Question

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?

Answer 1

Hint: We will find the mean by adding all numbers in the data set and then dividing by the number of values in the set. Then, median is the middle value when a data set is ordered from least to greatest. And, the mode is the number is the number that occurs most often in a data set. It is given that two new numbers are added to this data set, yet the mean does not change. Thus, the numbers must both either have the same value as the mean, thus by using this concept we will determine that two numbers.

Complete step by step answer:
1. Mean:
The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.
$mean = \dfrac{{sum\,of\,the\,numbers\,in\,the\,set}}{{total\,number\,of\,values\,in\,the\,set}}$
$\bar x = \dfrac{{{x_1} + {x_2} + \ldots + {x_n}}}{n}$
$\bar x = \dfrac{{2 + 5 + 5 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 9 + 10}}{{12}}$
$\bar x = \dfrac{{77}}{{12}}$
$\bar x = 6\dfrac{5}{{12}}$
Hence, the mean is $6\dfrac{5}{{12}}$.

2. Median:
 The median is the middle value when a data set is ordered from least to greatest.
Therefore, the given set when ordered least from greatest is, $2,5,5,6,6,6,7,7,7,7,9,10$.
Now, the middle term is the sixth and seventh term, we know that if we have two numbers as median then the average of the two numbers is taken as median.
Hence, the average of $6$ and $7$ is $6.5$.
Thus, the median is $6.5$.

3. Mode:
The mode is the number is the number that occurs most often in a data set.
Therefore, from the given data we can say that the most occurred number in the given data set is $7$.
Hence, the mode is $7$.

It is given that two new numbers are added to this data set, yet the mean does not change. Thus, the numbers must both either have the same value as the mean or their sum is $12\dfrac{5}{6}$ , thus their mean will be $6\dfrac{5}{{12}}$.
Hence, the two numbers can be $\dfrac{{5 + 7\dfrac{5}{6}}}{2} = 6\dfrac{5}{{12}}$ and $\dfrac{{2\dfrac{2}{6} + 10\dfrac{3}{6}}}{2} = 6\dfrac{5}{{12}}$.

Note: The term average is used to express an amount of data that is typical for a group of people or things. It summaries a large amount of data into a singular value. Indicate that there is some variability around this single value within the original data. In the language of mathematics there are three different definitions of average known as mean, median and mode.