Question

Calculate the mean, the median, and the mode of the numbers: $3,2,6,3,3,1,1,2$

Answer 1

Hint: First, we will see the definitions of mean, median, and mode.
Mean is known as the sum of all the observations divided by the total number of observations.
Median is defined as the middle point in any data set of observation, half of the observations are greater than this number, and half of them smaller.
Mode is defined as the most frequent or common observation occurring in the given data.
Formula used: Mean$ = \dfrac{{\sum\limits_{i = 1}^n {{X_i}} }}{n}$, n is the number of given data.

Complete step by step answer:
First, from the given data we will rewrite into the ascending order to solve the requirements easily, thus we get $3,2,6,3,3,1,1,2 \Rightarrow 1,1,2,2,3,3,3,6$ (lowest to greatest)
The mean of the given data is$\dfrac{{\sum\limits_{i = 1}^n {{X_i}} }}{n} = \dfrac{{1 + 1 + 2 + 2 + 3 + 3 + 3 + 6}}{8}$, where the numerator is the sum of the data and the denominator is the number of total data.
Thus, soling this we get, $\dfrac{{\sum\limits_{i = 1}^n {{X_i}} }}{n} = \dfrac{{1 + 1 + 2 + 2 + 3 + 3 + 3 + 6}}{8} \Rightarrow \dfrac{{21}}{8}$
Now by the division operation, we get, $\dfrac{{\sum\limits_{i = 1}^n {{X_i}} }}{n} = \dfrac{{21}}{8} \Rightarrow 2.625$which is the mean of the given data.
To find the median, we take the arranged data $1,1,2,2,3,3,3,6$ and since the median is to find the midpoint from the given data.
If the data is odd numbers, then we can easily get the midpoint like if $1,2,3$ is the odd terms in total number occurring.
Thus, the midpoint of the data is $2$.
If the even term occurs like from the arrangement $1,1,2,2,3,3,3,6$ it has $n = 8$ is even.
Now take the middle two numbers from the data, that is $2,3$which is in the center of the data.
Now adding the numbers and divided by two, we get $\dfrac{{2 + 3}}{2} = \dfrac{5}{2}$(denominator two is the total number which two and three)
Thus, we get the median for the data $1,1,2,2,3,3,3,6$is $2.5$
Now to find the mode for the given data, mode means most repeated values,
From the given data we have, $1,1,2,2,3,3,3,6$ here one is repeated two times, and two is repeated two times but three is repeated three times and hence the mode of the data $1,1,2,2,3,3,3,6$ is $3$

Note: For the median, we can als0 able to use the formula that, if n is even then $\dfrac{1}{2}(\dfrac{n}{{{2^{th}}term}} + (\dfrac{n}{2}) + {1^{st}}term)$
Also, note that for the mode if there are two most repeated terms like $1,1,1,2,2,2$ we have the mode as both one and two.