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Q - How else do the scores from each class differ?

Answer

1) Range
2) ______________
 
 
 
 
 

Measures of Dispersion

     = index of the degree to which scores ________________________________

    = measures degree each score differs ________________________________

    Error = ________________ of scores from mean/median
 
    = indicator of _________________measure of central tendency is (e.g., mean)

        Low error = ______________________________

        High error = ______________________________

 

Variance
    = degree to which scores deviate from ____________

    Error in terms of ___________________________

Why?
- or else the sum of the errors = 0
- to make bigger errors count more   Calculating Pop. Variance

Step 1: Find deviations (errors) from mean and square them
 
 
 
 
 
 
 
 
 

Step 2: Find sum of the squared errors (called Sum of Squares)
 
 
 
 
 
 
 
 
 
 

Step 3: Find the mean of the squared errors (Mean Squared Error)

= sum of sq. deviations/# of scores

 = Sum of Squares/N
 
= SS/N

 
 
 

  Variance = MSE =
 

s 2 = sum of sq. deviations from mean
                            N
 

s 2 =
 
 
 
 
 
 
 

 
Standard Deviation
    Widely used measure of dispersion

    "Standard" = typical

    "Deviation" = error
 

    = typical error of scores
 
    = the ____________________ of the variance

  Example: If variance = 4 what is S.D.? S.D. = variance

s =

s =
 
 

 Why Use Standard Deviation
 
 
 
 
 

        Undo the squaring to put the units back to normal

  Height example:

 

Standard Deviation Formula

    = the square root of the variance
 

    s 2 = S (Xi-m )2
                    N

    s =
 

Calculating Pop. S.D.

    Procedure Overview
        - Calculate variance
        - Find square root of variance

 
  Step 1: Find deviations (errors) from mean and square them  
 
 
 
 
  Step 2: Find sum of the squared errors (called Sum of Squares)  

 
 
 
 
 

 Step 3: Find the mean of the squared errors (Mean Squared Error) = sum of sq. deviations/# of scores

= Sum of Squares/N

= SS/N

=

 =

  Step 4: Find the square root of the Mean Squared Error

                                        =

 

Computational Formula for Pop. S.D.
 
 
 
 
 
 

 
Using the Com. Form. for Pop. S.D.
    Step 1: Find (S Xi) 2
 

 
 
 
 
 

    Step 2: Find S Xi 2

 
 
 
 
      Step 3: Calculate SS = S Xi2 - (S Xi) 2
                                                            N
 
 
 
 
 
 

    Step 4: Calculate
 
 
 

 

  Sample Variability

So far we've considered populations only

s 2 = population parameter for variance

s = population parameter for Std. Dev.

m = population parameter for the mean
 

What about when we only have a sample of all of the data?

  s2 = sample parameter for variance

s = sample parameter for Std. Dev.

(Xbar) = sample parameter for mean
 
 

 Sample Variance  
 
 
 
 
 
  Sample Standard Deviation  
 
 
 
 
 
 

Sample Variability
 
 
 
 
 
 
 
 

Why N-1

    s = estimated s

    s2 = estimated s 2

    N-1 = more accurate...accounts for sampling error
 

 
 
 
 

    Also notice as sample size é s more representative of s

 

Degrees of Freedom
     Why N - 1...why not N - 2 or N - 3

        Idea: There is only 1 degree of freedom

        Variance and S.D. are based on the deviations of scores from the _____________

 
        We have to calculate the mean to find either variance or S.D.

 
Let's say the mean for 5 numbers = 10

4 of the 5 numbers are free to be ANY value

1 of the 5 numbers is determined by the other 4

1 of the 5 numbers is not free

 

  
Comp. Formula for Sample S.D. 
 
 
 
 
 
 
 
 
 
 

Comp. S.D. Formula Comparisons
 
 
 

 
 
 
 
 
 

Homework Chapter 4: 7-13, 25, 29-33,37

 

 
 
 
 
 

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