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# 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.

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**.

## 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.

### 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**.

## 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.*