Arithmetic Mean in Continuous Series

The observations based upon any test can be an experiment for reading the changes in value and noted to vary between a range. The value for each experiment may not be identical. These values may be noted to be within a range of numbers. Thus, the range may not be helpful for all the scenarios. Few observations work on the range, but not all. 

In the statistical domain, the observation can be any set of values regardless of the experiment. A few scenarios include the height of people, marks of students, sales value per month, etc. Therefore, it becomes difficult to get all the numbers and note them. Missing out values can be a severe issue. Hence, the concept leads to the origin of a new variable denoting this unique value, representing the overall observation. 

The arithmetic mean was introduced to be a value that can represent the overall data for the taken observation. Supporting the experiment, one can easily find the value representing the observed values as a whole. 

Arithmetic Mean in Continuous Series

Assume that a sample experiment takes place such that the observed values are in a given range. Suppose a total of m readings were noted and analysed. The readings can have different values, wherein a few can be repeated. Now, the term mean denotes the overall experiment as a whole. Thus, we can find the mean to represent the whole lot. 

What is the mean? The answer to this question is the overall representation of data. Now, this is the definition we know from the above analysis. 

The mean is computed from the data by taking the average for each entry to the exact value. The Mean can be said to be the mid-value such that the total deviation is tending to zero from this unique represented value for the overall data. This calculation is similar to finding out the average for any set of values for any test. 

For any continuous series, we can compute the arithmetic mean using different methods wherein the procedure remains the same with assumptions.  

The experiment has different readings, and the values can be unique or repeated depending on our experiment type. Suppose the different values are given in the form of intervals with their respective frequencies. f1, f2, f3…. Thus, for any continuous series, evaluate the midpoints of each interval and take that as values. 

Values=upper limit + lower limit/2

Thus, using the direct method to evaluate the arithmetic mean, we would need to multiply the value and frequencies:

Arithmetic Mean=Σfm/Σf

Now, the values may be extremely large, thus we assume a value and we subtract the values to get an assumed mean. Therefore, the arithmetic mean is evaluated using the short-cut method,

Arithmetic Mean=A+Σfb/Σf and b=m-A

Now, the deviation from the assumed mean can be extreme. Thus, we make this value small by dividing it by a factor. This is the step deviation method, and the arithmetic mean computed as 

Arithmetic Mean=A+Σfb/Σf and b=m-Ac 

Example

A sample experiment was carried out in a company based on the number of working hours in a day for a set of workers. The noted observations were – 

For the given experiment, the working hours per worker for the whole day can be represented using the arithmetic mean. The above is a continuous series, and thus, from the formula, we can compute the interval values and mean. 

Thus, the workers represent the frequencies for hours. So, using the arithmetic mean formula, we can say that, 

Arithmetic Mean = (5*1.5+4.5*1+2*7.5+7*10.5)/(5+1+2+7) = 7.5+4.5+15+73.5/15 = 100.5/15 = 6.7

Thus, the overall lot of the worker taken into consideration can be said to work for 6.7 hours daily. 

Conclusion

The arithmetic mean of different observations for any set of tests or experiments can be used to represent the whole as one valued observation. This value can be part of the experimental observations or a unique value for the experiment. Depending on the number and value of the observations, the mean can have different values.