B

2. Biostatistics is the application of statistical principles to questions and
problems in medicine, public health or biology. One can imagine that it might
be of interest to characterize a given population (e.g., adults in Boston or all
children in the United States) with respect to the proportion of subjects who
are overweight or the proportion who have asthma, and it would also be
important to estimate the magnitude of these problems over time or perhaps
in different locations. In other circumstances in would be important to make
comparisons among groups of subjects in order to determine whether
certain behaviors (e.g., smoking, exercise, etc.) are associated with a
greater risk of certain health outcomes. It would, of course, be impossible to
answer all such questions by collecting information (data) from all subjects
in the populations of interest. A more realistic approach is to study samples
or subsets of a population. The discipline of biostatistics provides tools and
techniques for collecting data and then summarizing, analyzing, and
interpreting it. If the samples one takes are representative of the population
of interest, they will provide good estimates regarding the population overall.
Consequently, in biostatistics one analyzes samples in order to make
inferences about the population.

3. A fundamental task of biostatistics is to analyze samples in order to make inferences
about the population from which the samples were drawn. To illustrate this, consider
the population of Delhi in 2010, which consisted of 6,547,629 persons. One characteristic
(or variable) of potential interest might be the diastolic blood pressure of the population. It is
obviously not feasible to measure and record blood pressures for of all the residents, but
one could take samples of the population in order estimate the population's mean diastolic
blood pressure.
Despite the simplicity of this
example, it raises a series of
concepts and terms that need to be
defined. The
terms population, subjects, sample,
variable, and data elements

4. It is possible to select many samples from a given population. The simple
example above shows three small samples that were drawn to estimate the
mean diastolic blood pressure of Delhi residents, although it doesn't specify
how the samples were drawn. Note also that each of the samples provided a
different estimate of the mean value for the population, and none of the
estimates was the same as the actual mean for the overall population (78
mm Hg in this hypothetical example). In reality, one generally doesn't know
the true mean values of the characteristics of the population, which is of
course why we are trying to estimate them from samples. Consequently, it is
important to define and distinguish between:
• population size versus sample size
• parameter versus sample statistic.

5. Measures of Central Tendency
A measure of central tendency is a measure which indicates where the
middle of the data is. The three most commonly used measures of central
tendency are: The Mean, the Median, and the Mode.

6. Participant ID Systolic Blood
Pressure
Diastolic Blood
Pressure
Total Serum
Cholesterol
Weight Height Body Mass Index
1 141 76 199 138 63.00 24.4
2 119 64 150 183 69.75 26.4
3 122 62 227 153 65.75 24.9
4 127 81 227 178 70.00 25.5
5 125 70 163 161 70.50 22.8
6 123 72 210 206 70.00 29.6
7 105 81 205 235 72.00 31.9
8 113 63 275 151 60.75 28.8
9 106 67 208 213 69.00 31.5
10 131 77 159 142 61.00 26.8
Data Values for a Small Sample
In order to illustrate the computation of sample statistics,
we selected a small subset (n=10) of participants in the
Center of Heart Study. The data values for these ten
individuals are shown in the table below.

7. The first summary statistic that is important to report is the sample size. In this example the
sample size is n=10. Because this sample is small (n=10), it is easy to summarize the
sample by inspecting the observed values, for example, by listing the diastolic blood
pressures in ascending order:
62 63 64 67 70 72 76 77 81 81
Simple inspection of this small sample gives us a sense of the center of the observed
diastolic pressures and also gives us a sense of how much variability there is. However, for
a large sample, inspection of the individual data values does not provide a meaningful
summary, and summary statistics are necessary. The two key components of a useful
summary for a continuous variable are:
• a description of the center or 'average' of the data (i.e., what is a typical value?) and
• an indication of the variability in the data.

8. There are several statistics that describe the center of the data, but for now we will focus on the sample mean, which is computed by summing all of
the values for a particular variable in the sample and dividing by the sample size. For the sample of diastolic blood pressures in the table above, the
sample mean is computed as follows:
To simplify the formulas for sample statistics (and for population parameters), we usually denote the variable of interest as "X". X is simply a
placeholder for the variable being analyzed. Here X=diastolic blood pressure.
The general formula for the sample mean is:
The X with the bar over it represents the sample mean, and it is read as "X bar". The Σ indicates summation (i.e., sum of the X's or sum of the
diastolic blood pressures in this example).
When reporting summary statistics for a continuous variable, the convention is to report one more decimal place than the number of decimal places
measured. Systolic and diastolic blood pressures, total serum cholesterol and weight were measured to the nearest integer, therefore the summary
statistics are reported to the nearest tenth place. Height was measured to the nearest quarter inch (hundredths place), therefore the summary
statistics are reported to the nearest thousandths place. Body mass index was computed to the nearest tenths place, summary statistics are reported
to the nearest hundredths place.
Sample Mean

11. Measures of Dispersion
A measure of dispersion conveys information regarding
the amount of variability present in a set of data.
• Note:
1. If all the values are the same → There is no
dispersion .
2. If all the values are different → There is a dispersion:
a) If the values close to each other →The amount of
Dispersion small.
b) If the values are widely scattered → The Dispersion is
greater. Measures of Dispersion are :
1.Range (R).
2. Variance.
3. Standard deviation.
4.Coefficient of variation (C.V).

