Question

What is the mean, median, range and mode of $55,50,58,55,60,54,56,58,58,54$?

Answer 1

To solve this question we need to have the concept of statistics. In this question we need to find Mean, Median and Mode for the set of numbers given in the question. Mean refers to the average of the total numbers in a set. Median is the middle term from the set when the numbers are arranged in the order. Mode is the number with the highest frequency. Range is the difference between the largest and the smallest value in the set.

Complete step-by-step answer:
The question asks us to find the value of Mean, Median, Mode and Range for the numbers of $55,50,58,55,60,54,56,58,58,54$. To solve this question we will first find the Mean of the numbers given. Mean is the average of all the values given in the question. So for doing this, we will sum all the numbers by the total number of observations which is given in the question. Mathematically Mean will be:
$\text{mean}=\dfrac{\text{sum of all the observations}}{\text{Total number of observations}}$
Sum of all the observation is $55+50+58+55+60+54+56+58+58+54=558$
$\text{Total number of observations}$ $=10$
So now the mean will be:
$\Rightarrow \text{mean}=\dfrac{588}{10}$
The mean of the number is $55.8$.
The next step is to find the median of the number given in the question. To do this we will arrange the number in an order from low to high and then our medium will be valued at the ${{n}^{th}}$ position of the observation. On arranging the numbers in the order we get:
$\Rightarrow 50,54,54,55,55,56,58,58,58,60$
The median has two different formulas one for the odd number of observations and other for the even number of observations. Since in the given question we have $10$ observations which is an even number so the formula used will be:
$\text{median}=\dfrac{{{\left( \dfrac{\text{n}}{\text{2}} \right)}^{\text{th}}}+{{\left( \dfrac{n}{2}+1 \right)}^{\text{th}}}\text{observation}}{2}$
On writing the values in the formula we get:
$\text{median}=\dfrac{{{\left( \dfrac{10}{\text{2}} \right)}^{\text{th}}}+{{\left( \dfrac{10}{2}+1 \right)}^{\text{th}}}\text{observation}}{2}$
$\text{median}=\dfrac{{{\left( 5 \right)}^{\text{th}}}+{{\left( 6 \right)}^{\text{th}}}\text{observation}}{2}$
As per the above arrangement the ${{5}^{th}}$ and ${{6}^{th}}$ terms are $55$ and $56$ respectively.
$\text{median}=\dfrac{55+56}{2}$
$\text{median}=55.5$
The median for the given set of numbers is $55.5$.

The mode of a data set is the element that appears most frequently. In the question given to us the number $58$ is repeated the maximum number of times. The mode of the set is $58$.

Range is the difference between the largest and the smallest value in the set. On analysing the set of numbers given in the question, the largest number and the smallest number are $60$and $50$ respectively. So the range is:
$\Rightarrow 60-50$
$\Rightarrow 10$
So the range of the numbers are$10$.
$\therefore$ For the given set of numbers mean is $55.8$, median is $55.5$, mode is $58$ and the range is $10$.

Note: The median has two different formulas, one for the odd number of observations and other for the even number of observations. Formula for odd number of observations is:$\text{median}={{\left( \dfrac{n+1}{2} \right)}^{th}}\text{term}$
Formula for even number of observations is which means $n$ is an even number:
$\text{median = }\dfrac{{{\left( \dfrac{\text{n}}{\text{2}} \right)}^{\text{th}}}+{{\left( \dfrac{n}{2}+1 \right)}^{\text{th}}}\text{observation}}{2}$