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

May 16, 2017

Mean $= 19.133$
Median $= 19$
Mode $= 19$

#### Explanation:

The Mean is the arithmetic average, $19.133$

The Median is $\text{([the number of data points] + 1) ÷ 2}$

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 $\frac{15 + 1}{2} = 8 t h$ 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'.

Jun 3, 2017

Mode = $19$

Median = $19$

Mean = $19.13$

#### Explanation:

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

$5 \text{ "13" "13" "15" "15" "18" "19" "19" "19" "20" "22" "26" "27" "27" } 29$

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:
$\textcolor{b l u e}{5} \text{ "13" "13" "15" "15" "18" "19" "19" "19" "20" "22" "26" "27" "27" } \textcolor{b l u e}{29}$

Range = $\textcolor{b l u e}{29 - 5 = 24}$

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.

$5 \text{ "13" "13" "15" "15" "18" "color(red)(19" "19" "19)" "20" "22" "26" "27" "27" } 29$

Mode = $\textcolor{red}{19}$

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

$\frac{15}{2} = 7 \frac{1}{2}$

$15 = 7 + 1 + 7$

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

$\textcolor{g r e e n}{5 , 13 , 13 , 15 , 15 , 18 , 19} \text{ "color(magenta)(19)" } \textcolor{g r e e n}{19 , 20 , 22 , 26 , 27 , 27 , 29}$

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

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

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

$\text{Mean} = \frac{287}{15} = 19.13$

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