12. Standard Deviation
If there are no extreme or outlying values of
the variable, the mean is the most appropriate
summary of a typical value, and to summarize
variability in the data we specifically estimate
the variability in the sample around the
sample mean. If all of the observed values
in a sample are close to the sample mean,
the standard deviation will be small (i.e.,
close to zero), and if the observed values
vary widely around the sample mean, the
standard deviation will be large. If all of the
values in the sample are identical, the sample
standard deviation will be zero.
X=Diastolic Blood Pressure Deviation from the Mean
76 4.7
64 -7.3
62 -9.3
81 9.7
70 -1.3
72 0.7
81 9.7
63 -8.3
67 -4.3
77 5.7
Table - Diastolic Blood Pressures and Deviations from the Sample Mean
When discussing the sample mean, we found that the sample mean for
diastolic blood pressure = 71.3. The table below shows each of the
observed values along with its respective deviation from the sample
mean.

13. • The deviations from the mean reflect how far each individual's diastolic blood pressure is
from the mean diastolic blood pressure. The first participant's diastolic blood pressure is 4.7 units
above the mean while the second participant's diastolic blood pressure is 7.3 units below the
mean. What we need is a summary of these deviations from the mean, in particular a measure of how
far, on average, each participant is from the mean diastolic blood pressure. If we compute the mean
of the deviations by summing the deviations and dividing by the sample size we run into a
problem. The sum of the deviations from the mean is zero. This will always be the case as it is a
property of the sample mean, i.e., the sum of the deviations below the mean will always equal the sum
of the deviations above the mean. However, the goal is to capture the magnitude of these deviations
in a summary measure. To address this problem of the deviations summing to zero, we could take
absolute values or square each deviation from the mean. Both methods would address the
problem. The more popular method to summarize the deviations from the mean involves
squaring the deviations (absolute values are difficult in mathematical proofs). The table below
displays each of the observed values, the respective deviations from the sample mean and the
squared deviations from the mean.

14. X=Diastolic Blood Pressure Deviation from the Mean Squared Deviation from the Mean
76 4.7 22.09
64 -7.3 53.29
62 -9.3 86.49
81 9.7 94.09
70 -1.3 1.69
72 0.7 0.49
81 9.7 94.09
63 -8.3 68.89
67 -4.3 18.49
77 5.7 32.49

15. The squared deviations are interpreted as follows. The first participant's squared deviation is 22.09
meaning that his/her diastolic blood pressure is 22.09 units squared from the mean diastolic blood
pressure, and the second participant's diastolic blood pressure is 53.29 units squared from the
mean diastolic blood pressure. A quantity that is often used to measure variability in a sample is
called the sample variance, and it is essentially the mean of the squared deviations. The sample
variance is denoted s² and is computed as follows:
Why do we divided by (n-1) instead of n?
The sample variance is not actually the mean of the
squared deviations, because we divide by (n-1) instead of
n. In statistical inference we make generalizations or
estimates of population parameters based on sample
statistics. If we were to compute the sample variance by
taking the mean of the squared deviations and dividing by
n we would consistently underestimate the true population
variance. Dividing by (n-1) produces a better estimate of
the population variance. The sample variance is
nonetheless usually interpreted as the average squared
deviation from the mean.

16. In this sample of n=10 diastolic blood pressures, the sample variance is s2 = 472.10/9 =52.46.
Thus, on average diastolic blood pressures are 52.46 units squared from the mean diastolic blood
pressure. Because of the squaring, the variance is not particularly interpretable. The more common
measure of variability in a sample is the sample standard deviation, defined as the square root of
the sample variance:

18. Try to find:
- sample size
- mode
- median
- range
- sample mean
- sample variance
- sample standard deviation

19. • We usually don't have information about all of the subjects in a population of interest, so
we take samples from the population in order to make inferences about unknown
population parameters.
• An obvious concern would be how good a given sample's statistics are in estimating the
characteristics of the population from which it was drawn. There are many factors that
influence diastolic blood pressure levels, such as age, body weight, fitness, and heredity.
• We would ideally like the sample to be representative of the population. Intuitively, it
would seem preferable to have a random sample, meaning that all subjects in the
population have an equal chance of being selected into the sample; this would minimize
systematic errors caused by biased sampling.
• In addition, it is also intuitive that small samples might not be representative of the
population just by chance, and large samples are less likely to be affected by "the luck of
the draw"; this would reduce so-called random error. Since we often rely on a single
sample to estimate population parameters, we never actually know how good our
estimates are. However, one can use sampling methods that reduce bias, and the degree
of random error in a given sample can be estimated in order to get a sense of the
precision of our estimates.

20. A sample of 10 women seeking prenatal care at Almaty
Medical center agree to participate in a study to assess
the quality of prenatal care. At the time of study
enrollment, you the study coordinator, collected
background characteristics on each of the moms including
their age (in years).The data are shown below:
24 18 28 32 26 21 22 43 27 29
Try to find:
- sample size
- mode
- median
- range
- sample mean
- sample variance
- sample standard deviation