Question

A TV reporter was given a task to prepare a report on the rainfall of a city Dispur of India in a particular year. After collecting the data, he analyzed the data and prepared a report on the rainfall of the city. Using this report, he drew the following graph for a particular time period of 66 days.

Based on the above graph, answer the following questions.
1) Identify the less than type and more than type ogive from the given graph
2) Find the median rainfall of Dispur.
3) Obtain the mode of the data if mean rainfall is 23.4 cm.

Answer 1

Hint:
Here, we have to use the concept of statistics to give the answer to the following questions. So, in the first part, we will use the concept of the less than and more than type ogive to find out the respective curve. In the second part, we will use the concept of median to find out the median of the curves. In the third part, we will use the formula of the mean, mode and median to find out the value of the mode.

Formula Used: We will use the formula \[3{\text{Median}} = {\text{Mode}} + 2{\text{Mean}}\] to find the mode.

Complete Complete Step by Step Solution:
From the given two curves we have to identify the less than type and more than type given.
We know that in the less than type ogive, the cumulative frequency always increases and in case of more than type ogive, the cumulative frequency always decreases. So by this fact, we can easily say that:
Curve 1 is less than type ogive and Curve 2 is more than type ogive.

Here, we have to find out the median rainfall of Dispur. We know that the middle most term of the data is known as median. So, in this case where the curve 1 and curve 2 intersects is the middle part of the data. So by projecting the line on the $x$ -axis from that point we can clearly see the value of the median.
So, the value of the median is nearly 21.

Now, we have to calculate the mode of the data given. The mean is given as 23.4 cm and median is 21 as calculated earlier.
Now we will substitute the values of median and mean in the formula \[3{\text{Median}} = {\text{Mode}} + 2{\text{Mean}}\].
Substituting \[{\text{Mean}} = 23.4\] and \[{\text{Median}} = 21\] in the formula, we get
\[ \Rightarrow 3\left( {21} \right) = {\text{Mode}} + 2\left( {23.4} \right)\]
\[ \Rightarrow 63 = {\text{Mode}} + 46.8\]
Now by simplifying the equation, we get
\[ \Rightarrow {\text{Mode}} = 63 - 46.8 = 16.2{\text{cm}}\]
Hence, the mode of the data is \[16.2\] cm.


Note:
Statistics is the science of collecting some data in the form of the number and studying it to forecast or predict its future possibility. Some definitions we should know
Mean is equal to the ratio of sum of the total numbers and total count of the numbers. Mean is also known as the average of the numbers.
Mode is the most common or most repeating numbers in the given data.
Median is the middle value of the given list of numbers or it is the value which is separating the data into two halves i.e. upper half and lower half.