Question

Calculate the mode of the following using empirical formula:
17, 10, 12, 11, 10, 15, 14, 11, 12, 13.

Answer 1

We know:

$$\begin{gathered} \mathrm{Mean}=\frac{\mathrm{Sum}\mathrm{of}\mathrm{observations}}{\mathrm{Number}\mathrm{of}\mathrm{observations}} \\ =\frac{17+10+12+11+10+15+14+11+12+13}{10} \\ =\frac{125}{10}=12.5 \end{gathered}$$

$$\begin{gathered} \mathrm{Arranging}\mathrm{the}\mathrm{observations}\mathrm{in}\mathrm{ascending}\mathrm{form},\mathrm{we}\mathrm{get}: \\ 10,10,11,11,12,12,13,14,15,17 \end{gathered}$$

$$\begin{gathered} \mathrm{The}\mathrm{number}\mathrm{of}\mathrm{observations}\mathrm{is}10,\mathrm{which}\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \mathrm{Median}=\frac{1}{2}\left[{\left(\frac{\mathrm{n}}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{\mathrm{n}}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[{\left(\frac{10}{2}\right)}^{\mathrm{th}}\mathrm{term}+{\left(\frac{10}{2}+1\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[5^{\mathrm{th}}\mathrm{term}+{\left(6\right)}^{\mathrm{th}}\mathrm{term}\right] \\ =\frac{1}{2}\left[12+12\right] \\ =12 \end{gathered}$$

On applying the empirical formula, we have:
Mode = 3(Median) $-$ 2(Mean)
or, Mode = 3(12) $-$ 2(12.5)
= 11