difference between this formula and
the population standard deviation?
In the first case, all the Xs represent the
entire population. In the second case, the Xs
represent a sample.
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Standard Deviation Example
26.8
106.8

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Calculating S using the raw-score formula
To calculate SX2 you square all the scores first
and then sum them   To calculate (SX)2 you sum
all the scores first and then square them
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The raw-score formula example
?X 134
?X2 3698
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Estimating the population standard deviation
from a sample
S, the sample standard, is usually a little
smaller than the population standard deviation.
Why?
The sample mean minimizes the sum of squared
deviations (SS). Therefore, if the sample mean
differs at all from the population mean, then the
SS from the sample will be an understimate of the
SS from the population
Therefore, statisticians alter the formula of the
sample standard deviation by subtracting 1 from N
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Population Variance and Standard Deviation

• When we have data from the entire population we
  use ? to compute ?X using the same formulas (p.
  115)
• We usually need to estimate
• Variance and standard deviations of the sample
  are biased estimates of the population
• Limited in terms of how free the scores can vary
• Not all of the deviations in the sample are free
  to be random

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Estimates of Population Variability

• P. 117, 118, and 119
• Symbol s2X and sX or s-hat -- estimations
• Correction factor N-1
• Not all of the deviations in the sample are free
  to be random
• Degrees of freedom df
• With M 6 and scores of 1,5,7,and 9, then the
  only possibility is for the score to be 8
• More accurate estimate of population variability

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Formulas for s-hat (estimate)
Definitional formula
Raw-score formula
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The Estimate of the Variance
Remember what the variance is..
The standard deviation squares, or the number
that you took the square root of to get the
standard deviation
The variance is not a very useful descriptive
statistic, but it is very important value you
will use in other techniques (e.g., the analysis
of variance or ANOVA)
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Sum up

• Assuming a normal distribution
• Sample mean is a good estimate of population mean
• The estimate of the population variance and
  standard deviation tells us how spread out the
  scores are
• 68 of the scores are within 1 and 1 sX

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Application to Normal Distribution

• Knowing the standard deviation you can describe
  your sample more accurately
• Look at the inflection points of the distribution

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Transformations

• Adding or subtracting just shifts the
  distribution, without changing the variation
  (variance)
• Multiplying or dividing changes the variability,
  but it is a multiple of the transformation

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Variance is Error in Predictions

• The larger the variability, the larger the
  differences between the mean and the scores, so
  the larger the error when we use the mean to
  predict the scores
• Error or error variance average error between
  the predicted mean score and the actual raw
  scores
• Same for the population estimate of population
  variance

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Summarizing Research Using Variability

• Remember, the standard deviation is most often
  the measure of variability reported
• The more consistent the scores are (i.e., the
  smaller the variance), the stronger the
  relationship

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Proportion of Variance Accounted For

• Objective approach compute proportion of
  variance accounted for
• Can compute the overall mean and standard
  deviation, not taking into consideration the
  relationship with the levels of the IV
• It is the largest error we would accept
• When look at relationship we compute variance for
  each condition and average

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Computation

• Subtract the average error from the each of the
  conditions from the error of the total sample
• Divide that difference into the error from the
  total sample
• Gives proportion of error accounted for by the
  levels of the IV

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Thus, .

• Proportional improvement in predictions by using
  a relationship
• The stronger and more consistent the
  relationship, the greater proportion of variance
  we can account for