Statistics

Statistics

1. Descriptive statistics

   • Descriptive statistics has to do with the collection, organization, overview and presentation of sample data. 

2. Inferential statistics

   • Inferential statistics has to do with obtaining inferences or conclusions (via conjectures) about populations based on information taken from the samples.
   • Statistics
     • A statistic is a measure of overview that is calculated to describe a characteristic of a single sample of the population.   
   • Sample
     •  A sample is the portion of the population that is selected for analysis.   
   • Population
     • A population (or universe) is the totality of the elements or things under consideration.     
   • Parameter
     •  A parameter is a measure of overview that is calculated to describe a characteristic of a population.   

3. Random variables

   • Qualitative
     • Qualitative random variables produce categorical responses, i.e., to describe an element of a population.
   • Quantitative
     • Quantitative random variables produce numerical answers; quantify an element of the population.
     • Discrete
       • Discrete data are numerical responses arising from a process of counting.
     • Continous
       • Continuous data are responses arising from a process of measurement.   

4. Measurement Scales

   • Nominal
     • Only satisfies the identity property of measurement. Values assigned to variables represent a descriptive category, but have no inherent numerical value with respect to magnitude.   
   • Ordinal
     • The ordinal scale has the property of both identity and magnitude. Each value on the ordinal scale has a unique meaning, and it has an ordered relationship to every other value on the scale.   
   • Interval
     • The interval scale of measurement has the properties of identity, magnitude, and equal intervals.   
   • Ratio
     • The ratio scale of measurement satisfies all four of the properties of measurement: identity, magnitude, equal intervals, and a minimum value of zero.   
   • Properties
     • Identity
       • Each value on the measurement scale has a unique meaning.
     • Magnitude
       • Values on the measurement scale have an ordered relationship to one another. That is, some values are larger and some are smaller.   
     • Equal intervals
       • Scale units along the scale are equal to one another
     • A minimum value of zero
       • The scale has a true zero point, below which no values exist.   

5. Sampling methods

   • Non-probability samples
     • With non-probability sampling methods, we do not know the probability that each population element will be chosen, and/or we cannot be sure that each population element has a non-zero chance of being chosen.   
     • Voluntary
       • A voluntary sample is made up of people who self-select into the survey. Often,these folks have a strong interest in the main topic of the survey.  
     • Convenience
       • A convenience sample is made up of people who are easy to reach.  
   • Probability samples
     • With probability sampling methods, each population element has a known (non-zero) chance of being chosen for the sample.
     • Simple Random
       • Simple random sampling refers to any sampling method that has the following properties.    - The population consists of N objects.   - The sample consists of n objects.   - If all possible samples of n objects are equally likely to occur, the sampling method is called simple random sampling.   
     • Systematic Random
       • With systematic random sampling, we create a list of every member of the population. From the list, we randomly select the first sample element from the first k elements on the population list. Thereafter, we select every kth element on the list. i+(n-1)k  
     • Cluster
       • With cluster sampling, every member of the population is assigned to one, and only one, group. Each group is called a cluster. 
     • Stratified
       • With stratified sampling, the population is divided into groups, based on some characteristic. Then, within each group, a probability sample (often a simple random sample) is selected. 
     • Multistage
       • With multistage sampling, we select a sample by using combinations of different sampling methods. 

6. Visual displays of data

   • Displays
     • Line Graphs
     • Circle Graphs
     • Bar Graphs
     • Stem and Leaf
     • Histogram
   • Frecuency distribution
     • Lists the distinct data values (x) along with their frequencies (f).
     • Frecuency
       • Designates the number of times the corresponding item occurred in the data set.
     • Relative frecuency
       • This is the fraction, or percentage, of the data set represented by the item.
     • Grouped frecuency distribution

7. Measures of Central Tendency

   • Median
     • Divides a group of members into two parts, with half the numbers below the median and half above it.  Position of median = n + 1 / 2 = ∑f + 1 / 2
   • Weighted mean
     • The weighted mean of a group of (weighted) items is the sum of all products of items times weighting factors, divided by the sum of all weighting factors.  w = ∑ (x∙f) / ∑f
   • Mean
     • The mean of a set of data items is found by adding up all the items and then dividing the sum by the number of items. x̄ = ∑x / n
   • Mode
     • The mode of a data set is the value that occurs most often. 

8. Measures of dispersion

   • Standar deviation
     • s = sqrt [ ∑(x - x̄) ^2 / n - 1 ]
   • Range
     • The range of a data set is a straightforward measure of dispersion. Range = (gratest value in the set) - (least value in the set) 
   • Chevyshev's Teorem
     • 1 - 1/k^2
   • Coefficient of Variation
     • It expresses the standard deviation as a percentage of the mean. V = s / x̄ ∙100  

9. Measurement of position

   • The z-score
     • If x is a data item in a sample with mean x̄ and standard deviation s, then the z-score of x is calculated as follows: z = x-x̄ / s
   • Xctiles
     • Percentiles
       • If approximately n percent of the items in a distribution are less than the number x, then x is the nth percentile of the distribution, denoted Pn.
     • Deciles
       • Deciles are the nine values (denoted D1, D2..., D9) along the scale that divide a data set into ten (approximately= equal-sized parts.
     • Quartiles
       • Quartiles are the three values (Q1, Q2, and Q3) that divide a data set into four (approximately) equal-sized parts. Q1 the median of the lower values. Q2 median Q3 the median of the upper values.
   • Box and plot
     • A box plot involves the median (a measure of central tendency), the range (a measure of dispersion), and the first and thrid quartiles (measures of position), all incorpored into a simple visual display.