How to find mean of positively skewed data?

CFA Level I Quant
#1

Hey guys,

I have this distribution of data which is positively skewed, but I am having trouble finding the mean, can you help me out? I’m trying to prove mean > median > mode. As you can see, most of the data is clustered towards the lower side of the scale, but I am confused to know that the mean is to the right, maybe I am making the mistake, the mean is not affected by the frequency, but rather only by the income itself?

Frequency Income

3 Less than 31

50 31-40

70 41-50

87 51-60

70 61-70

40 71-80

30 81-90

20 91-100

11 101-110

9 111-120

5 121-130

3 131-140

2 141-150

#2

The mean is affected by the frequency.

Multiply the midpoint of each interval by the corresponding frequency (use whatever you want for the Less than 31 category), sum these products, then divide by the sum of the frequencies.

#3

Hey S2000, I did that, but I got a median > mean …

Frequency Income Sum

3 15,500 31,000

50 35,500 1,775,000

70 45,500 3,185,000

87 55,500 4,828,500

70 65,500 4,585,500

40 75,500 3,020,000

30 85,500 2,565,000

20 95,500 1,910,000

11 105,500 1,160,500

9 115,500 1,039,500

5 125,500 627,500

3 135,500 406,500

2 145,500 291,000

Sum of income: 25,483,500

Sum of frequency: 400

Mean --> 63,708

Median --> 1,775,000

Mode --> N/A

BUT, if I divide 25,483,500 by 13 (number of income observations), I will get that the Mean > median.

#4

MEDIAN --> you need to add the freq and obtain the location where the 200th observation lies … in the ascending order of the income.

also for the first bucket - your mean income is 15500 - not 30000 as you have it.

#5

51-60 contains the 200th element.

so median = 51 + (200-123)/87 * 10 = 59.85 …

low + (median element # - Cumulative Freq at prev class) / Current Class Frequency * Size of the Class.

#6

Woops! My bad, let me fix that! Now I have:

Mean: 25,440,000/400 => 63,600

Median: I still don’t understand how you got to the 51-60 element, are u sorting in ascending order the (income midpoint * frequency) row or the Income row on its own.

Thank you for the help on the Mean!!

#7

you have the raw income column

sum up the frequency till you get to the range that contains the 200th element.

then the above …