Variability

1
Chapter 4

• Variability

2

• Statistics means never having to say you're
  certain.

3
Variability

• The amount by which scores are dispersed or
  scattered in a distribution.
• Page 74 graphs

4
Range

• Difference between the largest and smallest
  scores.
• Problem large groups may have large range

5
Variance and Standard Deviation

• Standard Deviation - The square root of the
  variance.
• Or
• The square root of the mean of all the squared
  deviations from the mean!!???
• The value of a standard deviation can NOT be
  negative.

6
Standard Deviation

• A rough measure of the average (or standard)
  amount by which scores deviate on either side of
  their mean.

7
Progress Check 4.1 Page 80

• Employees of Corporation A earn annual salaries
  described by a mean of 90,000 and a standard
  deviation of 10,000.
• a. The majority of all salaries fall between what
  two values?
• b. A small minority of salaries are less than
  what value?
• c. A small minority of all salaries are more than
  what value?
• c. Answer parts (a), (b), and (c) for Corporation
  Bs employees, who earn annual salaries described
  by a mean of 90,000 and a standard deviation of
  2,000.

8
Standard Deviation

• Deviations from the mean.
• The sum of all the deviations equals the
  variance.
• To calculate the variance
• The sum of squares equals the sum of all squared
  deviation scores (p. 83)

9
Sum of Squares

• Calculation example of sample sum of squares (SS)
  using the computation formula (p. 83)
• SSSX2 (SX)2
• n

10
Standard Deviation

• Calculation example of sample standard deviation
  using the computation formula (p. 86)
• s vs2 v

SS2 n-1
11
Why n-1? (p88)

• This applies the sample estimate to the variance
  rather then the population estimate.
• If we use the population estimate we would
  underestimate the variability.
• In other words, this allows a more conservative
  and accurate estimate of the variance within the
  sample.

12
Degrees of freedom

• Degrees of freedom (df) refers to the number of
  values that are free to vary, given one or more
  mathematical restrictions, in a sample being used
  to estimate a population characteristic. (p. 90)

13
The value of the population mean mu (µ)

• Most of the time the population mean is unknown
  so we use the value of the sample mean and the
  degrees of freedom (df) n-1.

14
Standard Deviation calculation (p88)

1. Assign a value to n representing the number of X
   scores.
2. Sum all X scores.
3. Square the sum of all X scores.
4. Square each X score.
5. Sum all squared X scores.
6. Substitute numbers into the formula to obtain the
   sum of squares, SS.
7. Substitute numbers in the formula to obtain the
   sample variance, s2.
8. Take the square root of s2 to obtain the sample
   standard deviation, s.

15
Qualitative data and variance

• No measures of variability exist for qualitative
  data!
• However, if the data can be ordered, then the
  variability can be described by identifying
  extreme scores (ranks).

16
Progress Check

• Calculate the mean, median, mode, and standard
  deviation for the following height of students in
  inches.
• 64, 61, 73, 70, 71, 75, 69, 60, 63, 71, 65, 62