5 dispersion

Measures of Dispersion Dept of AGB Veterinary College, Hebbal, Bangalore

Central Tendency and Variation • Central tendency: Mode, Median, Mean, GM,HM • Variation: Range, Quartile deviation, Average(mean) deviation, Standard deviation, and Variance The idea is the figure out what is happening with your group

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Series

A 50

50

50

Series

B 56

56

48

Series

C

29

120

1

42

43

59

The mean of all these series is 50 But the distributions are widely distributed

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

52

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Measure of Dispersion Measures of dispersion are the measures of “spread” or “scatteredness” of observations about an average.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Measure of Dispersion The term dispersion is used to indicate 1. The fact that within a given group of data, the observations differ from one another in value (or) 2. That there is lack of uniformity in their size.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Objectives of dispersion • To determine the reliability of an average • To serve as a basis for control of the variability • To compare two or more series with regard to their variability

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Measure of Dispersion The degree to which numerical; data tend to spread about an average or central value is called the variation or dispersion of data.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Measures of dispersion help to study the important characteristics of the dispersion, by measuring the extents to which there are differences between individual observations and the measure of central tendency.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

• Absolute measure of dispersionDispersion expressed as units i.e. weight in kg, Height in cm • Relative measure of dispersion – Dispersion expressed in terms of a pure number, free from units of measurements It is the an absolute measure of dispersion divided by an average Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Absolute measure of dispersion ďƒ˜The measures of dispersion which are expressed in terms of original unit of a series are termed as absolute measures. • These are not suitable for comparing the variability of the two distributions which are expressed in different units of measurements.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Relative measure ďƒ˜Are obtained as ratios or percentages and are thus pure numbers independent of the unit of measurements. • For comparing variability of the two distributions (even if they are measured in the same unit), we compute relative measures of dispersion instead of absolute measure of dispersion Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

The various measures of dispersion are 1.

2.

3.

4.

Range Quartile deviation Mean deviation Standard deviation

Relative measures of dispersion 1.Coefficient of range 2.Quartile coefficient of dispersion 3.Mean coefficient of dispersion 4.Coefficient of variation Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

1. Range It is defined as the difference between the highest and the lowest values for the variable in distribution denoted by ‘R’.

Range= Highest value- Lowest value.

In case of grouped frequency distribution it is calculated as the difference between the upper limit of the highest class and the lower limit of the smallest class.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Merits and demerits of range  simplest but crude method.  It is not based on the entire set of data. It cannot be regarded as reliable measure of variability.  very much affected by fluctuations of sampling.

Its value varies widely from sample to sample.  If smallest and largest value is unaltered but all other observations are changed the range does not vary.  It is not suitable for mathematical treatment.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Coefficient of range • It is a relative measure of the range and estimated by using the formula:

L-S X100 Coefficient of Range= L+S L= Highest value in the series S= Smallest value in the series.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

2. Quartile Deviation (QD) • Quartile deviation is half the difference between the upper and lower quartiles in a distribution. • It is a measure of the spread through the middle half of a distribution.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

ďƒ˜The lower first quartile (Q1) divide the lower half of the distribution into two equal parts i.e it is the value below which 25% of the observation lie and above which 75% of the observation lie.

ďƒ˜Similarly, the upper or third quartile (Q3) divides the upper half of the distribution into two equal parts i.e it is the value below which 75% of the observations lie and above which 25% of the observation lie

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Uses of QD • It is useful because it is not influenced by extremely high or extremely low scores. • It is an ordinal statistic and is most often used in conjunction with the median

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Quartile coefficient of dispersion (QC)

Relative measure of QD

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

3. Mean deviation • Mean deviation or average deviation in a series is the AM of the deviations of the various items from an average (mean, median and mode) of the series taking all deviations

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Mean deviation

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Relative measure of MD

Coefficient of MD = MD about an average A X 100 A Where, A is the mean, median or mode

Note : ďƒ˜In actual practice mean or median is used but not mode ďƒ˜Median is preferred to mean Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

4. Standard deviation • It is the positive square root of the average of its squared deviations of observations from mean.

• It is denoted by σ.

• This was first suggested by Karl Pearson in 1893.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Standard Deviation • Standard deviation for population is denoted by ‘σ ’ and for sample as ‘s’.

• It is defined as the positive square root of the mean of the squares of deviations of the given observations from their arithmetic mean.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Standard deviation  

 X 

  2

N

 x  

2

 

x

2

n n 1

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Properties of SD • The standard deviation has the same unit of measurement as the variable.

• If a constant value is added or subtracted from each of the observations SD is unchanged.

• If the observations are multiplied by a positive constant value SD is multiplied by the same constant value.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Uses of SD • It is a pure number independent of unit of measurement and thus suitable for comparing the variability, homogeneity and uniformity of two or more distributions.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Coefficient of variation It is a relative measure of dispersion.

Coefficient of variation is the percentage variation from the mean with standard deviation being considered as the total variation. or It is a ratio between standard deviation and mean of sample expressed in percentage.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Coefficient of variation

CV 

 X

 100

σ = Standard deviation of the sample = mean of the sample

Lesser the CV more will be the consistency and vice versa.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Variance R.A.Fisher first introduced the term in 1913.

• Variance is the mean of the squared deviations about the mean of a series.

Variance is the square of standard deviation • It is denoted by S2 for sample variance • σ2 for population.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Square of SD is called variance

  X   

2

  2

N

Merits: 1.It is based on all observations.

2.It can be used for further mathematical treatment.

Range, variance and SD are all absolute measures of dispersion.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Standard error • The mean of random sample may be taken as representative of the population mean • The difference between the sample mean and the population mean is due to sampling or it is called as sampling error or standard error

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Standard error The standard deviation of the sampling distribution is called standard error.

S .E 

sd n

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

The following are the haemoglobin values 9g/100ml) of piglets receiving treatment for piglet anaemia: 9.1,10,11.4,12.4,9.8,8.3,9.9,9.1,7.5,6.7 .

Calculate the sample mean, variance, standard deviation and coefficient of variation.

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

X

X2

9.1

82.81

10 11.4 12.4 9.8 8.3 9.9 9.1 7.5 6.9 ∑X=94.2

100 129.96 153.76 96.04 68.89 98.01 82.81 56.25 44.89 ∑X2=913.89

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Mean= ∑X

=

n Variance=

S2

94.2

=

9.42

10 =

(∑X)2

∑X2

913.42- 8873.6 10

n

n-1 Standard Deviation=

= 2.90

10-1

2.90

=1.70

Coefficient of variation (C.V)= S

X 100

X =

1.70 X 100 = 18.046% 9.42

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Formulas for variability Note the similarity of these formulas

AD 

Average deviation

Variance

2 

Standard deviation



N

 X

 

X 

 

2

N

 X

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

 

N

2

Measures of dispersion

Merits

Demerits

Range

Easy to calculate

Not rigidly defined Not based on all observations Affected by extreme values Do not possess sampling stability Not amenable for further mathematical treatment

Quartile deviation

Easy to calculate Simple to understand Not affected by extreme items

Not rigidly defined Not based on all observations Do not possess sampling stability Not amenable for further mathematical treatment

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Measures of dispersion

Merits

Demerits

Mean deviation

It is based on all observation Rigidly defined and easy to calculate

Affeceted by extreme values Do not posses sampling stability Not amenable for further mathematical treatment

Standard deviation

Rigid formula Affected by extreme Based on all values observation Capable of further mathematical treatment Less affecetd by sampling Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,

Thank You

Dr. R. Jayashree, Asst. Prof.

(AGB), Veterinary College,