MEASUREMENT OF VARIABILITY Measurement of variability measure of

MEASUREMENT OF VARIABILITY

Measurement of variability � measure of spread or measure of dispersion. It is used to measure the degree of variation in a set of data The mean per se is of limited value as it gives no information regarding variability with which the observations are scattered around itself.

� � � There are some measures for this dispersion such as: 1) Range 2) Mean deviation 3) Variance 4) Standard deviation

1) Range: is the difference in values between the largest and smallest observations in the set. � Advantages : It is the simplest measure of dispersion, as it is easily to calculated. � Disadvantages: * Although the range is often of considerable interest, it is not satisfactory as an indicator of general variability since it is based on tow extreme values only and ignores the distribution of the observations within these limits. * It does not show the variation of the central values. �

II) Mean deviation (average deviation): It is the arithmetic average of all difference between the observations and their mean. We ignore the sign otherwise the total will be equal zero. � Please write the equation � Mean deviation=-In the above first group we find: Mean deviation= (20 - 20 + 15 - 20 + 30 - 20 + 20 - 20+ 15 – 20) / 5 We calculate the difference by ignoring the sign M. D= (0+5 10+ 0 + 5 ) /5 = 2/ 5 = M. D. = 4 years �

III) Variance � To get rid of the sign we try to square the difference between the observations and their mean, the sum of squared deviation when divided by the number of observations we will obtain the mean squared deviation.

� Coefficient of variation The ratio of the standard deviation of a distribution to its arithmetic mean. A statistical measure of the dispersion of data points in a data series around the mean. It is calculated as follows: � In probability theory and statistics, the coefficient of variation (CV) is a measure of dispersion of a probability distribution. It is defined as the ratio of the standard deviation o to the mean µ: C= α / µ.

Standard deviation � � � � In statistics, the average amount a number varies from the average number in a series of numbers. For example, it would be useful to know how much variation there is in reponse to a direct-mail package across several mailing lists. The standard deviation, represented by the Greek letter sigma ("S" for a population and "s" for a sample) is equal to the square root of the variance. The formula is: where n= number of values in the sample, xi = each value in the sample, -X 22 = mean (average) value of the sample.

� � It is a measure of how much the data in a certain collection are scattered around the mean. A low standard deviation means that the data are tightly clustered; a high standard deviation means that they are widely scattered. The advantage of a standard deviation calculation over a variance calculation is that it is expressed in terms of the same scale as the values in the sample.

Relationship between standard deviation � Relationship between standard deviation and mean The mean and the standard deviation of a set of data are usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point.

� The coefficient of variation of a sample is the ratio of the standard deviation to the mean. It is a dimensionless number that can be used to compare the amount of variance between populations with different means. Chebyshev's inequality proves that in any data set, nearly all of the values will be nearer to the mean value, where the meaning of "close to" is specified by the standard deviation. Rapid calculation methods

VI) Standard Deviation It is the positive square root of variance. � Standard deviation = � Coefficient of Variation: It is the standard deviation of the set of observations divided by their mean. C. V. = S / X It is used to compare the relative variability of the two set of data. �

Normal Distribution Curve � � This is a mathematical model which adequately describes many types of measurements in the fields of medicine and biology. It is described by its mean and standard deviation.

Properties of the curve : � 1) It is bell-shaped. � 2) It is symmetrical curve rising to its peak at the mean which is located at the midpoint of the base. � 3) It diminishes into small values above and below the mean , it approaches infinity on both direction but not reach or touch the base line. � 4) Areas under normal curve : Approximatelyv 68% of observations fall within one standerd deviation of mean. �