MEASURES OF DISPERSION Definition:-The degree to
Description: MEASURES OF DISPERSION Definition:-The degree to which the individual values of the variate scatter away from the average or the central value is called a dispersion. Various Measures of variation There are different measures of variation:
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slide1. MEASURES OF DISPERSION Definition:-The degree to which the individual values of the variate scatter away from the average or the central value is called a dispersion. Various Measures of variation
There are different measures of variation:
1.Range
2.Mean Deviation
3.Standard Deviation
4.Variance<br>
slide2. Mean Deviation:-The mean deviation is the average of the absolute value of the deviation from the mean.(or median or mode) Mean Deviation or MD =∑I x I /N(for ungrouped data) here MD=mean deviation. x=deviation from actual mean. ∑ x =sum of all deviation of a distribution. I I=not considering sign(+ v e or – v e) while summation is done. Deviation=score-mean or x=x-x
Procedure:-
1.Calculate mean.
2.calculate deviation from mean ignoring the sign( + or –).
3.Divide ∑ x by the N.(N is the total no of items)<br>
slide3. Example:-eggs laid by a species of birds were counted as 5,7,8,10,14,12,13,5,8,8.compute the mean deviation of the distribution. Mean= ∑ x /n MD = ∑ Ix I/N =90/10 =26/10 =9 =2.6<br>
slide4. Example:-The diastolic Blood pressure of 10 individuals was as follows;83,75,81,79,71,95,75,77,84 and 90.find the mean deviation. Mean= ∑ x /n MD = ∑ Ix I/N =810/10 =56/10 =81 =5.6<br>
slide5. Standard Deviation:-The standard deviation is the most frequently used measures of deviation. In simple terms, it is defined as “Root –Mean-Square-Deviation”. it is denoted by the Greek letter sigma( σ )or by the initial S.D. S.D= ∫ ∑(X-X)² / n(when n>30) = ∫ ∑(X-X)² / n-1(When n<30) The steps involved in calculating the standard deviation are as follows:-
1.First of all take the deviation of each value from the arithmetic mean. (X-x)
2.Then square each deviation (X-X)²
3.Add up the squared deviations. ∑ (X-X)²
4.Divide the result by the number of observation n or n-1(in case of n<30)
5.Then take the square root ,which gives the standard deviation.<br>
slide6. Example:-The diastolic Blood pressure of 10 individuals was as follows;83,75,81,79,71,95,75,77,84 and 90.find the standard deviation. S.D = ∫∑(X-X)² / n-1 = ∫482/10-1
=∫53.55
=7.31<br>
slide7. Example:- Hemoglobin percent g/100ml of liver fed wallagootter was recorded as 23,22,20,24,16,17,18,19 and 21.calculate the standard deviation. S.D = ∫∑(X-X)² / n-1 = ∫60/9-1
=∫7.5
=2.75<br>
slide8. Variance:-Variance of a distribution is defined as the square of the standard deviation. Variance = ( S.D )² coefficient of variance= S.D/mean*100<br>
slide9. Calculation of mean Deviation Discrete Series:-In discrete series the formula for calculating mean deviation is M.D = ∑ f ID I N Example:-calculate mean deviation,median from the following series:<br>
slide10. M.D = ∑ f ID I N
= 36 /48
= 0.75
Median = size of N + 1 th item
2
=48+1/2
= 24.5 th item
Size of 24.5 th item is 12,hence median =12.<br>
slide11. Calculation of Mean Deviation-Continuous Series:- For calculating mean deviation in continuous series the procedure remains the same as discussed above. The only difference is that here we have to obtain the mid-point of the various classes and take deviations of these points from median. The formula is same ,i.e., M.D = ∑ f ID I N Example- Find the median and mean deviation of the following data:<br>
slide12. Solution:-calculation of median and mean deviation<br>
slide13. Median = size of N/2 item =100/2 =50th item Median lies in the class 30-40
Median = L + N/2-c.f x i
f
L=30,N/2=50,c.f=37, f= 25 ,I =10
= 30 + 50-37 x 10
25
= 30 + 5.2
= 35.2
M.D = ∑ f ID I N
= 1314.8/100
=13.148<br>
slide14. Calculation of standard deviation –Discrete Series. S.D= ∫∑f x²/ N, where x = (X – X) Assumed Mean Method σ = ∫∑f d²/ N –(∑f d)² N Example- Calculate the standard deviation from the data given below.<br>
slide15. σ = ∫∑f d²/ N –(∑ f d) ² N ∑f d² =362, ∑f d= 128 ,N =217
σ = ∫ 362 –(128) ² 217 217
= ∫ 1.668 – 0.348
= 1.149<br>
slide16. Calculation of standard deviation –Continuous Series. σ = ∫∑f d²/ N –(∑f d)² x i N where d = (m – A) , i = class interval Example - Calculate the mean and standard deviation of the following frequency distribution of marks:<br>
slide18. Mean=A+ ∑f d/N X i =35 +118/200 x 10 =35+5.9 =40.9 σ = ∫∑f d²/ N –(∑ f d) ² x i N ∑f d² =510, ∑f d= 118 ,N =200
σ = ∫ 510 –(118) ²x10 200 200
= ∫ 2.55– 0.3481 x 10
= 1.4839 x10
=14.839<br>
There are different measures of variation:
1.Range
2.Mean Deviation
3.Standard Deviation
4.Variance<br>
slide2. Mean Deviation:-The mean deviation is the average of the absolute value of the deviation from the mean.(or median or mode) Mean Deviation or MD =∑I x I /N(for ungrouped data) here MD=mean deviation. x=deviation from actual mean. ∑ x =sum of all deviation of a distribution. I I=not considering sign(+ v e or – v e) while summation is done. Deviation=score-mean or x=x-x
Procedure:-
1.Calculate mean.
