Question

Marks obtained by four students are: 25, 35, 45, 55. The average deviations from the mean is
A) 10
B) 9
C) 7
D) None of these

Answer 1

Hint:
Here, we will find the mean marks of the four students by adding their respective marks and dividing the sum by the number of students. After finding the mean, we will find the deviation by subtracting the mean from each observation and neglecting the negative sign. Adding all the deviations together and dividing the sum by 4 will give us the required average deviations from the mean.

Formula Used:
We will use the following formulas:
1) Mean, $\overline x = \dfrac{{{S_N}}}{N}$, which states that mean is equal to sum of observations divided by total number of observations.
2) Mean Deviation $ = \dfrac{{\sum {\left| {x - \overline x } \right|} }}{N}$, where $x$ represents the respective observation, $\overline x $ represents the mean and $N$ represents the total number of observations.

Complete step by step solution:
According to the question, marks obtained by four students are: 25, 35, 45, 55.
Hence, total number of students, $N = 4$
Now, first of all, we will find the mean marks of these 4 students.
We know that, Mean is equal to sum of observations divided by total number of observations.
Substituting the given values in this formula $\overline x = \dfrac{{{S_N}}}{N}$, we get,
$\overline x = \dfrac{{25 + 35 + 45 + 55}}{4}$
$ \Rightarrow \overline x = \dfrac{{160}}{4} = 40$
Therefore, the mean of these observations is 40.
Now, we will find the deviations of each observation, using the formula: $\left| {x - \overline x } \right|$, where $x$ represents the respective observation and $\overline x $ represents the mean.
Hence,
1) Deviation of 25 is $\left| {25 - 40} \right| = \left| { - 15} \right| = 15$
2) Deviation of 35 is $\left| {35 - 40} \right| = \left| { - 5} \right| = 5$
3) Deviation of 45 is $\left| {45 - 40} \right| = \left| 5 \right| = 5$
4) Deviation of 55 is $\left| {55 - 40} \right| = \left| {15} \right| = 15$
Therefore, the average deviations from the mean can be written as: $\dfrac{{\sum {\left| {x - \overline x } \right|} }}{N}$
Hence, we will add all the four deviations in the numerator to find the required average deviations from the mean.
$\dfrac{{\sum {\left| {x - \overline x } \right|} }}{N} = \dfrac{{15 + 5 + 5 + 15}}{4}$
$ \Rightarrow \dfrac{{\sum {\left| {x - \overline x } \right|} }}{N} = \dfrac{{40}}{4} = 10$
Therefore, the average deviations from the mean is 10

Hence, option A is the correct answer.

Note:
Mean is also known as the average of the given numbers or observations. It is calculated by adding all the observations together and dividing the sum obtained by the total number of observations. Also, the mean deviation or the average absolute deviation is the average of the deviations or in simple terms, when we add all the deviations together and divide the sum by total number of observations, we get the required mean deviation.