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Statistical Analysis


                  Measures Of Variability And Inferential Statistics


Variability
     •        Measures of variability help us describe the degree of dispersion or
              variability of the scores.
     •        A= 24,24,25,25,26,26 X = 175/5 = 25
     •        B = 16,19,22,25,28,30,35 X = 175/5 = 25
     •        Means are the same but the scores are very different.


Range
     •        Measure of variability
     •        Difference between top and bottom score
     •        Sensitive to extremes
         –          should you count the bottom score of zero for an absent student ?


Range
             Range = Hs (high score) - Ls (low score) + 1
 100 - 60 = 40 + 1= 41
              High score = 100 and Low score = 60


Variance and Standard Deviation
     •        Variance and standard deviation are very strong indicators of variability.
     •        A deviation score is represented by subtracting the individual score from the
               mean
     •        x = X – mean


The Variance equals the sum of the squared deviations divided by n



To calculate the variance and standard deviation you need the following information.
Remember the following symbols:



Standard Deviation
     •        Common measure of variability
     •        Shows variation in scores
     •        The smaller the deviation the closer the scores to the mean
     •        If all scores the same standard deviation = 0
     •        Takes into account all scores


The Standard Deviation equals the square root of the variance



Example of Standard Deviation Calculation
     •        First take all of the scores and list them in the x column
     •        Next calculate the mean
     •        The second  column (each score minus the mean)
     •        The third column (each score minus the mean squared)
     •        Total the  third column
     •        You now have all of the information needed to calculate the standard deviation.
               The following is an example:




Practice Problems
     •        See  if you can calculate the mean and standard deviation for the following
               problem.


Standard Scores
                Allow us to make comparisons between individuals on the same test
               or different tests.  Standard scores also standardize the measuring units.



                                     SD
      x = raw score,
      X = mean score,
     SD = standard deviation


         Calculate Z scores for Joe on a Psychology test and a Math test.
         Psychology: mean = 85, SD = 10, Joe = 80
         Math: mean = 70, SD = 5, Joe = 90
         Psychology 
 

               Math 
             How well did they perform?  Plot on the normal curve.



Questions
     •        How well did Joe do compared to others on Psychology?
        –          He scored better than what percent of the students?
        –          For math how well did Joe do compared to the other students.
        –          Your answer should be in a percentage.





Sampling Error
     •        When selecting samples from the population there
        is always a certain amount of error.
     •        Sampling error is the difference between the
        population parameter and the sample statistic.
     •        e = X - µ
     •        Sample IQ = 90 and Population IQ = 100
     •        e = 10
     •        The larger the sample the smaller the e

Sampling Error
     •        Sampling error is a direct function of the
        Standard Deviation
     •        Sampling errors are distributed in a normal
        manner with an expected mean of 0
     •        Standard Error of the Mean


Inferential Statistics are Used to Determine Tests of Significance
     •        In a study first make or develop a null
        hypothesis (H0 )
     •        H0 = There is no significant difference
        between SAT-9 test scores of private vs.
        public 9th grade students.
     •        You randomly select 500 public and 500
        private students and administer the SAT-9.


Tests of Significance
     •        The results indicate a mean of 50 for public and a mean of 52 for private.
     •        Is there a difference between the two groups or is any difference related to possible
               error sources?
     •        The error could be chance or sampling error.
     •        Ho : µA = µB


Type 1 and Type II Errors
     •        After we accept or reject the null Ho we could be correct or we could possibly make
        the wrong decision.
     •        Type I error is called an Alpha Error
     •        Type II error is called a Beta Error
        –          Type I unwarranted decisions
        –          Type II Status Quo


Level of Significance
     •        The level of significance is the point at which the null Ho is rejected.
     •        The value is referred to as a p value or a probability value.
     •        P < 0.05 indicates the probability of making a type I error is less than 5 chances in 100.
     •        In education p < 0.05 is often used.
     •        Could be more stringent p < 0.01


Independent T -test
     •        The null Ho states that there is no difference between public vs private school SAT
        results.
     •        You test the Ho using an independent T –Test
     •        You conclude
cf. handout


Results
     •        The T value  of - .536 with 18 DF is not significant at the .05 level of significance
     •        Therefore the null Ho that there is no significant difference between private and
        pubic schools is accepted.
     •        The researcher concludes that any difference is related to chance.