Question

What are the mean, median, mode, and range of the following numbers?
$\left\{ {1,2,3,4,5,6,7,8,\left. 9 \right\}} \right.$

Answer 1

Hint: Generally, the mean, the median, and the mode are the most commonly used measure of central tendency.
The mean generally refers to the ratio of the sum of all observations to the total number of observations in the given collection of data
The median is the process of finding the mid-value in the given set of data. To calculate the median, we need to apply the appropriate formula. Before getting into the formula, we need to identify whether the data is even or odd.
The mode is the process of selecting the most frequently occurred values in the given collection of data.
The range of data is the difference between the largest value and the smallest value present in the given set of data.
Formula used:
For mean, $\overline x $ \[ = \dfrac{{{x_1} + {x_2} + ..... + {x_n}}}{n}\]
Where, $\overline x $is the mean, \[{x_i}\]$\left( {1 \leqslant i \leqslant n} \right)$ denotes each value in the set and$n$ is the number of values.
For median,
\[Median = {\left( {\dfrac{{n + 1}}{2}} \right)^{th}}observation\] , if $n$ is odd.
\[Median = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}observation + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}observation}}{2}\], if $n$is even.
Where, $n$is the number of observations.
For range,$R = H - L$
Where $R$ denotes the range of values, $H$is the highest value, and $L$is the lowest value.

Complete step by step answer:
Let us consider the given set of values $\left\{ {1,2,3,4,5,6,7,8,\left. 9 \right\}} \right.$.
i)Mean:
Here, the number of values $\left( n \right)$=$9$ .
Hence, by using the above mean formula,
Mean $\overline x = \dfrac{{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9}}{9}$
$\overline x = \dfrac{{45}}{9}$
$ \to \overline x = 5$
Therefore, the mean is$5$.
ii) Median:
Here, the number of values $\left( n \right)$=$9$, which is odd.
So, by the above-median formula,
\[Median = {\left( {\dfrac{{9 + 1}}{2}} \right)^{th}}observation\]
$Median = {5^{th}}observation$
Hence, Median is $5$.
iii) Mode:
Here, no number gets repeated.
So, there is no mode for the given data.
iv) Range:
Here, the highest value $H = 9$ and the lowest value $L = 1$ .
Hence, by using the range formula,
$R = 9 - 1 $
$R = 8$
So, the range is $8$ for the given data.

Note:
The central tendency is a single value that acts as a representative for the whole collection of data.
The median is also calculated simply. To calculate the median, we need to first arrange the values in order (increasing or decreasing) and then we have to count the number of values$\left( n \right)$present in the given data. It$\left( n \right)$is odd, directly write the center value as the median. It $\left( n \right)$is even, then finds the average of two middle numbers to obtain the median.