measures of dispersion - university of baghdad · measures of dispersion ... • sd is the most...
TRANSCRIPT
Measures of dispersion & variability
• They measure the variability in the values of observations in the set.
• They also called measures of variation, spread and scatter.
Measures of dispersion & variability
• If all values are the same the dispersion is zero.
• If the values are homogenous and close to each other the dispersion is small.
• If the value are so different the dispersion is large.
Measures of dispersion
• Range: Is the difference between the largest and smallest value
• R=XL- XS
• R=Range• XL= largest value,• XS= smallest value
Properties of the range:
ØSimple to calculate
ØEasy to understand
ØIt neglect all values in the center and depend on the extreme value, extreme value are dependent on sample size
Properties of the range:
ØIt is not based on all observations
ØIt is not amenable for further mathematic treatment
Øshould be used in conjunction with other measures of variability
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=3the difference of each value in the set from the mean:
1-3= -22-3= -13-3= 04-3= 15-3= 2
• The summation of the differences =zero• Summation of square of the differences is not zero
Variance:
• Population Variance (sigma squared)
2
2 ∑(X- μ)
• α =----------------N
2 2 2
• α =[ N ∑x – (∑ X) ] / N.N
2
α= sigma squared(pop.var)X=observation valueμ= population meanN=population size
2
∑x =summa�on of squared2
(∑ X)=squared of summa�on
Variance:• Sample Variance
_ 22 ∑ (X- X )
• S=---------------- ORn-1
2 22 [ n∑X – (∑X) ] s= ----------------------
n(n-1)
2
• S= sample variance• n= sample size
Variance:
• Variance can never be a negative value
• All observations are considered
• The problem with the variance is the squared unit
Standard deviation (SD):• It is the square root of the variance
• SD=√sigma square= ± sigma(α)---- for population
2
• Sd= √S = ± S----for sample
Standard deviation (SD):• 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
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
Coefficient of variation (CV):
• It expresses the SD as a percentage of the mean
• CV= S /mean X100 (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
Coefficient of variation (CV):
• It measures relative rather than absolute variation
• It takes in consideration all values in the set
2
XDistance(mile)(X)
Pat. no2
X
Distance(mile)(X)
Pat. no
1691392551497108192931112111322515129341441213144125225151416913625515144127
1575141Total3668
Variance & sd2 2
2 n∑X – (∑X)
s= ----------------------n(n-1)
2
=(15)(1575) – (141) / 15 x 142
=17.8 milesd= √17.8 = ± 4.2 mile
EXERCISE
The following are the hemoglobin values (gm/dl) of 10 children receiving treatment for hemolytic anemia:9.1,10.0, 11.4, 12.4, 9.8,8.3, 9.9, 9.1, 7.5, 6.7Compute the sample mean, median, variance, and standard deviation