The mean of discrete observations \[{y_1},{y_2},.....,{y_n}\] is given by
A. \[\dfrac{{\sum\limits_{i = 1}^n {{y_i}} }}{n}\]
B. \[\dfrac{{\sum\limits_{i = 1}^n {{y_i}} }}{{\sum\limits_{i = 1}^n i }}\]
C. \[\dfrac{{\sum\limits_{i = 1}^n {{y_i}{f_i}} }}{n}\]
D. \[\dfrac{{\sum\limits_{i = 1}^n {{y_i}{f_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }}\]
Answer
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- Hint: Discrete observations means the observations are distinct. There is a clear gap between the observations. Mean is a measure of the central tendency of a finite set of observations. It is also known as average. The average of some observations is obtained by dividing the sum of the observations by the total number of observations.
Formula used:
\[\text{ Mean of the observations} =\dfrac{\text{ (sum of the observations)}}{\text{(number of observations)}}\]
Complete step-by-step solution:
Here the observations are \[{y_1},{y_2},.....,{y_n}\]
Sum of the observations is \[{y_1} + {y_2} + ..... + {y_n}\], which can be expressed as the summation \[\sum\limits_{i = 1}^n {{y_i}} \], where \[i = 1,2,.....,n\]
and number of observations is \[n\].
Now, using the formula of mean, we get
\[m = \dfrac{{\sum\limits_{i = 1}^n {{y_i}} }}{n}\]
So, option A is correct.
Additional information:
The mean is the same as the average of the data. The sum of all observations divided by the number of observations is the mean of the data.
Discrete observation: If there is a separation between two observations, then data is known as discrete observations.
Note: \[\sum\limits_{i = 1}^n {{y_i}} \]represents the sum of the observations \[{y_1},{y_2},.....,{y_n}\], where \[i = 1,2,.....,n\]. You must have to write the starting value of \[i\] under the summation symbol \[\sum {} \]and the end value of \[i\] above the summation symbol. Since the value of \[i\] starts from \[1\] and ends at \[n\], so the number of values of \[i\] is equal to \[n\] and hence number of observations is \[n\].
Formula used:
\[\text{ Mean of the observations} =\dfrac{\text{ (sum of the observations)}}{\text{(number of observations)}}\]
Complete step-by-step solution:
Here the observations are \[{y_1},{y_2},.....,{y_n}\]
Sum of the observations is \[{y_1} + {y_2} + ..... + {y_n}\], which can be expressed as the summation \[\sum\limits_{i = 1}^n {{y_i}} \], where \[i = 1,2,.....,n\]
and number of observations is \[n\].
Now, using the formula of mean, we get
\[m = \dfrac{{\sum\limits_{i = 1}^n {{y_i}} }}{n}\]
So, option A is correct.
Additional information:
The mean is the same as the average of the data. The sum of all observations divided by the number of observations is the mean of the data.
Discrete observation: If there is a separation between two observations, then data is known as discrete observations.
Note: \[\sum\limits_{i = 1}^n {{y_i}} \]represents the sum of the observations \[{y_1},{y_2},.....,{y_n}\], where \[i = 1,2,.....,n\]. You must have to write the starting value of \[i\] under the summation symbol \[\sum {} \]and the end value of \[i\] above the summation symbol. Since the value of \[i\] starts from \[1\] and ends at \[n\], so the number of values of \[i\] is equal to \[n\] and hence number of observations is \[n\].
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