What is obtained by adding all the values and by dividing the total by the number of items?
In statistics, the Arithmetic Mean (AM) or called average is the ratio of the sum of all observations to the total number of observations. The arithmetic mean can also inform or model concepts outside of statistics. In a physical sense, the arithmetic mean can be thought of as a centre of gravity. From the mean of a data set, we can think of the average distance the data points are from the mean as standard deviation. The square of standard deviation (i.e. variance) is analogous to the moment of inertia in the physical model. Show
Say, for example, you wanted to know the weather in Shimla. 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 that could take into consideration a wide range of data. Instead of weather for every particular day, we use terms such as average (arithmetic mean), median and mode to describe weather over a month or so. 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 Read: Mean, Median and Mode What is 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 mean may be either
If any data set consisting of the values b1, b2, b3, …., bn then the arithmetic mean B is defined as: B = (Sum of all observations)/ (Total number of observation) \(\begin{array}{l}= \frac{1}{n}\sum_{i=1}^n b_i = \frac{b_1+b_2+b_3+….+b_n}{n}\end{array} \) If these n observations have corresponding frequencies, the arithmetic mean is computed using the formula \(\begin{array}{l}x = \frac{x_{1}f_{1}+x_{2}f_{2}+……+x_{n}f_{n}}{N}\end{array} \) and\(\begin{array}{l}\text{using Sigma notation} = \frac{\sum_{i=1}^{n}x_{i}f_{i}}{N}\end{array} \) where N = f1+ f2 + ……….+ fn. The above formula can also be used to find the weighted arithmetic mean by taking f1, f2,…., fn as the weights of x1, x2,….., xn. When the frequencies divided by N are replaced by probabilities p1, p2, ……,pn we get the formula for the expected value of a discrete random variable. X = x1p1 + x2p2 +…….+ xnpn. or \(\begin{array}{l}\text{using Sigma notation} = \sum_{i=1}^{n}x_{i}p_{i}\end{array} \) Properties of Arithmetic MeanSome important properties of the arithmetic mean are as follows:
Arithmetic Mean of Ungrouped DataFor ungrouped data, we can easily find the arithmetic mean by adding all the given values in a data set and dividing it by a number of values. Mean, x̄ = Sum of all values/Number of values Example: Find the arithmetic mean of 4, 8, 12, 16, 20. Solution: Given, 4, 8, 12, 16, 20 is the set of values. Sum of values = 4+ 8+12+16+20 = 60 Number of values = 5 Mean = 60/5 = 12 Arithmetic Mean of Ungroued DataIf x1, x2, x3 ,……,xn be the observations with the frequencies f1, f2, f3 ,……,fn, then the arithmetic mean is given by: x̄ = (x1f1+x2f2+……+xnfn) / ∑fi where ∑fi is the summation of all the frequencies. Let us understand the arithmetic mean of ungrouped data with the help of an example. Example: Find the mean of given distribution:
Solution: Let us find the value of xifi and ∑fi, for all the values of x and f respectively.
Hence, the required mean is: x̄ = 440/15 = 29.33 Merits of Arithmetic Mean
Demerits of Arithmetic Mean
Representative Values of DataWe 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 Shimla 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. Arithmetic Mean ExampleArithmetic means utilizes two basic mathematical operations, addition and division to find a central value for a set of values. Example: 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; The arithmetic mean of Virat Kohli’s batting scores also called his Batting Average is; Sum of runs scored/Number of innings = 661/10 The 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 11th innings are 70, the new average becomes; The 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 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. The same applies to the students with 90, in the 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 varied, 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. Arithmetic RangeRange, 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. 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. This gives us the extra information which is not getting through on average. Related Articles
Frequently Asked Questions on Arithmetic MeanThe arithmetic mean or mean is the simplest way to calculate the average for the given set of numbers. It is classified into two different types, namely simple
arithmetic mean and weighted arithmetic mean. The arithmetic mean is defined as the ratio of the sum of all the given observations to the total number of observations. For example, if the data set consists of 5 observations, the
arithmetic mean can be calculated by adding all the 5 given observations divided by 5. Add the two given numbers and then divide the sum by 2. For example, 2 and 6 are the two numbers, the arithmetic mean is calculated as
follows: To calculate the central tendency for the given data set, we use different measures like mean, median, mode and so on. Among all these measures, the
arithmetic mean or mean is considered to be the best measure, because it includes all the values of the data set. If any value changes in the data set, this will affect the mean value, but it will not be in the case of median or mode. Yes,
the arithmetic mean can be negative. The data can be distributed anywhere. So, the mean value can be negative or positive or zero. Learn more about the fascinating world of Mathematics by downloading BYJU’S – The Learning App. Join us and fall in love with learning. Is a value obtained by dividing the sum of all the values by their total number of values?The arithmetic mean is often known simply as the mean. It is an average, a measure of the centre of a set of data. The arithmetic mean is calculated by adding up all the values and dividing the sum by the total number of values.
What is the sum of the data divided by the total number of data?The mean of a set of numbers, sometimes simply called the average , is the sum of the data divided by the total number of data.
What is the name for the total of all the data values divided by the number of data points?The mode is the number in a set of numbers that appears the most often. The mean of a set of numbers is the sum of all the numbers divided by the number of values in the set. The mean is also known as the average.
What is the sum of the values of a group of items divided by the number of such items?The mean (also called arithmetic mean), in everyday language called the average, is the sum of the values of a group of numbers divided by the amount of numbers in the group.
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