Data Handling - Arithmetic Mean And Range

There is an immense amount of data in this world. The internet age has only compounded this problem. It would do well to remember that too much data is bad and to this end, we have introduced the concept of representative values in data handling. Say for example, you wanted to know the weather in Ooty. On the internet you would find the temperatures for a lot of days, data of the temperature in the past and the data of the temperature in the present and also the predictions of the temperature in the future. Wouldn’t all this be extremely confusing? Instead of this long list of data mathematicians decided to use representative values which could take into consideration a wide range of data. Instead of weather for every particular day, we use terms such as average (mean), median and mode to describe weather over a month or so. We see the use of representative value quite regularly in our daily life. When you ask about the mileage of the car, you are asking for the representative value of the amount of distance travelled to the amount of fuel consumed. This doesn’t mean that the temperature in Ooty in constantly the representative value but that overall it amounts to the average value. Average here represents a number that expresses a central or typical value in a set of data, calculated by the sum of values divided by the number of values.

Mean, median, mode

There are several types of Representative values that are used by mathematicians in Data handling, namely;

Out of the four above, mean, median and mode are types of average.

Arithmetic Sequence is defined as the sequence of numbers in a manner that the difference between the consecutive terms is same. For example, 5, 7, 9, 11 is an arithmetic sequence since the common difference between them is 2.

Arithmetic Mean

Arithmetic mean represents a number that is obtained by dividing the sum of the elements of a set by the number of values in the set. So you can use the layman term Average, or be a little bit fancier and use the word “Arithmetic mean” your call, take your pick -they both mean the same. The Arithmetic means utilizes two basic mathematical operations, addition and division to find a central value for set of values. If you wanted to find the arithmetic means of the runs scored by Virat Kohli in the last few innings, all you would have to do is sum up his runs to obtain sum total and then divide it by the number of innings. For example;

Innings 1 2 3 4 5 6 7 8 9 10
Runs 50 59 90 8 106 117 59 91 7 74

The arithmetic mean of Virat Kohli’s batting scores also called his Batting Average is;

Sum of runs scored/Number of innings  \(=\frac{661}{10}\)

Arithmetic mean of his scores in the last 10 innings is 66.1. If we add another score to this sum, say his 11th innings, the arithmetic mean will proportionally change. If the runs scored in the 11th innings is 70, the new average becomes;

                            \(=\frac{661+70}{10+1} = \frac{731}{11} = 66.45\)

The Arithmetic Mean formula is given by:

\(A = \frac{1}{n} .\sum_{i = 1}^{n}x_{i}\)

Where,

A is the arithmetic mean.

n is the number of items or numbers.

xi  is the value of every individual item being averaged.

Geometric Mean formula

\(\bar{x}_{geom} = \sqrt[n]{x_{1}. x_{2}…….x_{n}}\)<

Geometric mean is defined as the nth root of the product of n values in a data.

  • The geometric mean of two values is equal to the square root of their product.
  • The geometric mean of three values is equal to the cubic root of their product.

Average is a pretty neat tool, but it comes with its set of problems. Sometimes it doesn’t represent the situation accurately enough. I’ll show you what I mean. Let’s take the results of a class test for example. Say there are 10 students in the class and they recently gave a test out of 100 marks. There are two scenarios here.

First:  50, 53, 50, 51, 48, 93, 90, 92, 91, 90

Second: 71, 72, 70, 75, 73, 74, 75, 70, 74, 72

Why don’t you calculate the Arithmetic mean of both the sets above? You will find that both the sets have a huge difference in the value even though they have a similar arithmetic mean. In this respect, completely relying on Arithmetic mean can be occasionally misleading. At least from the point of view of students scoring 50’s/ 100, the second scenario is quite different. Same applies for the students with 90, in case of these students in the second set, the marks are reduced. So for both the classes the results mean something different but the average for both classes are the same. In the first class, the students are performing very variedly, some very well and some not so well whereas in the other class the performance is kind of uniform. Therefore we need an extra representative value to help reduce this ambiguity.

Range

Range as the word suggests represents the difference between the largest and the smallest value of data. This helps us determine the range over which the data is spread. Taking the previous example into consideration once again. There are 10 students in the class and they recently gave a test out of 100 marks. There are two scenarios here.

First:  50, 53, 50, 51, 48, 93, 90, 92, 91, 90

Second: 71, 72, 70, 75, 73, 74, 75, 70, 74, 72

The range in the first scenario is represented by the difference between the largest value, 93 and the smallest value, 48. The range therefore is,

Range in First set = 93 – 48 = 45

Whereas in the second scenario, the range is represented by the difference between the highest value, 75 and the smallest value, 70.

Range in the second set = 75 – 70 = 5

The difference in the value of Range between the two scenario enables us to estimate the range over which the values are spread, the larger the range, the larger apart the values are spread.  To learn more about this topic download Byju’s – The Learning App.


Practise This Question

There are 50 students in a class and the average mark scored by the class in Maths is 65 out of 100. The total marks scored by all the students except Karan and Arjun is 3080. If the marks scored by Karan & Arjun are same. What are the marks scored by Arjun?