) of the percentile (Py) in the array sorted in ascending order. The value of Lymay 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 Lyis 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(Q3or P75) for the 15 European equity markets given in Exhibit 12. According to Equation 8, the position of the 60th percentile is L60= (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 P60lies 60 percent of the distance between 6.81 percent and 7.20 percent.To summarize:■■When the location, Ly, is a whole number, the location corresponds to an actual observation. For example, if we were determining the third quartile, then Lywould have been L75= (15 + 1)(75/100) = 12, and the third quartile would be P75= X12, where Xiis defined as the value of the observation in the ith (i= L75) position of the data sorted in ascending order (i.e., P75=8.16).■■When Lyis not a whole number or integer, Lylies between the two closest integer numbers (one above and one below), and we use linear interpolationbetween those two places to determine Py. 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 P60for the equity returns, we found that Ly= 9.60; the next lower whole number is 9 and the next higher whole number is 10. Using linear interpolation, P60≈ X9+ (9.60 − 9) (X10− X9). As above, in the 9th position is the return to equities in Ireland, so X9= 6.81 percent; X10= 7.20 percent, the return to equities in Norway. Thus our estimate is P60≈ X9+ (9.60 − 9.00)(X10− X9) = 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 P60from below and above, respectively. Because (8)
Other Measures of Location: Quantiles ■Member Use Only37C F A I N S T I T U T E M E M B E R U S E O N LY9.60 − 9 = 0.60, using linear interpolation we move 60 percent of the distance from 6.81 to 7.20 as our estimate of P60. We follow this pattern whenever Lyis a non-integer: The nearest whole numbers below and above Lyestablish the posi-tions of observations that bracket Pyand 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 QuintilesThe 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.