11 Measures of central tendency Arithmetic meanGeometric meanMedianModeWeighted Mean
12 Mean, mx, or average value Mean of a r.v. X is its expected value.Sample estimate is the arithmetic average.Arithmetic mean of grouped data (k is number of groups, n is total number of observations, ni is the number of observations in group i, xi is the class mark of the ith group.
13 Geometric meanUsed when the ratio of two consecutive observations is either constant or nearly constant.The logarithm of the population geometric mean would be the expected value of the logarithm of X.
14 Median, XmdThe observation such that half of the values in the sample lie on either side of Xmd. The median may not exist.Population median, mmd would be the value satisfying:X continuousX discrete
15 Mode, mmoMost frequently occurring value. The sample or population may have none, one or more than one mode.Population mode is the value of X maximizing px(x).X continuousX discrete
16 Weighted meanUsed for describing the central tendancy of grouped data.
17 Measures of Dispersion Measures of the spread of the dataRangeVariance
18 Range Difference between the largest and smallest sample values. For a population this interval often ranges from - ∞ to ∞ or from 0 to ∞.The sample range is a function of only 2 of the sample values, but does convey some idea of the spread of the data.Disadvantage of range: does not reflect frequency or magnitude of values that deviate from the mean.Occasionally use the relative rangeRelative range =
19 Variance, s2 Defined as the second moment about the mean. The average squared deviation from the mean. For a discrete population of size n:Sample estimate of sx2 is sx2
20 Variance Two basic differences between population and sample variance. used instead of mn-1 is used as the denominator rather than n to avoid a biased estimate for sx2Variance of grouped data
22 Units of Variance Units of the variance are the same as units on X2. Units on its positive square root, the standard deviation, sx, are the same as the units of the random variable, X.A dimensionless measure of dispersion is the coefficient of variation, Cv.
23 Measures of Symmetry Many distributions are not symmetrical Tailing off to the right or the left is skewing the distribution.Tailing to the right-positively skewedTailing to the left-negatively skewed
25 Practical measurements of skewness One measure of absolute skew is to measure the difference between the mean and the mode.Not meaningful for comparison sake because it is dependent on units of measure.
26 Pearson’s first coefficient of skewness Relative measure of skewness more useful for comparison.Population skewnessSample skewness
27 Measures of Peakedness (Flatness) Kurtosis refers to the extent of peakedness of a probability distribution in comparison to the normal distribution.Kurtosis is the 4th moment about the mean.Calculate the coefficient of kurtosis, k
29 CovarianceMeasure of the linear relationship between two jointly distributed random variables, X and Y.Covariance is the 1,1 central moment
30 CovarianceIf X and Y are independent:Sample statistic:
31 Correlation Coefficient Normalized covarianceIf X and Y are independent
32 Correlation Coefficient Measure of how two variables vary together.A value of r equal to positive one implies that X and Y are perfectly related by Y=a+bX.Positive values indicate large (small) values of X tend to be paired with large (small) values of Y.Negative values indicate large (small) values of X tend to be paired with small (large) values of Y.Two values are uncorrelated ONLY IF r(x,y)=0.Correlation does NOT equal cause and effect.
33 Correlation coefficient: Linear dependence and functional dependence