In Mathematics, we come across statistics to measure central tendency and dispersion. The standard measures of central tendencies are mean, median, and mode. The dispersion consists of variance and standard deviation. In this section, we will discuss central tendency and learn what it means in math. Show Mean, or arithmetic mean, is the average of the given set of data or observations. Mean is measured when an intermediate value is required. This value lies between the extreme values of the presented collection of data. We can calculate it by dividing the sum of observations by the total number of observations. What is mean in math? This is a common question asked by many scholars. Mean is the most popular method of measuring central tendency. It is used mainly for continuous data but can also work for discrete data. Mean = (sum of observations)/(total number of observations). The other two methods are median and mode. The Median is the middle number in a sorted, ascending, or descending set. On the other hand, the mode is the most frequent score in our data set. Mean FormulaThe mean formula is as simple to understand as the mean definition. The mean (or average) of several observations is the sum of the values of all the observations divided by the total number of observations. The symbol x denotes it, read as ‘x̄’ Mean Formula = Sum of Given Data / Total number of data Calculation of meanIf we have n number of values in a data i.e., x 1,x2,x3,…..,,xn. The mean is given as x̄ = (x 1+x2+x3+…..+xn) / n Also, x̄ = ∑ x / n Arithmetic meanThe arithmetic mean is the simplest and most widely used method to calculate the mean. The following are some of the crucial applications of arithmetic mean:
Different types of mean The mean or average most commonly used is the arithmetic mean. However, there are some other types of the mean. The use depends upon the data available and the type of results required. Following are the kinds of mean:
Weighted mean = Σw.x / Σw where, Σ = summation, w = the weights, x = the value. How to use the formula:
a.Geometric mean: Indicates the central tendency of a set of data by using the product of their values rather than their sum. Geometric mean = √x 1.x2….xn where, n = the total number of observations. b.Harmonic mean: The mean is calculated by the reciprocal of values instead of the values themselves. If x 1,x2,x3,….., xn are the individual items up to n terms, then, Harmonic Mean, HM = n / [(1/ x 1)+(1/x2)+(1/x3)+…+(1/xn)] Harmonic Mean UsesThe main uses of harmonic means are as follows:
Merits and Demerits of Harmonic MeanThe following are the merits of the harmonic mean:
The demerits of the harmonic series are as follows:
Relation between Arithmetic, Geometric and Harmonic Means The three means namely arithmetic mean, geometric mean, harmonic mean are together known as Pythagorean mean. Following are the formulas: Arithmetic Mean = (a1+a2+a3+….+an) / n Harmonic Mean = n / [(1/a1)+(1/a2)+(1/a3)+…+(1/an)] Geometric Mean = n√a1.a2.a3…an Let G be the geometric mean, H harmonic mean, and A arithmetic mean, then the relationship between them is given by: G=√AH Or G² = A.H Why is Geometric Mean Better than an Arithmetic Mean?The geometric mean and arithmetic mean are the two methods to determine the average. The geometric mean is always less than the arithmetic means for any two positive unequal numbers. Sometimes, arithmetic mean works better, like representing average temperatures, etc. Let us Learn How to Calculate the Mean by a Few ExamplesExample 1: Seven people worked for 11, 8, 14, 21, 9, 12, and 16 hours, respectively, doing social work in their community in a week. Find the mean (or average) of their time for social work in a week. Solution: Mean= Sum of Given Data/ Total number of data = (11+ 8+ 14+ 21+ 9+ 12+ 16)/ 7 = 13 So, the mean time spent by these seven people doing social work is 13 hours a week. Example 2: Find the mean of the marks obtained by 12 students of Class 9 of a school out of 100. Marks are given below in the table:
Solution: Mean= Sum of Given Data/ Total number of data = (10+ 20+ 36+ 92+ 92+ 88+ 80+ 70+ 92+ 40+ 50+50)/12 = 60 Example 3: Find the harmonic mean and geometric mean for data 3, 6, 9, and 12. Solution: Given data: 3, 6, 9, 12 Harmonic Mean: Step 1: Finding the reciprocal of the values: 1/3 = 0.33 1/6 = 0.16 1/9 = 0.11 1/12 = 0.08 Step 2: Calculate the average of the reciprocal values from step 1. Here, the total number of data values is 4. Average = (0.33 + 0.16 + 0.11 + 0.08)/4 Average = 0.68/4 Step 3: Finally, take the reciprocal of the average value obtained from step 2. Harmonic Mean = 1/ Average Harmonic Mean = 4/0.68 Harmonic Mean = 5.88 Hence, the harmonic mean for the data is 5.88. Geometric Mean: Step 1: n = 4 is the total number of values. Now, find 1/n. 1/4 = 0.25. Step 2: Find geometric mean using the formula: (3× 6× 9× 12)0.25 So, geometric mean = 6.640 Question 4: Calculate the arithmetic mean of the given numbers: 2.5, 4.8, 2.7, 6.0, 3.1, 6.4, 7.2, 8.2, and 5.5. Solution: Given that, the numbers are 2.5, 4.8, 2.7, 6.0, 3.1, 6.4, 7.2, 8.2, and 5.5. Step 1: The count of numbers is 9. Calculate the sum of the numbers. 2.5 + 4.8 + 2.7 + 6.0 + 3.1 + 6.4 + 7.2 + 8.2 + 5.5 = 46.4 Step 2: Calculate the arithmetic mean of the given numbers. 46.4/9 = 5.15 Hence, the arithmetic mean of the given numbers is 5.15 Question 5: Calculate the harmonic mean for the following data:
Solution: The calculation for the harmonic mean is shown in the below table:
The formula for weighted harmonic mean is HMw = N / [(f1/x 1)+(f2/x2)+(f3/x3)+….(fn/xn )] HMw = 42 / 7.879 HMw = 5.331 Therefore, the harmonic mean, HMw is 5.331. When the sum of a set of observation divided by number of observation is?Arithmetic mean refers to the average amount in a given group of data. It is defined as the summation of all the observation is the data which is divided by the number of observations in the data.
When the sum of all items is divided by their number it is known as?The arithmetic mean, also known as arithmetic average, is the sum of all the values in a list of numerical values divided by the number of items in the list.
What is the formula of sum of observations?Arithmetic Mean, AM = Sum of all Observations/Total Number of Observations.
Is mean the sum of observations?Mean is defined as the ratio of the sum of all observations to the total number of observations. No worries!
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