Question

What is the mean, median and mode of 4, 5, 7, 10?

Answer 1

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 4. Now, for even number of observations provided apply the formula Median = $\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}$ term to calculate the median after arranging the observations in either ascending or descending order of their numerical value. Finally, to calculate the mode of the data set given, observe the data which is appearing most number of times. If each data is appearing only once then there will not be any mode.

Complete step by step solution:
Here we have been provided with the numbers 4, 5, 7, 10 and we are asked to calculate the mean, median and mode of these numbers.
(1) 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 4 so we have the value of n equal to 4. Substituting n = 4 in the formula for mean we get,

& \Rightarrow \bar{x}=\dfrac{\sum\limits_{i=1}^{4}{{{x}_{i}}}}{4} \\\ & \Rightarrow \bar{x}=\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}}{4} \\\ \end{aligned}$$ Substituting the values of given observations we get, $$\begin{aligned} & \Rightarrow \bar{x}=\dfrac{4+5+7+10}{4} \\\ & \Rightarrow \bar{x}=\dfrac{26}{4} \\\ & \therefore \bar{x}=6.5 \\\ \end{aligned}$$ Therefore, the mean of the given data set is 6.5. (2) Now, to calculate the median first we have to arrange the given numbers in either ascending or descending order of their numerical value. We can see that they are already arranged in ascending order. Therefore applying the formula of median for n = 4 terms (which is odd) we get, $\Rightarrow $ Median = $$\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}$$ $\Rightarrow $ Median = $$\dfrac{{{\left( \dfrac{4}{2} \right)}^{th}}term+{{\left( \dfrac{4}{2}+1 \right)}^{th}}term}{2}$$ $\Rightarrow $ Median = $$\dfrac{{{2}^{nd}}term+{{3}^{rd}}term}{2}$$ In the above arrangement we see that the ${{2}^{nd}}$ and ${{3}^{rd}}$ terms are 5 and 7 respectively, so we get, $\Rightarrow $ Median = $$\dfrac{5+7}{2}$$ $\Rightarrow $ Median = 6 Therefore, the median of the given data set is 6. (3) Now, Mode of a given data set is defined as the data which is appearing the most number of times. On observing the given data set we can say that each data is appearing only once, so we can conclude that there is no mode value for the given data set. **Note:** Note that in case we have odd number of terms then the median of the data set is given by the formula Median = $${{\left( \dfrac{n+1}{2} \right)}^{th}}$$ term. In case there is one mode of the data set then it is called unimodal data set, for 2 modes the data set is called bimodal and for more than two modes the data set is called multimodal. Do not use the empirical relation between mean, median and mode given as Mode = 3(median) – 2(Mean) because it is used for the moderately skewed distributions.