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.

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 Formula 

The 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 mean

If we have n number of values in a data i.e., x 1,x2,x3,…..,,xn. The mean is given as

= (x 1+x2+x3+…..+xn) / n

Also, = ∑ x / n

Arithmetic mean

The 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:

  • In algebraic treatment.
  • To calculate the average score in sports.
  • In economics
  • In anthropology
  • In history.
  • To measure the earth’s average temperature to measure global warming.
  • To measure the annual rainfall of a particular area.

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: The weighted arithmetic mean is similar to the arithmetic mean. The only difference is that each data point contributes equally to the final average. Some data points contribute more than others.

Weighted mean = Σw.x / Σw

where, 

Σ = summation,

w = the weights,

x = the value.

How to use the formula:

  1. Firstly find the product of the numbers in your data set and the weights.
  2. Add all the products in Step 1. Set this number aside.
  3. Now add up all of the weights.
  4. Divide the Summation of products by the Summation of weights.

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 Uses

The main uses of harmonic means are as follows:

  • It helps to find multiplicative or divisor relationships between fractions 
  • It is often used in averaging things like rates.

Merits and Demerits of Harmonic Mean

The following are the merits of the harmonic mean:

  • Rigidly confined.
  • All the items are involved in the calculation, i.e. no item is ignored.
  • It can advance the algebraic method.
  • It produces more accurate and reliable results as compared to other means.
  • It provides the highest weight to the smallest item of a series.
  • It can also be measured when a series holds any negative value.
  • It produces a skewed distribution of a normal one.
  • It creates a curve straighter than that of the A.M and G.M.

The demerits of the harmonic series are as follows:

  • The harmonic mean is greatly affected by the values of the extreme items
  • The calculation is not possible if any of the items is zero
  • The harmonic mean calculation is tiring, as it involves the analysis using the reciprocals of the number.

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 Examples 

Example 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:

10 20 36 92
92 88  80 70
92 40 50 50

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:

x 1 3 5 7 9 11
f 2 4 6 8 10 12

Solution: The calculation for the harmonic mean is shown in the below table:

(x) (f) (1/x) (f/x)
1 2 1 2
3 4 0.333 1.332
5 6 0.2 1.2
7 8 0.143 1.144
9 10 0.1111 1.111
11 12 0.091 1.092
N =42 Σ f/x = 7.879

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.