STATISTICS CHAPTER 2

STATISTICS CHAPTER 2

1. MEASURES OF CENTRAL TENDANCY AND LOCATION

   • First and basic question asked by researchers is whether the data in their sample is centered around a particular value
   • we'll present numerical measurements which provide the typical observation of the data and that are refferred to measures of central tendancy
   • MEAN, MEDIAN, MODE
     • ARITHMETIC MEAN
       • Sum of the data values divided by n° of observations
     • MEDIAN
       • Middle observation from a set of observations arranged in increasing/decreasing order
       • In case of even numbers, it will be the n° located in the 0.5(n+1) position
     • MODE
       • Most frequently occurring value
       • Unimodal, bimodal and multimodal observations can exist.
       • Most commonly used with categorical data
     • The decision on whether mean, median mode to use deoends on the context.
       • Categorical data best described by median or mode, not mean
   • SHAPE OF A DISTRIBUTION
     • Median is the prefferred measure to describe the distribution of incomes in a city, state or country
       • Distribution incomes often right skewed since it contains a relative small proportion of high values
         • It is NOT implied that the median should always be prefferred to the mean when population sample is skewed.
   • GEOMETRIC MEAN
     • Is the nth root of the product of n numbers
       • the geometric mean rate of return is the answer -1
       • eg. 20 and 5. ar. mean is 12.5, geom. mean is 10
   • PERCENTILES AND QUARTILES
     • Data must be arranged in ascending order
     • Percentiles separate large ordered data sets into 100ths, where the 50th is the median
     • Quartiles are descriptive measures that separate large datasets into four quartiles
       • Q1 separates smallest 25% of data
       • Q2 is the median
       • Q3 separates the s% of data from remaining largest 25% of data
       • In describing numerical data, we often refer to 5 number summary
         • 5 number summary refers to five descriptive measures: minimum, first quartile, media, third quartile, maximum
           • Minimum<Q1<median<Q3<Maximum

2. MEASURES OF VARIABILITY

   • Mean alone provides insufficient data
     • 2 sets of data may have the same mean but vary in individual observations
   • Variation exists in all areas
   • RANGE AND INTERQUARTILE RANGE
     • The range is the difference between the largest and smallest observations
       • The larger the spread of data, the larger the range will be
     • The Interquartile Range (IQR) measures the spread in the middle 50% of data.
       • Difference between Q3 and Q1
   • BOX AND WHISKER PLOT
     • A box and whisker plot is a graph describing the shape of the distribution in terms of the five number summary.
     • Inner box shows numbers ranging from first to third quartile
       • Line drawn in the box through median
     • 2 whiskers
       • 25th percentile to min.
       • 75th percentile to maximum
   • VARIANCE AND STANDARD DEVIATION
     • The variance is the sum of the squared differences between each observation and the population mean divided by population size.
     • The standard deviation is the square root fo the variance.
       • Used to compute risk
   • COEFFICIENT OF VARIATION
     • Is a measure of relative dispersion that expresses the standard deviation as a percentage of the mean
   • CHEBYSHEV'S THEOREM
   • Empirical rule
     • For many large populations, the empirical rule provides the estimate of the approximate percentage of observations that are contained within one, two or three standard deviations of the mean
   • Z-SCORE
     • Standardized value indicating the number of standard deviations a value is from the mean.
     • Z-Score >0 means value is greater than the mean
     • Defined as follows:

3. WEIGHTED MEAN AND MEASURES OF GROUPED DATA

   • Weighted mean of a set of data is
     • wi is weigth of ith observation and n=
   • APPROXIMATE MEAN AND VARIANCE FOR GROUPED DATA
     • Suppose the data are grouped into k classes with frequencies f1, f2, f3...fk.
       • If midpoints of these classes are m1, m2,...mk then the sample mean is:

4. MEASURES OF RELATIONSHIPS BETWEEN VARIABLES

   • COVARIANCE
     • Measure of the linear relationship between two variables.
       • Positive value=direct/increasinf linear relationship
       • Negative value indicates decreasing relationship
     • Population covariance is
     • DOES NOT PROVIDE A MEASURE FOR THE STRENGTH OF THE RELATIONSHIP BETWEEN TWO VARIABLES
   • CORRELATION COEFFICIENT
     • Computed by dividing the variance by the product of the standard deviations of the two variables
       • Population coefficient:
       • Sample correlation coefficient:
     • Ranges from -1 TO 1