Question

A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data.

Number of cars	Frequency
$0 - 10$	$7$
$10 - 20$	$14$
$20 - 30$	$13$
$30 - 40$	$12$
$40 - 50$	$20$
$50 - 60$	$11$
$60 - 70$	$15$
$70 - 80$	$8$

Answer 1

According to the question given in the question we have to determine the mode of the data when a student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given above. So, first of all we have to understand about the mode which is as explained below:
Mode: The mode is the value that appears most frequently in the given data set and a set of data may have one mode, more than one mode or no mode at all.
Now, from the table above, first of all we have to determine the frequency and then we have to obtain the maximum class that belongs for the interval in the given table.
Now, we have to use the formula to find the mode which is as explained below:

Formula used: $\Rightarrow$Mode$= l + \dfrac{{f - {f_1}}}{{2f - {f_1} - {f_2}}} \times h...............(A)$
Where, l is the lower class limit of the modal class, h is the size or height of the class interval, f is the frequency of the modal class and ${f_1}$is the frequency of the class preceding the modal class, and ${f_2}$ is the frequency of the class succeed the modal class. Hence, on substituting all the values in the formula (A) above, we can easily determine the mode for the given data.

Complete step-by-step solution:
Step 1: First of all as explained about the mode we have to determine the maximum frequency in the table which is as below:

Number of cars	Frequency
$0 - 10$	$7$
$10 - 20$	$14$
$20 - 30$	$13$
$30 - 40$	$12$
$40 - 50$	$20$
$50 - 60$	$11$
$60 - 70$	$15$
$70 - 80$	$8$

Hence, with the tale the maximum frequency is 20.
Step 2: Now, we have to determine the l which is the lower class limit of the modal class, h which is the size or height of the class interval, f which is the frequency of the modal class and ${f_1}$ which is the frequency of the class preceding the modal class, and ${f_2}$which is the frequency of the class succeed the modal class. Hence, with the help of the data we can determine the all as,
For modal class: 40-50
$l = 40,f = 20,{f_1} = 12,{f_2} = 11$ and $h = 10$
Step 3: Now, to determine the mode we have to use the formula (A) which is as mentioned in the solution hint, Hence, on substituting all the values in the formula (A),
$\Rightarrow$Mode$= 40 + \dfrac{{20 - 12}}{{2 \times 20 - 12 - 11}} \times 10$
On solving,
$= 40 + \dfrac{{80}}{{17}} \\\ = 40 + 4.71 \\\ = 44.7cars$
Final solution: Hence, with the help of the formula (A) as mentioned in the solution hint we have determined the mode for the given data which is 44.7 cars.

The mode for the given data is 44.7

Note: Other popular measures of central tendency include the mean, or the average of a set and the median, the middle value in the set.
Height can be obtained by finding the average for the obtained modal class in which we have to add the lower and higher limit then we have to divide it by 2.
Mode is the value that appears most frequently in the given data set and a set of data may have one mode, more than one mode or no mode at all.