# Mean|Definition & Meaning

## Definition

The mean, also known as the average, is the measure of the central tendency of a given data set. It is calculated by first adding the total values given in a data set and then dividing it by the total number of values.

Figure 1 shows the calculation of the mean of four numbers 3, 6, 9, and 12, given in a data set.

Figure 1 – Demonstration of Mean as a Measure of Central Tendency of a Given Data Set

Notice how the mean is located at the center of the data set.

## Mean’s General Formula

The mean is the sum of all the values in the given data divided by the total number of values, so its formula can be written as:

Mean or Average = Sum of Values / Total Number of Values

## Other Measures of Central Tendency

The other measures of central tendency are the mode and the median. Each measure of central tendency indicates a different central value of a given set of numbers.

### Median

The middle value of a data set is known as the median when the numbers in the data set are arranged in ascending order.

### Mode

The value most repeated in a given set of numbers is known as its mode.

Figure 2 shows the calculation of the mean, mode, and median from a given set of numbers.

Figure 2 – Calculation of Mean, Mode and Median for a Given Set of Values

## Relationship Between Mean, Mode, and Median

The relationship between mean, mode, and median, also known as the “empirical formula,” exists only for a skewed distribution. It is given as:

2 * Mean = 3 * Median – Mode

If the mode and median for a given skewed distribution, the mean can be calculated from the above formula by dividing it by 2 as:

Mean = (3 * Median – Mode) / 2

## Calculation of Mean for Different Types of Data

In statistics, there are two types of data: grouped data and ungrouped data. For both types of data, the formulas for calculating the mean are different.

### Grouped Data

To calculate the mean $\overline{Y}$ for grouped data, the following formula is used:

$\overline{Y}_{\text{grouped data}} = \frac{\sum f_i y_i}{ \sum f_i }$

Where f is the frequency of the values in the frequency distribution grouped data.

The numerator is calculated by first multiplying the values $y_1,\, y_2,\, \ldots,\, y_i$ with their respective frequencies $f_1,\, f_2,\,\ldots,\, f_i$. The products $f_{1}y_{1},\, f_{2}y_{2},\, \ldots,\, f_{i}y_{i}$ are added to give the numerator.

The denominator is obtained by adding the frequencies. Figure 3 shows the mean calculation for grouped data.

Figure 3 – Mean Calculation for a Grouped Data (Frequency Distribution)

### Ungrouped Data

For ungrouped data, only the values $y_1,\, y_2,\, \ldots,\, y_m$ are given. The mathematical formula for mean $\overline{Y}$ for ungrouped data is:

$\overline{Y}_{\text{ungrouped data}} = \frac{ y_1 + y_2 + \cdots + y_m}{m}$

Where “m” is the number of values. If some of the values in the ungrouped data are negative, the same formula is used to calculate the mean.

## Major Types of Mean

There are three main types of mean, arithmetic mean, geometric mean, and harmonic mean.

### Arithmetic Mean

The arithmetic is the sum of all the observations divided by the total number of observations. The A.M is given by the formula:

$A.M = \frac{ \sum y_m}{m}$

For example, the arithmetic mean of 2, 7, 1, and 5 is:

A.M = (2 + 7 + 1 + 5) / 4 = 3.75

Till now we have discussed the arithmetic mean, as by “mean” the arithmetic mean is usually considered.

### Geometric Mean

The geometric mean G.M of two numbers p and q is the product of p and q to the square root of its product. Mathematically, it is:

G.M = $\sqrt{p \times q}$

For example, the geometric mean of 6 and 9 is:

G.M = $\sqrt{6 \times 9}$ = $\sqrt{54}$ = 7.348

The geometric mean of “m” values $y_1,\, y_2,\, \ldots,\, y_m$ is the product of these values to the mth root or 1/m power. So, in general, the G.M is:

$G.M = \sqrt[m]{ y_{1} \times y_{2} \times \cdots \times y_{m} }$

### Harmonic Mean

The harmonic mean H.M of two numbers p and q is given by:

H.M = (2pq) / (p + q)

For example, the harmonic mean of 4 and 6 is:

H.M = 2(4)(6) / (4 + 6) = 48 / 10 = 4.8

For m values $y_1,\, y_2,\, \ldots,\, y_m$, the harmonic mean is:

$\text{H.M} = m \div \left[ \dfrac{1}{y_1} + \dfrac{1}{y_2} + \cdots + \dfrac{1}{y_m} \right]$

Figure 4 shows the three measures of central tendency with the further categorization of mean and their formulas for two numbers p and q.

Figure 4 – Classification of the Measures of Central Tendency and Types of Mean with their Formulas

## Relationship Between Arithmetic, Geometric and Harmonic Mean

For a given set of data, the relation between A.M, G.M, and H.M is:

A.M ≥ G.M ≥ H.M

The three means will be equal if all the values of the data set are the same.

## Other Types of Mean

Some less common types of the mean are given as follows:

### Contraharmonic Mean

For two numbers p and q, the contraharmonic mean C.M will be:

$\text{C.M} = \dfrac{p^2 + q^2}{p + q}$

For m numbers $y_1,\, y_2,\,\ldots,\,y_m$, the C.M is:

$C.M = \frac{ {y_1}^2 + {y_2}^2 + \cdots + {y_m}^2}{ y_1 + y_2 + \cdots + y_m }$

### Root Mean Square

The root mean square $Y_{rms}$ is used to make the negative values positive in a data set. Its formula is given as:

$Y_{rms} = \sqrt{ \frac{ {y_1}^2 + {y_2}^2 + \cdots + {y_m}^2 }{m} }$

Where $y_1,\, y_2,\, \ldots,\, y_m$ are the m number of values of a data set.

### A Function’s Mean

The mean $x_{avg}$ of the function g(x) is the area under the function’s curve from c to d divided by the total length the area is covering. Mathematically, it is written as:

$x_{avg} = \frac{1}{d-c} \int_{c}^{d} g(x) dx$

## Example

Calculate the A.M, G.M, and H.M of the given set of values:

6, 8, 3, 10, 12

Show that:

A.M ≥ G.M ≥ H.M

### Solution

The arithmetic mean is calculated as follows:

$A.M = \frac{ \sum y_m }{m}$

A.M = (6 + 8 + 3 + 10 + 12) / 5 = 39 / 5

A.M = 7.8

The geometric mean will be:

$G.M = \sqrt[5]{ y_{1} \times y_{2} \times y_{3} \times y_{4} \times y_{5} }$

G.M = ${ \sqrt[5]{6 \times 8 \times 3 \times 10 \times 12} }$

G.M = 7.03

The harmonic mean will be:

$\text{H.M} = m \div \left[ \dfrac{1}{y_1} + \dfrac{1}{y_2} + \dfrac{1}{y_3} + \dfrac{1}{y_4} + \dfrac{1}{y_5} \right]$

H.M = 5 ÷ [(1 / 6) + (1 / 8) + (1 / 3) + (1 / 10) + (1 / 12)]

H.M = 6.18

As:

7.8 ≥ 7.03 ≥ 6.18

Hence:

A.M ≥ G.M ≥ H.M

All the images are created using GeoGebra.