The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

178 Solutions to Selected Exercises

Chapter 3

1. What does a stem - and - leaf diagram show?
   A stem - and leaf diagram has the nice property of describing the shape of the data
   distribution in a way similar to a histogram but without losing information about
   the exact value of the cases with a histogram bin.


2. What is the difference between a histogram and a relative frequency
   histogram?
   A histogram has a bar height that equals the number of cases belonging to the bin
   interval, The relative frequency histogram has the same shape, but the height rep-
   resents the number of cases belonging to the bin interval divided by the total
   number n of cases in the entire sample. So the height of the bar represents a pro-
   portion or percentage of the data falling in the interval (or a frequency relative to
   the total).


3. What portion of the data is contained in the box portion or body of a box -
   and - whiskers plot?
   The bottom of the box is the 25th percentile and the top is the 75th percentile. So
   the box contains 50% of the data.


4. What relationship can you make to the three measures of location (mean,
   median, and mode) for right – skewed distributions?
   For unimodal distributions that are right skewed: mean < median < mode.


5. What is the defi nition of mean square error?
   The mean square error is the average of the squared deviations of the observations
   from their target. Note that the target is not always the mean. Using this defi nition
   one can show that Mean Square Error = B 2 + Variance, where B is the bias (the
   difference between the mean and the target). When the estimate is unbiased, B = 0 ,
   and Mean Square Error = Variance.

Chapter 4

1. What is a continuous distribution?
   A continuous distribution is a probability distribution with a density defi ned on an
   interval, the whole real line, or a set of disjoint intervals.


2. What is important about the normal distribution that makes it different from
   other continuous distributions?
   The normal distribution is a special continuous distribution because of the central
   limit theorem, which states that for most distributions (continuous or discrete)
   the average of a set of n independent observations with that distribution has a
   distribution that is approximately a normal distribution if n is large (usually 30 or
   more).

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