Measures of central tendency and data distribution

• • The point of any average is to present the most typical score within a set of data
  • It enables us to:
  • Describe/ summarise a data set
  • establish norms for a sample2
  • compare data sets
  • 3 key measures:
  • MEAN
  • MODE
  • MEDIAN
• The (Arithmetic) Mean:

  • The sum of all the scores in the sample / The total number of scores in the sample
• The Median:

  • The middle value in the sample
  • If there are an even number of data-points then take the middle of those
• The Mode:

  • The most commonly occurring value in the sample
• Distribution of Data:

  • A certain number of assumptions need to be met in order to use certain descriptive statistics and statistical tests
  • How the data are distributed is one important aspect
  • Need to know how data are distributed in order to decide:
  • how to describe the data (i.e. descriptive statistics)
  • what statistical tests to use (i.e. inferential statistics)
• Statistical variation and data distribution:

  • For continuous data (i.e. interval or ratio data) you can assess the distribution within a sample by plotting a histogram
  • Expect most people to cluster around central value
  • Expect the numbers to decrease in more or less a symmetrical manner in each direction
• The Normal Curve:

  • If you plotted a histogram of a variable for the whole population it would look like a bell shaped curve
  • Smooth, symmetrical bell shaped curve indicates that data are normally distributed
  • Larger the sample the smoother the bell shaped curve
• Central Limit Theorem

  • given a *sufficiently large sample size from a population, the mean of all samples from the same population will be approximately equal to the mean of the population
  • *sufficiently large random samples from the population
  • Often our samples are neither large nor randomly chosen
  • So there is usually a fair risk that our data are not normally distributed
• Skewed Distribution:

  • Mean can be distorted by extreme values, particularly where samples are small
  • e.g. very low values or very high values
  • Results in a skewed distribution
• Outliers:

  • An observation point that is distant to other observations
  • May be due to variability within the sample
  • May be due to error
  • May result in a skewed distribution
  • May be real
• Skewed results

  A) negative

  B) positive