Arithmetic Mean for Ungrouped Data

The observations based upon any test which happened, it can be any experiment for reading the changes in value, can be 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 useful for all the scenarios. Few observations work on range, but not all. 

In the statistical domain, the observation can be any set of values regardless of the experiment. Few scenarios can be height of people, marks of students, sales value per month, and many more. Therefore, it becomes abruptly difficult to get all the values and note them. Missing out values can make a serious issue. Hence, the concept leads to the origin of a new variable denoting this unique value such that it represents the overall observation. 

The arithmetic mean was introduced to be a value which 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 for Ungrouped Data

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. Now, the readings can have different values, wherein few can be repeated. Now, the term denotes the overall experiment as a whole. Thus, we can find the mean for the whole lot as one to represent. 

The experiment had m readings, and the values can be unique or repeating depending on the type of experiment we had. Suppose, the different values are m1, m2, m3…. and so on. 

Now, the mean will represent the overall data which is in the form of ungrouped data from the experiment carried out. 

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. The calculation for this is similar to finding out the average for any set of values for any test. 

Now, when we find the average, we initially observe the values we have from the experiment. These different values can be added together to get a single value. This summation of the observation is taken into consideration for finding out the mean to represent as a whole. Now, this value is divided by the total number of observed values to get the average value for the experiment. This value represents the whole lot uniquely and this is known as the mean for any given data. The arithmetic mean represents the mean for the given arithmetic observations. 

Thus, one can say that, for any ungrouped data,

Arithmetic Mean = m1+m2+m3+…..m

Where, 

The different observations are m1, m2, m3….

And m represents the number of values we noted from carrying out the experiment.

This formula can be used on any set of observations for a sample experiment. Statistics uses this in different domains to carry out the representation of the central tendency. This is way far the most prominent formula for evaluating a value to represent the overall experiment as a whole. 

Example

In a company, there was a sample experiment carried out based on the number of working hours in a day for a set of workers. The observations noted were as – 4,8,2,7,1,3,6,5,6,3. For the given experiment, the working hours for the whole lot for a day per worker can be represented using the arithmetic mean.

Thus, the observation is for 10 workers of the company. The data given is the ungrouped data, hence, now, using the definition, we compute the summation of the values.

Hence, Summation = 4+8+2+7+1+3+6+5+6+3=45

Now, the mean of the given set of experiment can be computed, 

Mean=SummationNumber of observation=4510=4.5

Thus, the overall lot of the worker taken into consideration can be said to work for 4 and half hours daily. 

Conclusion

The arithmetic mean of different observations for any set of tests or experiments which is ungrouped can be used to represent the whole as a 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. 

Note that, if we add or subtract a value from the observation, the mean value deviates from the computed value.