Answer
Verified
455.1k+ views
Hint: This question is based on statistics. In this question a number of observations are given and we have to find the mean, median and mode for this observation. The mean of a data also known as the average is defined by the ratio of the sum of all the observations to the number of observations. The median of a data is the middle value when the data is arranged in ascending or descending order. For example, if there are $n$ number of terms then the formula for the median is given by –
${\rm{Median = }}\left( {\dfrac{{{\rm{n + 1}}}}{{\rm{2}}}} \right){\text{th observation}}$
When $n$ is an odd number
And,
${\rm{Median = }}\dfrac{{\left( {\dfrac{n}{2}} \right)th + \left( {\dfrac{n}{2} + 1} \right)th}}{2}{\rm{observation}}$
When $n$ is an even number.
The mode is defined as the observation having the highest number of occurrences.
Complete step-by-step answer:
Given:
The number of students $ = 15$
The marks of the students
$41,39,48,52,46,62,54,40,96,52,98,40,42,52,60$
The mean of the data given can be calculated in the following way –
${\rm{Mean = }}\dfrac{{{\rm{Sum of all the marks of the students}}}}{{{\rm{Number of the students}}}}$
Substituting the values in the formula we get,
$
{\rm{Mean}} = \dfrac{{41 + 39 + 48 + 52 + 46 + 62 + 54 + 40 + 96 + 52 + 98 + 40 + 42 + 52 + 60}}{{15}}\\
= \dfrac{{822}}{{15}}\\
= 54.8
$
Now to calculate the median of the data given, rearranging the data given in the ascending order, we get,
$39,40,40,41,42,46,48,52,52,52,54,60,62,96,98$
So, total number of the data observations $n = 15$
Since the number of observations $n$ is an “odd number” we use the formula for the median of the odd number of observations. The formula is given by –
${\rm{Median = }}\left( {\dfrac{{{\rm{n + 1}}}}{{\rm{2}}}} \right){\text{th observation}}$
Substituting the value of $n$ in the formula we get,
$
{\rm{Median = }}\left( {\dfrac{{15 + 1}}{2}} \right){\text{th observation}}\\
{\text{ = 8th observation}}
$
And we know from the data given in the ascending order that the $8{\rm{th}}$ observation is the number 52
So,
${\rm{Median = 52}}$
Now to calculate the mode of this data we have to find the number with the maximum number of occurrences or the frequency.
By observation we found that -
The number with the highest frequency is 52 (total 3 times).
So, the mode of these observations is –
${\rm{Mode = 52}}$
Therefore, the mean of the data is 54.8, the median for the data is 52 and the mode for the data is 52.
Note: It should be noted that depending upon the distribution of the data in the given number of observations the value for the mean, median and mode changes. For example, if the data distribution is symmetrical about the centre then it is a normal distribution and for this distribution the value of mean, median and mode is equal.
So, for normal distribution –
Mean = Median = Mode
${\rm{Median = }}\left( {\dfrac{{{\rm{n + 1}}}}{{\rm{2}}}} \right){\text{th observation}}$
When $n$ is an odd number
And,
${\rm{Median = }}\dfrac{{\left( {\dfrac{n}{2}} \right)th + \left( {\dfrac{n}{2} + 1} \right)th}}{2}{\rm{observation}}$
When $n$ is an even number.
The mode is defined as the observation having the highest number of occurrences.
Complete step-by-step answer:
Given:
The number of students $ = 15$
The marks of the students
$41,39,48,52,46,62,54,40,96,52,98,40,42,52,60$
The mean of the data given can be calculated in the following way –
${\rm{Mean = }}\dfrac{{{\rm{Sum of all the marks of the students}}}}{{{\rm{Number of the students}}}}$
Substituting the values in the formula we get,
$
{\rm{Mean}} = \dfrac{{41 + 39 + 48 + 52 + 46 + 62 + 54 + 40 + 96 + 52 + 98 + 40 + 42 + 52 + 60}}{{15}}\\
= \dfrac{{822}}{{15}}\\
= 54.8
$
Now to calculate the median of the data given, rearranging the data given in the ascending order, we get,
$39,40,40,41,42,46,48,52,52,52,54,60,62,96,98$
So, total number of the data observations $n = 15$
Since the number of observations $n$ is an “odd number” we use the formula for the median of the odd number of observations. The formula is given by –
${\rm{Median = }}\left( {\dfrac{{{\rm{n + 1}}}}{{\rm{2}}}} \right){\text{th observation}}$
Substituting the value of $n$ in the formula we get,
$
{\rm{Median = }}\left( {\dfrac{{15 + 1}}{2}} \right){\text{th observation}}\\
{\text{ = 8th observation}}
$
And we know from the data given in the ascending order that the $8{\rm{th}}$ observation is the number 52
So,
${\rm{Median = 52}}$
Now to calculate the mode of this data we have to find the number with the maximum number of occurrences or the frequency.
By observation we found that -
The number with the highest frequency is 52 (total 3 times).
So, the mode of these observations is –
${\rm{Mode = 52}}$
Therefore, the mean of the data is 54.8, the median for the data is 52 and the mode for the data is 52.
Note: It should be noted that depending upon the distribution of the data in the given number of observations the value for the mean, median and mode changes. For example, if the data distribution is symmetrical about the centre then it is a normal distribution and for this distribution the value of mean, median and mode is equal.
So, for normal distribution –
Mean = Median = Mode
Recently Updated Pages
Identify the feminine gender noun from the given sentence class 10 english CBSE
Your club organized a blood donation camp in your city class 10 english CBSE
Choose the correct meaning of the idiomphrase from class 10 english CBSE
Identify the neuter gender noun from the given sentence class 10 english CBSE
Choose the word which best expresses the meaning of class 10 english CBSE
Choose the word which is closest to the opposite in class 10 english CBSE
Trending doubts
A rainbow has circular shape because A The earth is class 11 physics CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Change the following sentences into negative and interrogative class 10 english CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write a letter to the principal requesting him to grant class 10 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What organs are located on the left side of your body class 11 biology CBSE