Measures of Dispersion/ VariabilityDr Faris Al Lami

MB ChB PhD

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

EXERCISE

• For the same 15 patients in the previous example , calculate measures of dispersion.

2

XDistance(mile)(X)

Pat. no2

X

Distance(mile)(X)

Pat. no

1691392551497108192931112111322515129341441213144125225151416913625515144127

1575141Total3668

Range

R=XL- XS

=15-3

=12 mile

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

Standard Error

• SE=SD/√ n

• =4.2/√15 = 4.2/3.87 = 1.085 mile

Coefficient of Variation

• CV= S /mean X100

• = 4.2 mile/ 9.4 mile X 100%

• =44.7%

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

EXERCISE

• A sample of 11 patients admitted to a psychiatric ward experienced the following lengths of stay, calculate measures of central tendency and dispersion.

lengthNo.lengthNo.

287291

148142

189113

2210244

1411145

total146