2.calculate deviation from mean ignoring the sign( + or –).
3.Divide ∑ x by the N.(N is the total no of items)<br>
slide3. Example:-eggs laid by a species of birds were counted as 5,7,8,10,14,12,13,5,8,8.compute the mean deviation of the distribution. Mean= ∑ x /n MD = ∑ Ix I/N =90/10 =26/10 =9 =2.6<br>
slide4. Example:-The diastolic Blood pressure of 10 individuals was as follows;83,75,81,79,71,95,75,77,84 and 90.find the mean deviation. Mean= ∑ x /n MD = ∑ Ix I/N =810/10 =56/10 =81 =5.6<br>
slide5. Standard Deviation:-The standard deviation is the most frequently used measures of deviation. In simple terms, it is defined as “Root –Mean-Square-Deviation”. it is denoted by the Greek letter sigma( σ )or by the initial S.D. S.D= ∫ ∑(X-X)² / n(when n>30) = ∫ ∑(X-X)² / n-1(When n<30) The steps involved in calculating the standard deviation are as follows:-
1.First of all take the deviation of each value from the arithmetic mean. (X-x)
2.Then square each deviation (X-X)²
3.Add up the squared deviations. ∑ (X-X)²
4.Divide the result by the number of observation n or n-1(in case of n<30)
5.Then take the square root ,which gives the standard deviation.<br>
slide6. Example:-The diastolic Blood pressure of 10 individuals was as follows;83,75,81,79,71,95,75,77,84 and 90.find the standard deviation. S.D = ∫∑(X-X)² / n-1 = ∫482/10-1
=∫53.55
=7.31<br>
slide7. Example:- Hemoglobin percent g/100ml of liver fed wallagootter was recorded as 23,22,20,24,16,17,18,19 and 21.calculate the standard deviation. S.D = ∫∑(X-X)² / n-1 = ∫60/9-1
=∫7.5
=2.75<br>
slide8. Variance:-Variance of a distribution is defined as the square of the standard deviation. Variance = ( S.D )² coefficient of variance= S.D/mean*100<br>
slide9. Calculation of mean Deviation Discrete Series:-In discrete series the formula for calculating mean deviation is M.D = ∑ f ID I N Example:-calculate mean deviation,median from the following series:<br>
slide10. M.D = ∑ f ID I N
= 36 /48
= 0.75
Median = size of N + 1 th item
2
=48+1/2
= 24.5 th item
Size of 24.5 th item is 12,hence median =12.<br>
slide11. Calculation of Mean Deviation-Continuous Series:- For calculating mean deviation in continuous series the procedure remains the same as discussed above. The only difference is that here we have to obtain the mid-point of the various classes and take deviations of these points from median. The formula is same ,i.e., M.D = ∑ f ID I N Example- Find the median and mean deviation of the following data:<br>
slide12. Solution:-calculation of median and mean deviation<br>
slide13. Median = size of N/2 item =100/2 =50th item Median lies in the class 30-40
Median = L + N/2-c.f x i
f
L=30,N/2=50,c.f=37, f= 25 ,I =10
= 30 + 50-37 x 10
25
= 30 + 5.2
= 35.2
M.D = ∑ f ID I N
= 1314.8/100
=13.148<br>
slide14. Calculation of standard deviation –Discrete Series. S.D= ∫∑f x²/ N, where x = (X – X) Assumed Mean Method σ = ∫∑f d²/ N –(∑f d)² N Example- Calculate the standard deviation from the data given below.<br>
slide15. σ = ∫∑f d²/ N –(∑ f d) ² N ∑f d² =362, ∑f d= 128 ,N =217
σ = ∫ 362 –(128) ² 217 217
= ∫ 1.668 – 0.348
= 1.149<br>
slide16. Calculation of standard deviation –Continuous Series. σ = ∫∑f d²/ N –(∑f d)² x i N where d = (m – A) , i = class interval Example - Calculate the mean and standard deviation of the following frequency distribution of marks:<br>
slide18. Mean=A+ ∑f d/N X i =35 +118/200 x 10 =35+5.9 =40.9 σ = ∫∑f d²/ N –(∑ f d) ² x i N ∑f d² =510, ∑f d= 118 ,N =200
σ = ∫ 510 –(118) ²x10 200 200
= ∫ 2.55– 0.3481 x 10
= 1.4839 x10
=14.839<br>