Question

How do you find the mean and median of 136, 129, 128, 122, 134, 120?

Answer 1

Hint: Use the formula \[\bar{x}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}\] to calculate the mean of the given numbers, where \[\bar{x}\] denotes the mean and n is the number of observations. Substitute the value of n equal to 6. Now, for even number of observations provided apply the formula Median = \[\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}\] to calculate the median after arranging the observations in ascending order of their numerical value.

Complete step by step solution:
Here we have been provided with the numbers 136, 129, 128, 122, 134, 120 and we are asked to calculate the mean and median of these numbers.
Now, we know that the mean of n observations is given by the formula \[\bar{x}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}\], i.e. the ratio of sum of all the observations to the number of observations. In the above formula \[\bar{x}\] denotes the mean. Clearly we can see that the number of observations is 6 so we have the value of n equal to 6. Substituting n = 6 in the formula for mean we get,
\[\begin{align}
  & \Rightarrow \bar{x}=\dfrac{\sum\limits_{i=1}^{6}{{{x}_{i}}}}{6} \\
 & \Rightarrow \bar{x}=\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}+{{x}_{6}}}{6} \\
\end{align}\]
Substituting the values of given observations we get,
\[\begin{align}
  & \Rightarrow \bar{x}=\dfrac{136+129+128+122+134+120}{6} \\
 & \Rightarrow \bar{x}=\dfrac{769}{6} \\
 & \therefore \bar{x}=128.167 \\
\end{align}\]
Therefore, the mean of the given numbers is 128.167.
Now to calculate the median first we have to arrange the given numbers in ascending order of their numerical value. So on arrangement we get 120, 122, 128, 129, 134, 136. Therefore applying the formula for median of n = 6 numbers (which is even) we get,
$\Rightarrow $ Median = \[\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}\]
$\Rightarrow $ Median = \[\dfrac{{{\left( \dfrac{6}{2} \right)}^{th}}term+{{\left( \dfrac{6}{2}+1 \right)}^{th}}term}{2}\]
$\Rightarrow $ Median = \[\dfrac{{{\left( 3 \right)}^{rd}}term+{{\left( 4 \right)}^{th}}term}{2}\]
In the above arrangement we see that ${{3}^{rd}}$ and ${{4}^{th}}$ terms are 128 and 129 respectively, so on substituting these numbers in the above relation we get,
$\Rightarrow $ Median = \[\dfrac{128+129}{2}\]
$\Rightarrow $ Median = \[\dfrac{257}{2}\]
$\therefore $ Median = 128.5
Therefore, the median of the numbers is 128.5.

Note: Note that here we have even number of terms whose median we were asked to find and that is why the formula \[\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}\] for the median is applied. In case the number of observations is odd then the formulas of the median becomes \[{{\left( \dfrac{n}{2}+1 \right)}^{th}}term\]. So you need to count the number of terms properly before applying the formula. Also, here we have arranged the terms in ascending order. You may also arrange them in descending order, in that case also you will get the same answer.