Describing Data A Single Variable The essence of descriptive statistics

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Describing DataA Single VariableThe essence of
descriptive statistics

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Exposed versus Control
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Exposed Subjects
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Comparison of Two Systems
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Shapes of distributions

• Number of peaks unimodal (most common), bimodal,
  etc.
• Unimodal distributions
• Symmetrical
• Skewed
• Positively skewed long tail at the upper end
• Negatively skewed long tail at the lower end

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Skewed Distributions
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Measures of dispersion

• Rangethe difference between the highest and
  lowest values. Easily understood, but ignores
  all the values in between
• Interquartile range
• The variance (S)
• How far each individual observation is away from
  the (arithmetic mean)

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Degrees of freedom

• The divisor n-1 in S2.
• The number of independent quantities making up
  the statistics that are free to vary
• The number of n original observations and their
  mean, which, of course, is not independent of
  them

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Measures of dispersion

• The standard deviation
• The square root of variance (SD or S), also
  called the root mean square deviation
• If one distribution is more spread out than
  another, then it has a larger standard deviation
• Commonly used to compare the variability of some
  measurement in two groups

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Normal Distribution

• Also called Gaussian
• It is not common or typical, but approximate
  normality of some distribution can be a great
  help
• Unimodal, symmetrical and bell-shaped
  (not all such distributions are normal, though)
• Mean, median and mode are equal in value

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Normal Distribution

• All its properties are characterized completely
  by its mean and standard deviation
• i.e. two normal distributions with the same means
  and SDs are identical

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• Mean m
• SD s
• 50 below, 50 above the mean
• 68.27 between
• m-s and ms
• 95 within m1.96s
• 2.5 in each of the two tails

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Three different distributions

• 1. The distribution of the variables in the
  sample
• e.g. 100 patients observed, mean
• 2. Underlying distribution in the population of
  all such patients
• we dont know what this distribution looks
  like, but assume that the sample was
  representative, to estimate population mean m
• 3. The sampling distribution of the mean
• theoretical distribution

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Standard Error

• Examine the distribution of all the possible
  means from a sample of a given size n
• All possible sample means have a normal
  distribution, and their mean is equal to the
  unknown population mean m
• The standard deviation, called the standard error
  of the mean is equal to

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• Normal distributions are rare in medical
  applications
• Non-normality is not a major problem
• As long as the distribution of the original
  variable is not very skewed, the sampling
  distribution of its mean is approximately normal
• This approximation improves as the sample size
  increases
• We can still use normal distribution for sampling
  distribution of the mean

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• As n increases, SE decreases
• SE increases with increasing variability in the
  population
• If the distribution is normal
• MeanSD contains about 68 of
  observations
• Mean1.96SD 95 of
  observations
• In general
• MeanSE 68 confidence interval for the
  mean
• Mean1.96SE 95 confidence interval for the mean

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Coefficient of Variation

• Used to compare variability of some measurement
  in two groups when the two groups have very
  different means
• The variation relative to the mean

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Important concepts
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• Variables and data come in two basic forms
• Qualitativealso known as categorical or nominal
• Quantitativealso known as metric or numerical
• A qualitative variable is called
• Ordinal if there is an order in the categories
• Binary, dichotomous or attribute if it has two
  categories

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• Presenting qualitative data
• Count the number of persons in each category
• Use a table, bar chart or pie chart

• A numerical distribution can be summarized by
  giving descriptions or measures of
• Its shape
• Where its centre is
• How spread out it is

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• Shapes of quantitative distributions
• Symmetrical or skewed
• Unimodal or bimodal (if symmetrical)

• Measures of the centre of a quantitative
  distribution
• Arithmetic mean (grouped and weighted mean)
• Median (used for skewed data)
• Mode
• Geometric mean (used for positively skewed data)

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• Measures of the spread in variability of
  distribution
• Range
• Interquartile range
• Variance
• Standard deviation
• Coefficient of variation

• The normal distribution is
• Is a bell-shaped distribution
• Has certain area properties
• In particular, 95 of observation in a normal
  distribution lie within mean 1.95 (standard
  deviations)

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• The sampling distribution of a statistic
• Is the distribution of potential values of that
  statistic that could be obtained in repeated
  samples of the same size from a population
• The standard error of a statistic
• Is the standard deviation of the sampling
  distribution of that static
• Measures the precision of the sample value of
  that statistic as an estimator of the true
  population value
• The smaller the standard error is, the more
  precise the estimate
• Decreases with increasing sample size

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• A confidence interval (CI)
• Gives the precision of a sample estimate
• Puts a error on a sample statistic
• (but sometimes the CI cannot be expressed simply
  as statistic error, like with log-normal
  distribution)
• Is a range of values surrounding a sample
  statistic within which, at a given level of
  confidence, the true value of the population
  parameter is likely to be found

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• Statistic Formula
  
• Arithmetic mean
• Weighted mean
• Median middle number
• Mode most frequent number
• Geometric mean
• Range maximum-minimum
• Interquartile range IQR75th-25th percentile
• Variance
• Standard deviation
• Coefficient of correlation