Transcript BIOSTATISTICS -III

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
=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 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 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THANK YOU
FOR APPRECIATING
LOGIC OF
BIOSTATISTICS