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Statistics for the Behavioral Sciences

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					PSY 307 – Statistics for the
Behavioral Sciences



Chapter 3-5 – Mean, Variance,
Standard Deviation and Z-scores
Measures of Central Tendency
(Representative Values)

   Quantitative data:
       Mode – the most frequently occurring
        observation
       Median – the middle value in the data
       Mean – average
   Qualitative data:
       Mode – can always be used
       Median – can sometimes be used
       Mean – can never be used
Mode
   The value of the most frequently
    occurring observation.
   In a frequency distribution, look for
    the highest frequency.
   In a graph, look for the peaks or
    highest bar in a histogram.
   Distributions with two peaks are
    bimodal (have two modes).
       Even if the peaks are not exactly the
        same height.
Median

   The middle value when observations
    are ordered from least to most, or
    vice versa.
       Half the numbers are higher and half
        are lower.
   When there is an even number of
    observations, the median is the
    average of the two middle values.
Mean
   The most commonly used and most
    useful average.
   Mean = sum of all observations
            number of all observations

          =   X
               n
   Observations can be added in any
    order.
Notation

   Sample vs population
       Population notation = Greek letters
   Individual value = x (lower case)
   Sample mean = x or M
   Population mean = m
   Summation sign = 
   Sample size = n
   Population size = N
Mean as Balance Point

   The sum of the deviations from the
    mean always equals zero.
       The mean is the single point of
        equilibrium (balance) in a data set.
   The mean is affected by all values
    in the data set.
       If you change a single value, the mean
        changes.
       Demo
The Most Descriptive Average
   When a distribution is not skewed
    (lopsided), the mean, median &
    mode are similar.
   When a distribution is skewed, the
    mean is closer to the extreme
    values, mode is farthest.
       Report both the mean and median for a
        skewed distribution.
   The mean is the preferred average.
Ranked Data

   Mean and modal ranks are not
    informative.
       The mean always equals the median
        (middle) rank, so use the median.
       The mode occurs when there is a tie in
        the data, but doesn’t mean much.
   Find the median by finding the
    middle rank (or the average of the
    two middle ranks).
Qualitative Data Averages
   The mode can always be used.
   The median can only be used when
    classes can be ordered.
       The median is the category that
        contains 50% in its cumulative
        frequency.
   Never report a median with
    unordered classes.
   Never report the mean.
Measures of Variability
   Range – difference between highest
    and lowest value.
   Variance – the mean of the squared
    deviations (differences) from the
    mean.
   Standard Deviation – square root of
    the variance.
       The average amount that observations
        deviate from the mean.
Interquartile Range (IQR)
   The range for the middle 50% of
    observations.
       Distance between the 25th and 75th
        percentiles.
   Remove the highest and lowest
    25% of scores then calculate the
    range for the remaining values.
   Used because it is insensitive to
    extreme observations.
    Using IQR (from Holcomb)

       In Rio, what percentage had been
        injecting from 4.5 to 14 years?
       Median Year Injecting = 10
       IQR is 4.5-14 (from text).
                        IQR




0           4.5                      14
     25%          25%          25%        25%   100%

                    Median = 50%
More Notation

   Sample variance = S2
   Population variance = s2
   Sample standard deviation = S or
    SD
   Population standard deviation = s
   Interquartile range = IQR
What Does Variance Describe?
   Variance and standard deviation
    describe the amount that actual
    observations differ from the mean.
       How spread out are the scores?
   The range doesn’t tell us how
    scores are distributed between the
    high and low values.
   Because the mean is the balance
    point, the mean of the unsquared
    deviations is always zero.
  An example using dogs.
   First calculate the height of the dogs.




         Mean = 600 + 470 + 170 + 430 + 300 = 1970 = 394 mm
                             5                  5


Source of example using dogs: http://www.mathsisfun.com/standard-deviation.html
Next, compare their heights to the
mean.




 The green line shows the mean. Subtract the mean from
 each dog’s height. Because some dogs are taller and
 others are shorter, some of the differences will be positive
 and some negative numbers. These differences will
 cancel each other out because the mean is the balance
 point in the distribution of dog heights.
Square the differences and take
the mean.




σ2 = 2062 + 762 + (-224)2 + 362 + (-94)2 = 108,520 = 21,704
                   5                          5
Take the square root to return to
the original units of measure.

   σ = √21,704 = 147
   Which dogs are within one
    standard deviation of the mean?




    Rottweillers are unusally tall dogs. And Dachsunds
    are a bit short.
Standard Deviation
   The variance is expressed in
    squared units (e.g., squared lbs)
    which are hard to interpret.
   Taking the square root of the
    variance expresses the average
    deviation in the original units.
   The square root of the variance
    gives a slightly different result than
    taking the average of the absolute
    deviations.
Interpreting the SD

   For most distributions, the majority
    of observations fall within one
    standard deviation of the mean.
       A very small minority fall outside two
        standard deviations.
   This generalization is true no matter
    what the shape of the distribution.
       It works for skewed distributions.
A Measure of Distance
   The mean shows the position of the
    balance point within a distribution.
   The standard deviation is a unit of
    distance that is useful for
    comparing scores.
   Standard deviations cannot have a
    negative value.
       They can measure in both positive and
        negative directions from the mean.
Definition Formula

   Definition formula – easier to
    understand conceptually.

         s    ( X  X )2    ( X  X )2
                   N

   The numerator is also called the
    Sum of the Squares (squared
    differences), abbreviated SS
Computation Formula

   Computation formula – easier to
    use, especially with large data sets.

     s2     X 2  ( X ) 2
                               s
                                    SS
                   n                N

   The computational and definition
    formulas produce the same result.
Population vs Sample
   The formulas are different
    depending on whether a sample or
    a population is being measured.
   Use n-1 in the denominator when
    using s or s2 to estimate s or s2 for
    a population.
   Using n-1 more accurately
    estimates the variability in a
    population.
Formulas

   Variance for sample:
              SS
         s 
          2

             n 1


   Variance for population:
              SS
          s  2

              N
Z-Score

   Indicates how many SDs an
    observation is above or below the
    mean of the normal distribution.
   Formula for converting any score to
    a z-score:

                      m  mean
        Z= X – m      s  std. deviation
             s
Properties of z-Scores

   A z-score expresses a specific value
    in terms of the standard deviation
    of the distribution it is drawn from.
       The z-score no longer has units of
        measure (lbs, inches).
   Z-scores can be negative or
    positive, indicating whether the
    score is above or below the mean.
Standard Normal Curve

   By definition has a mean of 0 and
    an SD of 1.
   Standard normal table gives
    proportions for z-scores using the
    standard normal curve.
   Proportions on either side of the
    mean equal .50 (50%) and both
    sides add up to 1.00 (100%).
Other Distributions
   Any distribution can be converted to
    z-scores, giving it a mean of 0 and
    a standard deviation of 1.
   The distribution keeps its original
    shape, even though the scores are
    now z-scores.
       A skewed distribution stays skewed.
   The standard normal table cannot
    be used to find its proportions.
Transformed Standard Scores

   Z-scores are useful for converting
    between different types of standard
    scores:
       IQ test scores, T scores, GRE scores
   The z-scores are transformed into
    the standard scores corresponding
    to standard deviations (z).
   New score = mean + (z)(std dev)

				
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