MPA-205-REPORT.pptx
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Standard deviation () and root mean square deviation (s) are measures of dispersion in a data set. is calculated as the positive square root of the average of the squared deviations from the mean, while s is the positive square root of the average of the squared deviations from an arbitrary value A. It is shown that s2 = 2 + (x - A)2, where x is the mean and d is the difference between x and A. This indicates that will always be less than or equal to s, and reaches its minimum value when A equals x. Therefore, standard deviation provides the smallest possible value of root mean square deviation.
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- 1. MEASURES OF DISPERSION, SKEWNESS AND KURTOSIS STANDARD DEVIATION () AND ROOT MEAN SQUARE DEVIATION (S) RELATION BETWEEN AND S Chapter 3 Reporters: JAZER B. LEUTERIO RINA CRIS DE MESA JULIAH ANGELICA B. CRISTOBAL MPA STUDENTS
- 2. STANDARD DEVIATION () AND ROOT MEAN SQUARE DEVIATION (S) Standard deviation, usually denoted by the Greek letter small sigma (), is the positive square root of the arithmetic mean. For the frequency distribution xi /, = 1, 2, , , = 1 癌 ( ) 2 where is the arithmetic mean of the distribution and =
- 3. STANDARD DEVIATION () AND ROOT MEAN SQUARE DEVIATION (S) The step of squaring the deviations (xi - x) overcomes the drawback of ignoring the signs in mean deviation. Standard deviation is also suitable for further mathematical treatment ( 3.7.3). Moreover of all the measures, standard deviation is affected least by fluctuations of sampling.
- 4. STANDARD DEVIATION () AND ROOT MEAN SQUARE DEVIATION (S) Thus we see that standard deviation satisfies almost all properties laid down for an ideal measure of dispersion except for the general nature of extracting the square root which is not readily comprehensible for non- mathematical person.
- 5. STANDARD DEVIATION () AND ROOT MEAN SQUARE DEVIATION (S) It may also be pointed out that standard deviation gives greater weight to extreme values and such has not found favor with economists or businessmen who are more interested in the results of the modal class.
- 6. STANDARD DEVIATION () AND ROOT MEAN SQUARE DEVIATION (S) Taking into consideration the pros and cons and also the wide applications of standard deviation in statistical theory, we may regard standard deviation as the best and the most powerful measure of dispersion.
- 7. STANDARD DEVIATION () AND ROOT MEAN SQUARE DEVIATION (S) The square of standard deviation is called the variance and is given by 2 = 1 (ヰ - )2
- 8. STANDARD DEVIATION () AND ROOT MEAN SQUARE DEVIATION (S) Root mean square deviation, denoted by s is given by = 1 (ヰ- )2 where A is any arbitrary number. 2 is called mean square deviation.
- 9. RELATION BETWEEN AND By definition we have = 1 (ヰ- )2 = 1 (ヰ - + - )2 = 1 [ (ヰ - )2 + ( - )2 + 2( - A) (ヰ - ) ] = 1 [ (ヰ - )2 + ( - )2 1 + 2 ( - A) [ (ヰ - )
- 10. RELATION BETWEEN AND ( - A), being constant is taken outside, the summation sign. But [ (ヰ - )= 0, being the algebraic sum of the deviations of the given values from their mean. Thus. 2 = 2 + ( - )2= 2 + 2, where d= - A
- 11. RELATION BETWEEN AND Obviously s2 will be least when d=0, i.e., = A. Hence mean square deviation and consequently root mean square deviation is least when the deviations are taken from A= , i.e., standard deviation is the least value of root mean square deviation.
- 12. RELATION BETWEEN AND The same result could be obtained alternatively as follows: Mean square deviation is given by 2 = 1 ( ヰ - )2 It has been shown in 2.5.1property 2 that ( ヰ - )2 is minimum when
- 13. RELATION BETWEEN AND A= . Thus mean square deviation is minimum when A= and its minimum value is (2 ) min = 1 [ (ヰ - )2 = 2 Hence variance is the minimum value of mean square deviation or standard deviation is the minimum value of root mean square deviation.
- 14. Thank you