Arithmetic Mean And Range

In statistics, arithmetic mean is the average of the given set of numbers or observations. The arithmetic range is the difference between the highest value and lowest value in a set of observations. Arithmetic mean is also called average or simply mean.

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. Using the internet, you would find the temperatures for many 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. 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.

Arithmetic Mean, Range, Median and Mode

There are several types of representative values that are used by mathematicians in data handling, namely;

• Arithmetic Mean (Average)
• Range
• Median
• Mode

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 the 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 fancier and use the word “Arithmetic mean“. Arithmetic means utilizes two basic mathematical operations, addition and division to find a central value for a set of values. It is usually denoted by a bar (x̄).

Arithmetic Mean Formula

Arithmetic mean = Sum of observations/Number of observations

Example

If you wanted to find the arithmetic means of the runs scored by a cricketer in the last few innings, all you would have to do is sum up his runs to obtain total and then divide it by the number of innings. For example;

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

Sum of runs scored/Number of innings  

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

The Arithmetic Mean formula is given by:

Where,

A is the arithmetic mean.

n is the number of items or numbers.

x_i  is the value of every individual item being averaged.

Geometric Mean

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.

The average is a pretty neat tool, but it comes with its set of problems. Sometimes it doesn’t represent the situation accurately enough. Let’s take the results of a class test, for example. Say there are ten 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 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/ 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, there is a huge difference in the student’s performance, some students scored 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.

Arithmetic 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.

Range Formula

Arithmetic Range = Highest value – Lowest value

Take the previous example into consideration once again. There are ten 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 scenarios enables us to estimate the range over which the values are spread. The larger the range, the larger apart the values are spread.

Solved Examples on Arithmetic Mean and Range

Q.1: What is the arithmetic mean of the first 10 natural numbers?

Solution: The first 10 natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Arithmetic mean = Sum of all values ÷ Total number of values

= (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) ÷ 10

= 55 ÷ 10

= 5.5

Q.2: Find the range of the given data set.

23, 2, 7, 12, 21, 33, 5

Solution: Given the data set is 23, 2, 7, 12, 21, 33, 5.

Range = Highest value – Lowest value

Highest value = 33

Lowest value = 2

Range = 33 – 2 = 31

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