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Biostatistics Standard deviation and variance

Standard deviation and variance are measures used to quantify the dispersion of a set of data. Standard deviation is represented by the Greek symbol σ and is calculated as the square root of variance. Variance is represented by σ2 and is calculated as the sum of the squared deviations from the mean divided by the number of values. Both standard deviation and variance are important statistical concepts used to analyze the spread of data distributions.

Education◦
Standard Deviation &Variance Dr. Harinatha Reddy Sri Krishnadevaraya University Department of microbiology
Standard deviation: • In statistics, the standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data. • It is represented by Greek symbol (s) and in short form S or SD. • It also known as root mean square deviation. This formula for ungrouped data: SD OR
SD for Ungrouped data: Example 1: A hen lays eight eggs. Each egg was weighed and recorded as follows: 60, 56, 61, 68, 51, 53, 69, 54 S.NO Variance (X) Deviation (d or dx)= X-X ̅ (X-X ̅ )2 or dx2 1 60 60-59= 1 1×1=1 2 56 56-59= -3 3×3=9 3 61 61-59= 2 2×2= 4 4 68 68-59= 9 81 5 51 51-59= -8 64 6 53 53-59= -6 36 7 69 69-59= 10 100 8 54 54-59= -5 25 N=8 X=432 ∑dx2=320 Mean(X ̅ )= ∑X/N = 432/8 =59 SD= SD = 320/8 SD= 6.32
Exercise 1: Calculate SD for ungrouped data: • 10,15,20,25,4,8
Standard deviation for grouped data (discrete variables): Where n = ∑f
Example: Standard deviation for grouped data (discrete variables): Workers (X) Frequency (f) 0 1 1 1 2 2 3 3 4 6 5 5 6 4 7 3 8 3 9 2 First calculate mean for discrete data: Mean formula for discrete data is Mean (X ̅ )= ∑fx / ∑f
Workers (X) Frequency (f) (fx) 0 1 0 1 1 1 2 2 4 3 3 9 4 6 24 5 5 25 6 4 24 7 3 21 8 3 24 9 2 18 ∑f=30 ∑fx=150 Mean formula for discrete data is Mean (X ̅ )= ∑fx / ∑f = 150/30 = 5 Mean (X ̅ )= 5
Workers (X) Frequency (f) (fx) (X-X ̅ ) (X-X ̅ ) 2 (X-X ̅ ) 2 f 0 1 0 0-5=-5 25 25×1= 25 1 1 1 1-5=-4 16 16×1=16 2 2 4 2-5=-3 9 9×2= 18 3 3 9 -2 4 12 4 6 24 -1 1 6 5 5 25 0 0 0 6 4 24 1 1 4 7 3 21 2 4 12 8 3 24 3 9 27 9 2 18 4 16 32 ∑f=30 ∑fx=150 ∑(X-X ̅ ) 2 f =152 SD
Exercise 1: Calculate SD for following discrete data Blood cells No of days 90 5 55 9 60 5 70 4 80 10 100 20
Standard deviation for grouped Data (Continuous serious)
Standard deviation for grouped Data Hours Number of students 10 -14 2 15 -19 12 20 -24 23 25-29 60 30-34 77 35-39 38 40 -44 8 Table 1. Number of hours per week spent watching television First calculate Mean (x ̅ )= ∑f.m / ∑f M= Middle values of class (It also mention as X)
Hours Number of students 10 -14 2 15 -19 12 20 -24 23 25-29 60 30-34 77 35-39 38 40 -44 8 First calculate Mean (x ̅ )= ∑f.m / ∑f M= Middle values of class (It also mention as X)
Hours Midpoint (x) Frequenc y (f) fx 10 to 14 12 2 24 15 to 19 17 12 204 20 to 24 22 23 506 25 to 29 27 60 1,620 30 to 34 32 77 2,464 35 to 39 37 38 1,406 40 to 44 42 8 336 ∑f= 220 ∑fx=6,560 Mean (x ̅ )= ∑f.m / ∑f = 6560/220 =29.82
Hours Midpoint (M or X) Frequenc y (f) fm or fx (M-X ̅ ) (M-X ̅ ) 2 (M-X ̅ ) 2 f 10 -14 12 2 12×2=24 12-29.82 = -17.82 (17.82 ) 2 =317.6 317.6×2 = 635.2 15 -19 17 12 17×12=204 17-29.82= -12.82 (12.82 ) 2 =164.4 164.4×12 =1,972.8 20 -24 22 23 22×23=506 -7.82 61.2 1,407.6 25 -29 27 60 27×60=1,620 -2.82 8.0 480.0 30 -34 32 77 32×77=2,464 2.18 4.8 369.6 35 -39 37 38 37×38=1,406 7.18 51.6 1,960.8 40 -44 42 8 42×8=336 12.18 148.4 1,187.2 ∑f= 220 ∑fx=6,560 ∑(M-X ̅ ) 2 f= 8,013.2 Mean (x)==29.82 SD Where n = ∑f M written as X in the equation (Middle point (M) of the class also called as X) SD
Coefficient of standard deviation
Coefficient of standard deviation Coefficient of standard deviation= Standard deviation (SD) Arithmetic mean (𝐗̅ ) = SD 𝐗̅
Hours Midpoint (M or X) Frequenc y (f) fm or fx (M-X ̅ ) (M-X ̅ ) 2 (M-X ̅ ) 2 f 10 -14 12 2 12×2=24 12-29.82 = -17.82 (17.82 ) 2 =317.6 317.6×2 = 635.2 15 -19 17 12 17×12=204 17-29.82= -12.82 (12.82 ) 2 =164.4 164.4×12 =1,972.8 20 -24 22 23 22×23=506 -7.82 61.2 1,407.6 25 -29 27 60 27×60=1,620 -2.82 8.0 480.0 30 -34 32 77 32×77=2,464 2.18 4.8 369.6 35 -39 37 38 37×38=1,406 7.18 51.6 1,960.8 40 -44 42 8 42×8=336 12.18 148.4 1,187.2 ∑f= 220 ∑fx=6,560 ∑(M-X ̅ ) 2 f= 8,013.2 Mean (X ̅ )==29.82 Coefficient of SD = SD 𝐗̅ =6.03/29.2 = 0.206
Uses of Standard deviation: • Standard deviation is based on all the observations. • Of all the measures of dispersion, standard deviation is best because it is least effected by fluctuations. • It used in the finding of standard error.
