Descriptive statistics

• Statistics are all around us, every day
• Sample statistics are estimates of population parameters
• Samples are limited to measuring characteristics of a portion of the population
• Inferential statistics

  • Used to make convincing statements & confident conclusions based on results
  • Statistical tests calculate the probability (p) that the observed effects/relationships happened by random chance
  • Probability = effects/relationships due to natural variability
  • Probability < effects/relationship enough to be real
• Mean: Sum of all observations divided by the number of observations
• Median: The middle measurement in an ordered set of data, less sensitive to extreme values than mean - more robust measure in some cases
• Applications of statistics

  • Advertising campaigns
  • Trends
  • Evaluating crime patterns
  • GDP
• Descriptive statistics

  • Used to summarise data
  • Measure of central tendency: Mean, median, mode
  • Measure of spread or dispersion: Standard deviation, variance, standard error, range
  • Help describe a sample from a population
• Population parameters describe characteristics of a statistical population
• As sample size (n) increases, sample statistics become more accurate - estimates of the true population
• Mean is sensitive to extreme values in a population
• Mode can also be used as a measure of central tendency for nominal and ordinal data
• Sample variance (s2)

  Based on the sum of squares (SS), which is the sum of the squared deviations of each observation from the sample mean
• Population variance (σ2) is the mean of SS divided by the mean square, where N is the population size
• SD is the most commonly reported measure of variability for the mean when describing a sample, reported as mean ± SD
• Median is the middle measurement in an ordered set of data, having an equal number of observations on either side and is less sensitive to extreme values than mean
• Mode


  • Most common value in a dataset, not sensitive to extreme values
• When reporting the mean, one must also report one of these measures of variability: Sample Variance (s2), Standard deviation of the sample (SD), Standard error of the sample mean (se), 95% confidence limits
• Sample variance (s2) is the mean of SS divided by the mean square
• Sample Standard deviation (SD) is the average deviation of observations from the sample mean and has the same units as the original measurements
• Standard error (SE) calculation

  se = SD/√n
• Coefficient of Variation (V)

  Used to compare variability (SD) between samples that have different means
• SD is the most commonly reported measure of variability for the mean when describing a sample
• Degrees of Freedom (d.f.) often = n-1 when the sample variance is calculated
• Range is the difference between the smallest and largest value observed
• SD


  Square root of the variance
• Measures of variability for the median

  • Range
  • Interquartile range
• Interquartile Range is the range of values in the 2nd and 3rd quartiles (middle 50% of the values)
• Standard error (SE) is a measure of the precision or uncertainty around the estimate of the population mean
• 95% Confidence limits of the Mean are a measure of the precision of the estimate of the mean
• Interquartile range calculation


  1. Lowest 25% of the values = 1st quartile

  2. Next 25% = 2nd quartile

  3. Next 25% = 3rd quartile

  4. Last 25% = 4th quartile
• Interquartile range

  Range of values in the 2nd and 3rd quartiles (middle 50% of the values)
• If a firm increases advertising, their demand curve shifts right, increasing the equilibrium price and quantity
• Frequency or number of each observed value/category in a dataset is used to estimate the probability of events happening
• Mean, Median & Mode are similar in value or approach the same value in large datasets
• If you add up marginal utility for each unit, you get total utility
• Marginal utility

  Additional utility (satisfaction) gained from the consumption of an additional product
• Median

  Middle-most value & forms the boundary between 2nd and 3rd quartiles
• Frequency distributions of ratio/interval data often take the shape of a normal distribution, approximating to a symmetrical and bell-shaped curve
• Data presentation: Boxplots