# How do you find the mean, median, and mode of 2, 5, 5, 6, 6, 6, 7, 7, 7, 7, 9, 10? Two new numbers are added to this data set, yet the mean does not change. What do you know about the two numbers?

Mar 15, 2017

MEDIAN = $\frac{6 + 7}{2} = 6.5$
MODE = $7$
MEAN =$\text{ } \frac{77}{12} = 6 \frac{5}{12}$
The added numbers must either BOTH be $6 \frac{5}{12}$
or their sum should be $12 \frac{5}{6}$

#### Explanation:

The mean, mode and median are all different types of averages.
We are given 12 values arranged in order.

$2 , \text{ "5," " 5," " 6," " 6," " 6," " color(red)(7," " 7," " 7" ", 7)," "9," } 10$

The Mode is the value with the highest frequency, ie. the one that occurs the most often: 7 occurs four times. MODE = $7$

The median is the value exactly in the middle of a set of values arranged in order. There are 12 values. $12 \div 2 = 6$
Split the values into the bottom 6 and the top 6:

$\textcolor{b l u e}{2 , \text{ "5," " 5," " 6," " 6," " 6,)" "color(green)(7," " 7," " 7" ", 7," "9," } 10}$
$\textcolor{w h i t e}{m m m m m m m m m m m m . m m} \uparrow$
The median lies exactly in the middle between the 6th and 7th values: between the last 6 and the first 7. Find the mean of these numbers: MEDIAN = $\frac{6 + 7}{2} = 6.5$

The mean is what is usually called the 'average'.

MEAN = $\frac{2 + 5 + 5 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 9 + 10}{12}$

MEAN =$\text{ } \frac{77}{12} = 6 \frac{5}{12}$

If 2 more numbers are added, for the mean to stay the same, the numbers must either BOTH have the same value as the mean, or their sum is $12 \frac{5}{6}$ so their mean will be $6 \frac{5}{12}$

for example:$\frac{5 + 7 \frac{5}{6}}{2} = 6 \frac{5}{12}$

$\frac{2 \frac{2}{6} + 10 \frac{3}{6}}{2} = 6 \frac{5}{12}$