As there are #12# data-points between #10# and #19# and #19# data-points between #20# and #29#. As such we have #12+19=31# data-points say up to #29# and hence cumulative frequency up to #29#. And similarly #31+31=62# data points up to #39#. This way we can derive cumulative frequency for other class intervals (C.I.) , as given below and final cumulative frequency is #113#, which is equal to sum of all the frequencies.
Note that classes are #10-19#, #20-29#,... means that while #19# comes in first C.I., #20# comes in second C.I. There is nothing common between the two as is usually when classes are defined as #10-20#, #20-30# and so on. This is as data points take only discreet points. To rectify this let us term C.I.'s as #9.50-19.50#, #19.50-29.50#, #29.50-39.50# and so on.
As the total data points are #113#, the middle one representing #(113+1)/2=57^(th)# item lies in C.I. #29.50-39.50# (marked with red colored arrow) and lower bound for this C.I. i.e. #L# is #29.50#. This is the median class. Other variable used in formula are #f_1#, the frequency of median class, which is #31#, #n=sumf#, which is #113# and #cf_1#, cumulative frequency of the class just before median class, which too is #31#.
C.I. #color(white)(XXXXXXX)#Frequency#color(white)(XXX)#Cum. Frequency
#09.50-19.50color(white)(XXXXXXXx)12color(white)(XXXXXXXXXx)12#
#19.50-29.50color(white)(XXXXXXxx)19color(white)(XXXXXXXXXX)31#
#29.50-39.50color(white)(XXXXXXxx)31color(red)(->)color(white)(XXXXXXXX)62#
#39.50-49.50color(white)(XXXXXXxx)27color(white)(XXXXXXXXXX)89#
#49.50-59.50color(white)(XXXXXXxx)16color(white)(XXXXXXXXXx)105#
#59.50-69.50color(white)(XXXXXXXX)8color(white)(XXXXXXXXxx)113#
Formula for Median is #Median=L+(n/2-cf_1)/(f_m)xxi#, hence
#Median=29.50+(113/2-31)/31xx10#
#=29.50+(113-62)/(2xx31)xx10=29.50+510/62#
#=29.50+8.23=37.73#
