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## BIOSTATISTICS -III

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**BIOSTATISTICS -III**GENERALIZATION OF RESULTS OF A SAMPLE OVER POPULATION**RECAP**• Types of data, variables , and scales of measurement • Types of distribution of data , the concept of normal distribution curve and skewed curves • Measures of central tendency (mean, median, mode) • Measures of data dispersion or variability, concept of variance and standard deviation, standard normal curve with standard deviation**STANDARD ERROR-DEFINITION**• Standard error is the measure of extent to which the sample mean deviates from true population mean. • It helps in determining the confidence limits within which the actual parameters of population of interest are expected to lie • It is used as a tool in tests of hypothesis or tests of significance.**STANDARD ERROR-CONCEPT**• Estimation of population parameters from results/ statistics of sample mean involves two factors • Standard deviation of the population of interest & • Sample size • The relationship of population standard deviation to sample size is STANDARD ERROR (SE) SE= SD/√n**FORMULAE FOR ESTIMATION OF STANDARD ERROR(SE) OF SAMPLE**• 1. SE of sample mean= SD/ √n • 2. SE of sample proportion(p) = √pq/n • 3. SE of difference between two means[SE(d)]=√SD1/ n1+ SD2/n2 • 4. SE of difference between two proportions= √p1q1/n1+ p2q2/n2**SE/ SEM (standard error of mean)**• SE is inversely related to square root of sample size ( the larger the sample ,closer the sample mean to population true mean) • Z scores can be calculated in terms of standard error by which a sample mean lies above or below a population mean • Z = x - µ / σ**REFERENCE RANGES**• The 95% limits( REFER TO 2 Std deviations on either side of mean) and are referred to as REFERENCE RANGE • For many biological variables they define what is regarded as the NORMAL RANGE OF THE NORMAL DISTRIBUTION**CONFIDENCE INTERVAL**• As standard error(the relation between sample size and population standard deviation) is used for estimation of population mean µ, formula is µ = X ± 2 SE • the variation in distribution of the sample means can also be quantified in terms of MULTIPLES OF STANDARD ERROR(SE)**Conventionally!!!!!!!!**• 1.96 /2 SE on either side of mean is taken as the limit of variability. • These values are taken as CONFIDENCE LIMITS with intervening difference being THE 95% CONFIDENCE INTERVAL which Gives an estimated range of values which is likely to include an unknown” POPULATION PARAMETER” .**WIDTH OF CONFIDENCE INTERVAL**• Reflects how uncertain we are about an unknown parameter • A wider confidence interval may indicate need for collection of more data before commenting on the population parameter**Reference range vsconfidence interval**• Reference range refers to individuals in populations with standard deviations • Confidence interval refers to standard error in data estimated from samples**Confidence interval for difference between two means**• It specifies the range of values within which the means of the two populations being compared would lie as they are estimated from the respective samples • If confidence interval includes “ZERO” we say, “THERE IS NO SIGNIFICANT DIFFERENCE BETWEEN THE MEANS OF THE TWO POPULATIONS AT A GIVEN LEVEL OF CONFIDENCE**THE 95 % CONFIDENCE INTERVAL**• Means we are 95% sure or confident that the estimated interval in sample contains the true difference between the two population means (the basic concept remains one of capturing 95% of data within 2 standard deviations of the standard normal curve of distribution of data in nature) • Alternately, 95% of all confidence intervals estimated in this manner (by repeated sampling ) will include the true difference**Sample of 100 women , Hb 12 gmstandard deviation( 0- 2gm)**• µ= X ± 2 SE OR X ± 2 SD/√N • µ (ci)= 12±[ 2x 2/√100 • =12±[4/10or0.4] • µ (ci)= 12± 0.4 • =11.6- 12.4 • INTERPRET ????**ROLE OF SAMPLE SIZE AND SD**• µ= X ± 2 SE OR X ± 2 SD/√N • µ (ci)= 12±[ 2x 2/√9 • =12±[4/3or 1.33] • µ (ci)= 12±1.33 • =10.66- 13.33 • INTERPRET ????**LARGER SD OF 4 GM% ?**• µ= X ± 2 SE OR X ± 2 SD/√N • µ (ci)= 12±[ 2x4/√9 • =12±[8/3or 2.66] • µ (ci)= 12±2.66 • =9.33- 14.66 • INTERPRET ????**SMALLER SD 0F 0.5 GM Hb**• µ= X ± 2 SE OR X ± 2 SD/√N • µ (ci)= 12±[ 2x0.5/√9 • =12±[1/3or 0.33] • µ (ci)= 12±0.33 • =11.6- 12.33 • INTERPRET ????**Comment about sample authenticity if true population mean**is known(11.2gm) • µ= X ± 2 SE OR X ± 2 SD/√N • µ (ci)= 12±[ 2x 2/√100 • =12±[4/10or0.4] • µ (ci)= 12± 0.4 • =11.6- 12.4 • What about sample mean’s predictive value • ?????? Representative of population under study or not?????????**Difference of proportion5200 workers in total population of**10000,(52%) sample of 100 individuals with 0.4 or 40% workers • What is the possible range of workers we expect to find in the sample of 100 with 95% confidence? • What conclusions/comments will be drawn about authenticity of sample under consideration?**Standard error of proportionp= probability of being workerq=**probability of being non worker • P(in pop)= 52% q(in pop)= 48% !!!!!! • SE for proportion= √pq/n= √52x48/100=√25=5 • P (CI)= p ± 2 SE = 52± 2 x5 = • 42% -62% • { sample’s proportion of workers = 40%} • COMMENT ????????????????**difference between two proportions**• Proportion of measles infection after vaccination with vacc A(p1) = 22/90=0.244(24.4%) q1= 100-2.44= 75.6% • Proportion of measles infection after vaccination with vacc B (p2) = 14/86 = 0.162(16.2%) q2= 100- 16.2= 83.3% Difference p1-p2= 24.4-16.2= 8.2**Standard error of difference between two proportions**• SE =√p1q1/n1 +p2q2/n2 = √24.4x75.6/90+16.2x83.8/86 = √ 20.79 +15.76 = √ 36.27 = 6 Difference p1-p2= 24.4-16.2= 8.2 FOR CI REMEMBER 2±SE SO SE= 4- 8 ( what about 8.2????) COMMENT !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!**THANK YOU**FOR APPRECIATING LOGIC OF BIOSTATISTICS