Variance
Variance: • The variance is the arithmetic mean of the squares of sum the deviations for the mean value of the data. • It is represented by s2 or σ2 • Formula for the ungrouped data= s2 or σ2 =
• Ascending order: 16,17,18,19,20,21,22,23,24.
Example: Variance for ungrouped data: • 23,22,20,24,16,17,18,19,21, • Ascending order: 16,17,18,19,20,21,22,23,24. • Mean: 180/9= 20 S.No Observation s (X) Deviation from mean (Dx or d= X-X ̅ ) Square of deviation (X-X ̅ )2 or X 2 1 16 16-20= -4 42 = 16 2 17 17-20= -3 32 = 9 3 18 18-20= -2 4 4 19 19-20= -1 1 5 20 20-20= 0 0 6 21 21-20= 1 1 7 22 22-20= 2 4 8 23 23-20= 3 9 9 24 24-20= 4 16 N= 9 ∑X= 180 ∑X-X ̅ )2 = 60 S2 = S2 =60/9-1 S2 = 7.5
Exercise 1 : Variance for ungrouped data: • 10,2,8,6,15,20,4,5 • Calculate variance for ungrouped data:
Variance for grouped data (continuous series): Variance of grouped data formula: X̅ : Mean M or X: Mid point of class interval. N= frequency The first step in the variance for grouped data is to calculate mean: Mean (X ̅ )= ∑fm / ∑f
Example: Variance for grouped data: Age H1N1 patients 31-35 2 36-40 3 41-45 8 46-50 12 51-55 16 56-60 5 61-65 2 66-70 2 The first step in the variance for grouped data is to calculate mean: Mean (X ̅ )= ∑fm / ∑f
Class interval Mid point (M or X) Frequency (f) Fm or fx 31-35 33 2 66 36-40 38 3 114 41-45 43 8 344 46-50 48 12 576 51-55 53 16 848 56-60 58 5 290 61-65 63 2 126 66-70 68 2 136 ∑f= 50 ∑fm= 2500 Mean (X ̅ )= ∑fm / ∑f = 2500/50 Mean (X ̅ ) = 50
Mean (X ̅ ) = 50 Class interval Mid point (M or X) Freque ncy (f) Fm or fx (X-X ̅ ) or (m-X ̅ ) (X-X ̅ )2 or (m-X ̅ )2 f(X-X ̅ )2 Or f(m-X ̅ )2 31-35 33 2 66 33-50= -17 172 = 289 289×2= 578 36-40 38 3 114 38-50= -12 144 144×3= 432 41-45 43 8 344 -7 49 392 46-50 48 12 576 -2 4 48 51-55 53 16 848 3 9 144 56-60 58 5 290 8 64 320 61-65 63 2 126 13 169 328 66-70 68 2 136 18 324 648 ∑f= 50 ∑fm= 2500 ∑(X-X ̅ )2 = 1052 ∑f(X-X ̅ )2 = 2900 = 2900/50-1 =2900/49= 59.18 SD or S = 59.18 SD or S= 7.69
Co-efficient of Variation (CV) Co-efficient of Variation (CV)= Standard deviation × 100 Mean
Calculate the Variance, standard deviation and co-efficient of variance for the data. Yield of wheat per hectare No of wheat fields 10-20 22 20-30 5 30-40 2 50-60 12 60-70 16 70-80 10 First find SD and followed by CV Co-efficient of Variation (CV)= Standard deviation × 100 Mean
Class interval Mid point (M or X) Freque ncy (f) Fm or fx (X-X ̅ ) Or (m-X ̅ ) (X-X ̅ )2 Or (m-X ̅ )2 f(X-X ̅ )2 Or f(m-X ̅ )2 10-20 22 20-30 5 30-40 2 50-60 12 60-70 16 70-80 10 ∑f= ∑fm= ∑(X-X ̅ )2 = ∑f(X-X ̅ )2 =
Significance of Variance: • It is easy to calculate. • It indicates the variability clearly. • The variance is the most informative among the measures of dispersion for populations. • It is most frequently used measure of variation in data especially with normal, binomial or Poisson distribution.
Biostatistics Standard deviation and variance