Properties of arithmetic mean

Central Tenancies are measures of location that summarise a dataset by giving a “single quantitative value” within the range of the data values. This tendency describes its (value’s) location relative to the entire data. Some common measures are: • • • Harmonic mean • • The most common, from all the measures that are mentioned above, is the Arithmetic mean. You would probably have heard your teacher saying “ this time the average score of the class is 70” or your friend saying “I get 10 bucks a month on average”. At that time, they are referring to the arithmetic mean. So, What is the arithmetic mean (A.P) and How to calculate it? To know the answer to these questions and to learn more about A.P, keep reading. Arithmetic Mean Definition Arithmetic mean and Average are different names for the same thing. It is obtained by the sum of all the numbers divided by the number of observations. It is used very often in daily life. But in day-to-day life, people often skip the word arithmetic or simply use the layman term “average”. Arithmetic Mean Formula Arithmetic mean is very easy to compute. But regardless of this fact, it does have a formula. A.P = Sum of all the observations / No. of observations For ungrouped data Here, n is equal to n 1 + n 2 + ... + n n. The drawback of A.P and Weighted Arithmetic Mean Arithmetic mean is not always practical to use. It is because it is highly skewed by the Also, the arithmetic mean fails to give a satisfactory average of the grouped data. ...

What are the properties of arithmetic mean? The properties are explained below with suitable illustration. Property 1: If x is the arithmetic mean of n observations x 1, x 2, x 3, . . x n; then (x 1 - x) + (x 2 - x) + (x 3 - x) + ... + (x n - x) = 0. Now we will proof the Property 1: We know that x = (x 1 + x 2 + x 3 + . . . + x n)/n ⇒ (x 1 + x 2 + x 3 + . . . + x n) = n x. ………………….. (A) Therefore, (x 1 - x) + (x 2 - x) + (x 3 - x) + ... + (x n - x) = (x 1 + x 2 + x 3 + . . . + x n) - n x = (n x - n x), [using (A)]. = 0. Hence, (x 1 - x) + (x 2 - x) + (x 3 - x) + ... + (x n - x) = 0. Property 2: The mean of n observations x 1, x 2, . . ., x n is x. If each observation is increased by p, the mean of the new observations is ( x + p). Now we will proof the Property 2: x = (x 1 + x 2 +. . . + x n)/n ⇒ x 1 + x 2 + . . . + x n) = n x …………. (A) Mean of (x 1 + p), (x 2 + p), ..., (x n + p) = /n, [using (A)]. = p x. Hence, the mean of the new observations is p x. Property 5: The mean of n observations x 1, x 2, . . ., x n is x. If each observation is divided by a nonzero number p, the mean of the new observations is ( x/p). Now we will proof the Property 5: x = (x 1 + x 2 + ... + x n)/n ⇒ x 1 + x 2 + ... + x n) = n x …………… (A) Mean of (x 1/p), (x 2/p), . . ., (x n/p) = (1/n) ∙ (x 1/p + x 2/p + …. x n/p) = (x 1 + x 2 + ... + x n)/np = (n x)/(np), [using (A)]. = ( x/p). To get more ideas students can follow the below links to understand how to solve various types of problems using th...

The observations based on any test conducted–be it 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 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 include people’s height, students’ marks, sales value per month, and more. Therefore, it becomes abruptly difficult to obtain all the values and note them. Missing values can cause serious issues. 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 that can represent overall data for the taken observation. Supporting the experiment, one can easily find the value representing observed values as a whole. Arithmetic Mean and its Characteristics 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. The experiment had m readings, and the values can be unique or repeating depending on our experiment type. Suppose the different values are m 1 , m 2 , m 3 …. and so on. Now, the me...

One of the characteristics of any given frequency distribution is central tendency. The characteristic by virtue of which the values of a variable tend to cluster around at the central part of the frequency distribution is called central tendency. A measure of central tendency is called an average. An average which is used to represent the whole series should neither have the lowest value nor the highest value in the group, but a value somewhere between two limits, possibly in the centre, where most of the items of the group cluster. In other words an average is a represented value of the whole set of observations, e.g., when we say “Germans are taller than Indians” we mean the average height of Germans is more than the average height of Indians. Sometimes a measure of central tendency is called a measure of location because it locates the position of the frequency distribution on the axis of the variable. An ideal measure of central tendency should have the following characteristics: • It should be rigidly defined so as to avoid different people choosing different values for the same measure of central tendency. • It should be easily comprehensible and easy to calculate. • It should be based upon all observations. • It should be amenable for further mathematical treatment. • It should be affected as little as possible by the presence of extreme values. • It should be least affected by sampling fluctuation, i.e., an ideal measure of central tendency should not vary in its ...