What is the mode, median and mean of 5, 27, 29, 13, 18, 19, 15, 19, 19, 27, 15, 22, 13, 26, 20?

The Mean is the arithmetic average, #19.133#

The Median is #"([the number of data points] + 1) ÷ 2"#

or the PLACE value equidistant (numerically) from the range extremes in an ordered set.
This set contains #15# numbers, arranged in order as #5,13,13,15,15,18,19,19,19,20,22,26,27,27,29#.

So the middle place is #(15+1)/2 = 8th# position.
The number at that location is 19.

The Mode is the most common value(s) in a set. In this case it is #19#, with three occurrences in the set.

The closeness of all three of these measures means that the data is 'normally distributed'.

A good starting place is to arrange the given values in order:

#5" "13" "13" "15" "15" "18" "19" "19" "19" "20" "22" "26" "27" "27" "29#

If you were asked the range. (which is not done in this case), it is now a simple matter to find the difference between the biggest and smallest values:
#color(blue)(5)" "13" "13" "15" "15" "18" "19" "19" "19" "20" "22" "26" "27" "27" "color(blue)(29)#

Range = #color(blue)(29-5 = 24)#

The MODE is the value with the highest frequency, the one that occurs the MOST often.
Because the values are all in order, we can see that #19# occurs three times.

#5" "13" "13" "15" "15" "18" "color(red)(19" "19" "19)" "20" "22" "26" "27" "27" "29#

Mode = #color(red)(19)#

The MEDIAN is the value exactly in the middle of a set of values arranged in order:
There are #15# numbers.

#15/2 = 7 1/2#

#15 = 7+1+7#

So count #7# from each side and the number in the middle is the median:

#color(green)(5, 13, 13, 15, 15, 18, 19)" "color(magenta)(19)" "color(green)(19, 20, 22, 26, 27, 27, 29)#

To find the MEAN (better known as the "average")

Add all the values together and divide by how many there are:

#5+13+13+15+15+18+19+19+19+20+22+26+27+27+29= 287#

#"Mean" = 287/15 = 19.13#

Remember that the mode, median and mean are ALL types of averages, they just give different information.