1. Mathematical Presentation of Data Measures of Dispersion

2. Quintiles, Centiles & Quartiles
A quintile is a value below which a certain proportion of observations occurred in the ordered set of data values.
A centiles are values, in a series of observations, arranged in ascending order of magnitude, which divide the distribution into 100 equal parts (10th Percentile, 3rd, 97th, and the 50th (median) percentile).

3. Quintiles, Centiles & Quartiles
Quartiles are the observations in an array that divide the distribution into four equal parts.
lower Quartile: the value below which 25% of observations lie in an ordered array
2nd quartile = Median = 50th percentile
Upper Quartile = 75th percentile
Interquartile Range: is the middle 50% of all observations

4. Distance travelled in Miles
Villages
7.7
7.4
7.3
7.1
6.8
6.7
6.6
6.5
Village (1) 10
9.3
8.5
6.2
5.8
5.4
4.2
Village (2)
Village (2)
Village (1)
Measures of Central Tendency
7.15
Mean
7.2
Median
7.7
Mode

5. Dot plots of Distance Travelled
Village (1)
Village (2)
Even though the measures of center tendency are all the same, it is obvious from the dot plots of each group of data that there are some differences in the ‘spread’ (or variation) of the data

6. Consider these means for weekly candy bar consumption
= ( )/8
= 7
Mean = {7, 8, 6, 7, 7, 6, 8, 7}
= ( )/8
= 7

7. Measures of Dispersion
As well as measures of central tendency we need measures of how variable the data are.
Dispersion is a key concept in statistical thinking.
The basic question being asked is how much do the scores deviate around the Mean?
Measures of Dispersion; These are
The range
The Variance
Standard Deviation
Standard Error
Coefficient of Variation

8. Measures of Dispersion; The Range
The range is an important measurement
Range
Highest Value
Lowest Value
However, they do not give much indication of the spread of observations about the mean
Simple to calculate
Easy to understand
It neglect all values in the center and depend on the extreme value, extreme value are dependent on sample size
It is not based on all observations
It is not amenable for further mathematic treatment
should be used in conjunction with other measures of variability

9. Variance: The mean sum of squares of the deviation from the mean.
e.g. if the data is: 1,2,3,4,5.
The mean for these data=3
the difference of each value in the set from the mean:
1-3= -2
2-3= -1
3-3= 0
4-3= 1
5-3= 2
The summation of the differences =zero
Summation of square of the differences is not zero

10. Another formula for the variance
Variance can never be a negative value
All observations are considered
The problem with the variance is the squared unit
Another formula for the variance

11. The standard deviation is the square root of the variance
The standard deviation measured the variability between observations in the sample or the population from the mean of that sample or that population.
The unit is not squared
SD is the most widely used measure of dispersion

12. Standard Error of the mean(SE)
It measures the variability or dispersion of the sample mean from population mean
It is used to estimate the population mean, and to estimate differences between populations means
SE=SD/√ n

13. Coefficient of variation (CV):
It expresses the SD as a percentage of the mean
CV= (S /mean) x (mean of the sample)
It has no unit
It is used to compare dispersion in two sets of data especially when the units are different
It measures relative rather than absolute variation
It takes in consideration all values in the set

14. Group Work

15. Group 1 Exercise (1) 141 Pat. no 13 9 5 1 7 10 2 3 11 15 12 4 14 6 T 8
Distance
(mile)(X)
Pat. no 13 9 5 1 7 10 2 3 11 15 12 4 14 6
141 T 8
A sample of 15 patients making visits to a health center traveled these distances in miles, calculate measures of Dispersion.

16. Group 1 Exercise (2)
Calculate the measures of dispersion for 16 fasting blood glucose levels (mmol/l) In a series of 15 patients which were recorded as follows:
5·8, 4·3, 25·9, 5·2, 6·1, 3·9, 4·4, 5·6, 5·3, 4·5, 4·6, 3·8, 5·1, 5·4, 4·6

17. Group 1 Exercise (3)
Calculate the measures of dispersion in a series of 20 patients from a renal unit, whom hemoglobin levels (mg/dl) were recorded as follows:
8·4, 11·9, 6·3, 8·2, 9·5, 9·0, 7·6, 9·0, 10·1, 9·8, 8·9, 10·4, 7·6, 8·8, 11·5, 10·6, 8·2, 8·7, 9·0, 8·8

18. Group 1 Exercise (4) No. 28 7 29 1 14 8 2 18 9 11 3 22 10 24 4 5 total
length
No.
Length 28 7 29 1 14 8 2 18 9 11 3 22 10 24 4 5
total 6
A sample of 11 patients admitted to a psychiatric ward experienced the following lengths of stay, calculate measures of dispersion.