Standard deviation

The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population.

1. Seminar on Standard Deviation Presented by: Jiban Ku. Singh M. Sc Part-I (2015-16) P. G. Department of Botany BERHAMPUR UNIVERSITY BHANJA BIHAR, Berhampur- 760007 GANJAM, Odisha, INDIA E-mail- [email protected]

2. Summary Measures Central Tendency Mean Median Mode Summary Measures Variation Quartile deviation Standard Deviation Mean Deviation Range

3. Standard Deviation While looking at the earlier measures of dispersion all of them suffer from one or the other demerit i.e. Range –it suffer from a serious drawback considers only 2 values and neglects all the other values of the series. Quartile deviation considers only 50% of the item and ignores the other 50% of items in the series. Mean deviation no doubt an improved measure but ignores negative signs without any basis . 3

4. Standard Deviation The concept of standard deviation was first introduced by Karl Pearson in 1893. Karl Pearson after observing all these things has given us a more scientific formula for calculating or measuring dispersion. While calculating SD we take deviations of individual observations from their AM and then each squares. The sum of the squares is divided by the Total number of observations. The square root of this sum is knows as standard deviation . The standard deviation is the most useful and the most popular measure of dispersion. It is always calculated from the arithmetic mean , median and mode is not considered. 4

5. Definition : Standard Deviation is the positive square root of the average of squared deviation taken from arithmetic mean. The standard deviation is represented by the Greek letter (sigma ). Formula. Standard deviation = =  

6. Formula Standard deviation = = Alternatively =            

7. CALCULATION OF STANDARD DEVIATION-INDIVIDUAL OBSERVATION Two Methods:- By taking deviation of the items from the actual mean. By taking deviation of the items from an assumed mean.

8. CASE-I. When the deviation are taken from the actual mean. DIRECT METHOD Standard deviation = = or =value of the variable of observation, = arithmetic mean = total number of observations.  

9. Example : Find the mean respiration rate per minute and its standard deviation when in 4 cases the rate was found to be : 16, 13, 17 and 22. Solution: Here Mean =   16 13 17 22 16 13 17 22 Standard deviation = = = =   -1 -4 5 1 16 25 1 16 25

10. Short-Cut Method Standard deviation = =   CASE-II. When the deviation are taken from the Assumed mean.

11. = = = = = 16.398   Example: Blood serum cholesterol levels of 10 persons are as under: 240, 260, 290, 245, 255, 288, 272, 263, 277, 251. calculation standard deviation with the help of assumed mean. Value A=264 240 260 290 245 255 288 272 263 277 251 240 260 290 245 255 288 272 263 277 251 576 16 676 361 81 576 64 1 169 169 576 16 676 361 81 576 64 1 169 169 -24 -4 26 -19 -9 24 8 -1 13 13 -24 -4 26 -19 -9 24 8 -1 13 13 Here, Mean = = = 9 = 263.9 is a fraction.  

12. CALCULATION OF STANDARD DEVIATION- DISCERETE SERIES OR GROUPED DATA Three Methods Actual Mean Method or Direct Method Assumed Mean Method or Short-cut Method Step Deviation Method

13. a) Actual Mean Method or Direct Method The S.D. for the discrete series is given by the formula. = Where is the arithmetic mean, is the size of items, is the corresponding frequency and  

14. b) Assumed Mean Method or Short-cut Method Standard deviation= = Where is the assumed mean, is the corresponding frequency and  

15. Example: Periods: 10 11 12 13 14 15 16 No. of patients: 2 7 11 15 10 4 1 Solution: Periods:(x) No. of patients ( ) 10 11 12 13 14 15 16 2 7 11 15 10 4 1 Total N= =50 Periods:(x) 10 11 12 13 14 15 16 2 7 11 15 10 4 1 Total -3 -2 -1 1 2 3 -6 -14 -11 10 8 3 =-10 -6 -14 -11 10 8 3 9 4 1 1 4 9 18 28 11 10 16 9 = 92 18 28 11 10 16 9 Mean= = = 13 = 12.8 is a fraction.   = = = = = 1.342  

16. c) Step Deviation Method We divide the deviation by a common class interval and use the following formula Standard deviation= = × Where common class interval, is assumed mean f is the respective frequency.  

17. = × = × = × = × = 1.235 × =4.94 mm Hg.   Example: B.P.(mmHg): 102 106 110 114 118 122 126 No. of days: 3 9 25 35 17 10 1 Solution : B.P.( mmHg ) No. of days ( ) 102 106 110 114 118 122 126 3 9 25 35 17 10 1 Total N=100 B.P.( mmHg ) 102 106 110 114 118 122 126 3 9 25 35 17 10 1 Total N=100 -3 -2 -1 1 2 3 -9 -18 -25 17 20 3 =-12 -9 -18 -25 17 20 3 27 36 25 17 40 9 = 154 27 36 25 17 40 9 M = × mm Hg  

18. CALCULATION OF STANDARD DEVIATION- CONTINUES SERIES S.D. of Continues Series can be calculated by any one of the methods discussed for discrete frequency distribution But Step Deviation Method is mostly used. Standard deviation= = × Where common class interval, is assumed mean f is the respective frequency.  

19. Example: I.Q. 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No. of students: 5 12 15 20 10 4 2 Solution: I.Q. No. of students: ( ) Mid-value 10-20 20-30 30-40 40-50 50-60 60-70 70-80 5 12 15 20 10 4 2 Total =N=68 I.Q. 10-20 20-30 30-40 40-50 50-60 60-70 70-80 5 12 15 20 10 4 2 Total -3 -2 -1 1 2 3 -15 -24 -15 10 8 6 =-30 -15 -24 -15 10 8 6 45 48 15 10 16 18 = 152 45 48 15 10 16 18 Standard deviation= = = =   15 25 35 45 55 65 75

20. CALCULATION OF COMBINED STANDARD DEVIATION It is possible to compute combined mean of two or more than two groups. Combined Standard Deviation is denoted by = Where c ombined standard deviation ,  

21. a) Combined S.D. = c ombined Mean = = = = 55   The following are some of the particulars of the distribution of weight of boys and girls in a class: Find the standard deviation of the combined data which of the two distributions is more variable Boys Girls Numbers 100 50 Mean weight 60 kg 45 kg Variance( ) 9 4   = = b) C.V (Boys)= C.V (Girls)=  

22. MERITS OF STANDARD DEVIATION Very popular scientific measure of dispersion From SD we can calculate Skewness , Correlation etc It considers all the items of the series The squaring of deviations make them positive and the difficulty about algebraic signs which was expressed in case of mean deviation is not found here. 22

23. DEMERITS OF STANDARD DEVIATION Calculation is difficult not as easier as Range and QD It always depends on AM Extreme items gain great importance The formula of SD is = Problem : Calculate Standard Deviation of the following series X – 40, 44, 54, 60, 62, 64, 70, 80, 90, 96   23

24. USES OF STANDARD DEVIATION It is widely used in biological studies . It is used in fitting a normal curve to a frequency distribution. It is most widely used measure of dispersion. 24

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