1. Dispersion

2. Basic problem
Population 1: 3,3,3,3,3
Population 2 : 1,2,3,4,5
Population 3: 0,0,15,0,0
Population 4: -294,-24,3,30,300
What is the similarity and the difference among the examples?
How can we measure dispersion?

3. Dispersion
The differencies between the values and the value’s deviaton from the measures of central tendency
Difference: xi-xj
Deviation:

4. How can we measure the dispersion
Difference
Range
Gini ’s average absolute difference
Deviation
Standard deviation
Variance
Coefficient of variation

5. Range
Range
Advantages?
Disadvantages?
Interpretation?

6. How can we measure the dispersion
Difference
Range
Gini ’s average absolute difference
Deviation
Standard deviation
Variance
Coefficient of variation

7. Variance Nominator: total sum of squares (TSS)
Calculation from ungrouped data:
Nominator: total sum of squares (TSS)
Denominator: number of elements (N)
Example

8. Standard deviation
Calculation from ungrouped data
Interpretation: how much do the individuals deviate on average from the mean.

9. Example
Ages (year) 20, 20, 20, 25, 25, 32, 33, 33, 33, 33

10. Calculation from frequency distribution
Ages 20, 20, 20, 25, 25, 32, 33, 33, 33, 33
Calculation from frequency distribution
Ages, year
Nr of Ind. (fj)
Ratio, gj 20 3
0,3 25 2
0,2 32 1
0,1 33 4
0,4
Total 10
1,0

11. Coefficient of variation
Interpretation: how much do the individuals deviate on average from the mean, measured as percentage of the mean.
Advantages?

12. Example
Ages 20, 20, 20, 25, 25, 32, 33, 33, 33, 33

13. Calculation of dispersion from intervals
Start from the midpoints (Xi0 + Xi1)/2
Ages, year
Number of flats; fi
Midpoints, Xi
-11
6,0
12-21
17,0
22-31
27,0
32-41
37,0
42-56
49,5
57-81
69,5
82-
94,5
Total -

14. xi fi
fi(x- )2
fixi2
6,0
17,0
27,0
37,0
49,5
69,5
94,5
Total

15. Properties of Dispersion
What happens if we add the same number to all of our values or multiply all of them by the same number?
Measures of dispersion
Increasing all values by the same A number
Multiplying all values by the same A(≠0) number
Range
Remains unchanged
Range is multiplied by A
Variance
Variance is multiplied by A
Standard deviation
Standard deviation is multiplied by A
Coefficient of variation