Question

Find the mode of the following data 90, 40, 68,90,50,60. \[\]
A.40\[\]
B.68\[\]
C.90\[\]
D.50\[\]

Answer 1

Hint: We arrange the given data sample 90, 40, 68,90,50,60. in ascending order. We find the data values greater than any data value and the data value that occurs most of the time. If they are different we take the data value that occurs the most number of times as mode and if all the data occur the same number of times then we take the greatest data value as the mode. \[\]

Complete step-by-step solution:
We know that mode is a measure of the central tendency of the data or population. It expresses just like mean and median, the most important value towards which all the data points show a tendency. The highest value in the data is called mode. If there are $n$ data points say ${{x}_{1}},{{x}_{2}},...,{{x}_{n}}$ then $M$ is will be the mode and if only if
\[M>{{x}_{i}},i=1,2,3...n\]
It may also happen that the highest value of $M$ may occur several times. So in accordance, we call the data unimodal if $M$ occurs only once, bimodal if $M$ occurs twice, trimodal if $M$ occurs thrice, and so on.\[\]
If we draw the histogram of the data if the data is multimodal say with $n$ modes then it will show $n$ number of peaks with equal heights.\[\]
The other definition of mode is the data value that occurs most number of times. If the number of occurrences of the data values ${{x}_{1}},{{x}_{2}},...,{{x}_{n}}$ is${{f}_{1}},{{f}_{2}},...{{f}_{n}}$ respectively then mode is given by
\[M={{x}_{k}},\text{if }{{f}_{k}}>{{f}_{i}}\forall i=1,2,...n\]
Let us observe the given data in question 90, 40, 68,90,50,60. Let us arrange the data in ascending order. We have 40, 50, 68, 90, and 90. We see that 90 is the data value which is greater than any other data value in the data sample and it occurs twice. None of the other data occurs more than once. So 90 is the highest data value and most occurring data value at the same time. So the mode is 90 and the correct option is C. \[\]

Note: We note that the data at sample is bimodal. We can find the mode of class interval data with the formula $M=L+\dfrac{{{f}_{m}}-{{f}_{m-1}}}{\left( {{f}_{m}}-{{f}_{m-1}} \right)\left( {{f}_{m}}-{{f}_{m+1}} \right)}\times w$ where L is the lower boundary,$w$ is the width of the modal interval${{I}_{m}}$. The frequency${{f}_{m-1}}$ is the frequency corresponding to the interval right before ${{I}_{m}}$ that is ${{I}_{m-1}}$ and the frequency ${{f}_{m+1}}$ is the frequency corresponding to the interval right after ${{I}_{m}}$ that is ${{I}_{m+1}}$.