Descriptive statistics

[MG1:Chp2, p7-p18]

 

 

Summaries of sample data (statistics) are defined by Roman letters (sample mean)

Summaries of population data (parameters) are defined by Greek letters (mu, variance)

 

 

 

Central tendency = The extent that observations cluster

Degreee of dispersion = The spread of the observations about a central location

 

Measures of central tendency

• Mode = The most common value
• Median = The middle value
• (Arithmetic) Mean = The average value

 

Degree of dispersion

• Range = Difference between the maximum and minimum value
• Percentile = Rank observations into 100 equal parts
  * Mean = 50th percentile
  * Interquartile range = 25th to 75th percentile
• Sample Variance = Sum of squares divided by degree of freedom
  * Sum of squares = sum of the square of each differences (between each observation and the mean)
  * Degree of freedom = number of observation minus 1
• Population variance = Sum of squares divided by number of observation
• Standard deviation = Square root of variance
• Coefficient of variation (CV) = SD / mean x 100%

NB:

• Degree of freedom is used when calculating the variance of a sample
  * Because each observation is free to vary except for the last one which must be a defined value in order for the mean match the fixed sample mean value

Sources of variability

• Biological variability
• Measurement imprecision
  --> Resulting in random error
• Mistakes or biases in measurement
  --> Systemic error

Standard error (SE)

[MG1:p9]

• Standard error (SE)
  = aka standard error of the mean
• SE = SD / square root of n
• SE is NOT meant to be used to describe variability of sample data
• SE is a measure of precision (of how well sample data can be used to predict population mean (a population parameter))
  * Used to calculate confidence interval
  * Often derived from one sample
  * Reliability of sample mean in predicting population mean [Chris Flynn]
• SE is the standard deviation of the sample means
• Increasing sample size can be a way of reducing SE
  * But need to increase sample 4 times to reduce SE by half

Confidence interval

• Derived from SE
• 95% confidence interval of the mean = sample mean +/- (1.96 x SE)
• 99% confidence interval = sample mean +/- (2.58 x SE)
• Definition of 95% CI
  = The range within which there is 95% probability the true population mean may lie

NB:

• In a normal distribution, 95% of the observations lie within 1.96 standard deviation of the mean

Frequency distributions

• Kurtosis describes how peaked the distribution is
  * Kurtosis of a normal distribution = 0
• Median is a better measurement of central tendency in a skewed distribution
  * Skew to the right, median will be smaller than the mean
• Bimodal distribution = Distribution with two peaks
  --> Suggests that the sample is not homogeneous and may represent two different populations

Normal distribution

• Sometimes referred to as a Gaussian distribution
• Two parameters define the curve, mu (the mean), and sigma (the standard deviation)
• Mode = median = mean
• Formula at [MG1:p13]

NB:

• Mean +/- 1 SD includes 68% of total area 
• Mean +/- 1.96 SD includes 95% of total area
• Mean +/- 2 SD includes 95.4% of total area
• Mean +/- 3 SD includes 99.7% of total area

Z distribution

In a STANDARD normal distribution

• Mean = 0
• Standard deviation = 1
• aka the z distribution
• A z transformation converts any normal distribution curve (with different mean and SD) to a standard normal distribution curve (mean = 0, SD = 1)
  * z = (x - mu)/SD

Central limit theorem

[MG1:p14]

• As the number of observations increase (n>100)
  --> The shape of a sampling distribution will approximate a normal distribution curve
  * Even if the distribution of the variable is not normal

Binomial distribution

[MG1:p14-p15]

Formula at [MG1:p15]

A binomial distribution exists if a population contains items which belong to one of two mutually exclusive categories
* e.g. gender, complication

Conditions include:

• Fixed number of observations (trials)
• Only two outcomes are possible
• Trials are independent
• Constant probability for occurrence of each event

Poisson distribution

• A binomial distribution approximates Poisson distribution when
  * The number of observation is very large, AND
  * Probability of an event is small (<0.05)
• A single parameter (lamda) which is both mean and the variance

Conditions:

• Events occur randomly
• Events occur independently
• Events occur uniformly (same probability) and singly

Example used in [MG1:p15] is for calculation of probability of more than one admission on late night admission

Incidence and prevalence

• Incidence = the number of individuals who develop a condition (i.e. new cases) in a given time period
  --> An estimation of probability of developing a disease in a specified time period
• Prevalence = the number of individuals with a condition at a point of time (i.e. total cases, pre-existing and new)

Presentation of data

[MG1:p17]

• For a normal distribution, mean and standard deviation are the best statistics to describe data
  * But mean can be affected by extreme values
• A bimodal distribution is best described with mode
• Ordinal data should be described with mode or median

Box and whisker plot

• Used to depict mean, interquartile range and range
• Middle line = median
• Box = 25th to 75th percentiles
• Whiskers = minimum and maximum, or 5th and 95th percentiles