of the percentile P y in the array sorted in ascending order The value of L y

# Of the percentile p y in the array sorted in

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) of the percentile ( P y ) in the array sorted in ascending order. The value of L y may or may not be a whole number. In general, as the sample size increases, the percentile location calculation becomes more accurate; in small samples it may be quite approximate. As an example of the case in which L y is not a whole number, suppose that we want to determine the 60th percentile of annual returns for the past five years as of 30 October 2018( Q 3 or P 75 ) for the 15 European equity markets given in Exhibit 12. According to Equation 8, the position of the 60th percentile is L 60 = (15 + 1)(60/100) = 9.60, or between the 9th and 10th items in Exhibit 14, which ordered the returns into ascending order. The 9th item in Exhibit 14 is the return to equities in Ireland, 6.81 percent. The 10th item is the return to equities in Norway, 7.20 percent. Reflecting the “0.60” in “9.60,” we would conclude that P 60 lies 60 percent of the distance between 6.81 percent and 7.20 percent. To summarize: When the location, L y , is a whole number, the location corresponds to an actual observation. For example, if we were determining the third quartile, then L y would have been L 75 = (15 + 1)(75/100) = 12, and the third quartile would be P 75 = X 12 , where X i is defined as the value of the observation in the i th ( i = L 75 ) position of the data sorted in ascending order (i.e., P 75 =8.16). When L y is not a whole number or integer, L y lies between the two closest integer numbers (one above and one below), and we use linear interpolation between those two places to determine P y . Interpolation means estimating an unknown value on the basis of two known values that surround it (lie above and below it); the term “linear” refers to a straight-line estimate. Returning to the calculation of P 60 for the equity returns, we found that L y = 9.60; the next lower whole number is 9 and the next higher whole number is 10. Using linear interpolation, P 60 X 9 + (9.60 − 9) ( X 10 X 9 ). As above, in the 9th position is the return to equities in Ireland, so X 9 = 6.81 percent; X 10 = 7.20 percent, the return to equities in Norway. Thus our estimate is P 60 X 9 + (9.60 − 9.00)( X 10 X 9 ) = 6.81 + 0.60 [7.20 − 6.81] = 6.81 + 0.60(0.39) = 6.81 + 0.23 = 7.04 percent. In words, 6.81 and 7.20 bracket P 60 from below and above, respectively. Because (8)
Other Measures of Location: Quantiles Member Use Only 37 C F A I N S T I T U T E M E M B E R U S E O N LY 9.60 − 9 = 0.60, using linear interpolation we move 60 percent of the distance from 6.81 to 7.20 as our estimate of P 60 . We follow this pattern whenever L y is a non-integer: The nearest whole numbers below and above L y establish the posi- tions of observations that bracket P y and then interpolate between the values of those two observations. Example 9 illustrates the calculation of various quantiles for the dividend yield on the components of a major European equity index. EXAMPLE 9 Calculating Percentiles, Quartiles, and Quintiles The EURO STOXX 50 is an index of 50 publicly traded companies, which provides a blue-chip representation of supersector leaders in the Eurozone. Exhibit 22 shows the market capitalization on the 50 component stocks in the index in November 2018. The market capitalizations are ranked in ascending order.