Descriptive Statistics

Descriptive Statistics

1. Basic Concepts

   • Population
     • A Population is a complete set of all possible outcomes from sone measurable process that we are interested in.  e.g: local taxes paid per household in UK Incomes of families in the UK Starting salaries of Graduates
   • Sample
     • A subset of all the possible values of a population  e.g Local taxes paid per household for 1000 households in the Uk Incomes of 100 families in the Uk Starting salaries of Lancaster Graduates
   • Random Sample
     • Most reliable way to pick a truly representative sample is to pick values from the whole population at random. A sample in which each member of the population has an equal chance of occuring in the sample
   • Probability & Statistics
     • Probability: shows how far a sample characteristic can be from a population characteristic when we know the population Statistics: shows what population characteristics are consistent with the sample 

2. Graphical Methods

   • Histograms
     • Estimate of the probability density function 
     • Frequency
       • Number in sample between x and y 
     • Relative Frequency
       • Relative freq of data in sample between x & y/ Number in sample 
     • Density
       • Number in category/ Number in sample x size (in units) of category
   • Ogive
     • A plot of the cumulative frequency of values less than or equal to sample outcomes 

3. Measures of location

   • describe the central point at which the data is distributed around i.e location of the data
   • Sample Mean
     • Sum of the sample values divided by the total number of values
   • Sample median
     • Value for which half of the sample are less than/ equal to this value  when n is odd = (n+1)/ 2  when n is even = n/2 & n+2/2
   • Sample Mode
     • The mode of a sample is the value that occurs most frequently.
   • Sample Quantiles
     • A pth sample quantile is the value for which the proportion of the sample is less than or equal to this value p
     • Upper Quartile
       • 0.75 x (n+1)
     • Lower Quartile
       • 0.25 x n + 1

4. Measures of Spread

   • Sample Variance
     • average of the squared differences of the data from the sample mean. s2
   • Sample Standard Deviation
     • Square root of the sample variance
   • IQ Range
   • Range
     • The range of a sample is the difference between the largest and smallest data values