GENERALIZATION OF RESULTS OF A SAMPLE OVER POPULATION 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 vs confidence 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

Practice and clarification time!!!!

Sample of 100 women , Hb 12 gm standard 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 =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 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 proportion 5200 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 proportion p= probability of being worker q= 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 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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