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Biostatistics- Measure of Dispersion, Standard deviation,Coefficient of variation

MEASURES OF DISPERSION:

Dispersion is the degree of spread or variation of the variable about a central value.

USES:

To determine the reliability of an average.
To serve as basis for control of variability.
To compare two or more series in relation to their variability.
Facilitate further statistical analysis.

RANGE:

It is the simplest method, Defined as the difference between the value of the smallest item and the value of the largest item.
This measure gives no information about the values that lie between the extremes values.
Subject to fluctuations from sample to sample.

MEAN DEVIATION:

It is the average of the deviations from the arithmetic mean.

M.D = (ΣX – Xi)/n ,

where Σ( sigma ) is the sum of, X is the arithmetic mean, Xi is the value of each observation in the data, n is the number of observation in the data.

STANDARD DEVIATION(SD):

Most important and widely used.
Also known as root mean square deviation, because it is the square root of the mean of the squared deviations from the arithmetic mean.
Greater the standard deviation, greater will be the magnitude of dispersion from the mean.
A small SD means a higher degree of uniformity of the observations.

CALCULATION:

For ungrouped data:
- Calculate the mean(X) of the series.
- Take the deviations (d) of the items from the mean by : d=Xi – X, where Xi is the value of each observation.
- Square the deviations (d²) and obtain the total (∑ d²)
- Divide the ∑ d² by the total number of observations i.e., (n-1) and obtain the square root. This gives the standard deviation.
- Symbolically, standard deviation is given by:

SD= √ ∑ d² /(n-1)

For grouped data with single units for class intervals:

S = √∑(Xi – X) x f i / (N -1)

Xi is the individual observation in the class interval

fi is the corresponding frequency

X is the mean

N is the total of all frequencies

For grouped data with a range for the class interval:

S =√ ∑(Xi – X) x f i / (N -1)

Xi is the midpoint of the class interval

Confidence intervals:

Lower limit & upper limit estimates for statistic given by:
- Lower Limit: statistic – C x SE (statistic)
- Upper Limit: statistic + C x SE.
Confidence coefficient:
- C = Confidence coefficient = 1.65 for 90% confidence interval
- C= 1.96 for 95% confidence interval
- C= 2.58 for 99% confidence interval
- C = 3.29 for 99.9% confidence interval

COEFFICIENT OF VARIATION(C.V.):

A relative measure of dispersion.
To compare two or more series of data with either different units of measurement or marked difference in mean.

C.V.= (Sx100)/ X

Where, C.V. is the coefficient of variation

S is the standard deviation

X is the mean

Higher the C.V. greater is the variation in the series of data

Exam Important

Mean deviation is Measure of dispersion

Square root of deviation is also called as Standard deviation

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Biostatistics- Measure of Dispersion, Standard deviation,Coefficient of variation