In the early morning while reading a newspaper, have you observed the daily temperature reports. Well, the temperature varies all day still how can a single temperature indicate the condition for the entire day? Or when you get your scorecard in exams, instead of analyzing your performance based on percentage in all subjects, the performance is based upon the aggregate percentage. The significance of indicating a single value for a large amount of data in real life makes it easy to study and analyze the collection of data and deduce important information out of it.
This condensation of a large amount of data into a single value is known as measures of central tendency. These measures of central tendency enable us to make a statistical summary of the huge organized data. One such method of measure of central tendency is the arithmetic mean. Let us discuss the arithmetic mean and its applications.
The most common measure of central tendency is the arithmetic mean. In layman terms, mean of data indicates an average of the given collection of data. It is equal to the sum of all the values in the group of data divided by the total number of values.
For n values in a set of data namely as x1, x2, x3 ………. xn, the mean of data is given as:
It can also be denoted as:
For calculating the mean when the frequency of the observations is given, such that x1, x2, x3 ………. xn are the recorded observations and f1, f2, f3 ………. fn are the respective frequencies of the observations then;
This can be expressed briefly as:
The above method of calculating the arithmetic mean is used when the data is ungrouped in nature. For calculating the mean of grouped data we calculate the class mark. For this, the midpoints of the class intervals is calculated as:
After calculating the class mark, the mean is calculated as discussed earlier. This method of calculating the mean is known as the direct method. Let us look into an example to understand this clearly,
Example: In a class of 30 students marks obtained by students in mathematics out of 50 is tabulated below. Calculate the mean of the data.
The mean of the data given above is,
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