The Mean — Definitions, Types, Formulae [Worked Examples]

by Jack Bodeley on September 07, 2021

The mean is a measure of central tendency because at the mean, the sum of deviations in the positive direction is equal to the sum of deviations in the negative direction.

Arithmetic Mean

Arithmetic mean is obtained by summing up the values of all items of a series and dividing this sum by the number of items.

Characteristics

1. The sum of the deviations of the individual items from the arithmetic mean is always zero i.e. $(x-\bar{x})=0$, where $x$ is the value of an item and $\bar{x}$ is the arithmetic mean.
2. The sum of squared deviations of the individual items from the arithmetic mean is always minimal i.e. sums of squared deviations obtained from values other than the arithmetic mean are always larger.
3. It is based on all items in a series. Meaning, a change in the value of any item will lead to a change in the value of the arithmetic mean.
4. The arithmetic mean gets distorted in a highly skewed distribution with few extreme values. In that case it ceases to be a represensative characteristic of the distribution.

Types

There are two types of arithmetic mean—

• Simple arithmetic mean
• Weighted arithmetic mean

Formula: Ungrouped, Unweighted Data (Simple Arithmetic Mean)

For individual series data, values of the variable are summed up and then the sum is divided by the total number of items.

$$\mu=\frac{\sum~x}{n}$$

$$or;$$

$$\bar{x}=\frac{\sum~x}{n}$$

$\mu$ is used to denote the mean of the population, $\bar{x}$ the sample mean where $n$ is the total number of observations.

Formula: Ungrouped, Weighted Data (Weighted Arithmetic Mean)

When arithmetic mean is calculated, equal importance is given to all items in the series. In cases where items have a relative degree of importance, weights are assigned and a weighted arithmetic mean is calculated instead.

If $w_1$, $w_2$... $w_n$ are assigned to $x_1$, $x_2$... and $x_n$ respectively, the weighted average mean is computed with the following formula:

$$\bar{x}=\frac{w_1x_1+w_2x_2+...+w_nx_n}{w_1+w_2+...+w_n}=\frac{\sum wx}{\sum w}$$

The simple arithmetic mean for grouped data can be calculated using three methods—

1. Direct method
2. Short-cut method
3. Step-deviation method

Formula: Grouped Data (Direct Method)

$$\bar{x}=\frac{\sum fm}{n}$$

Where $m$ and $f$ are the mid-point and the frequency of the class, respectively, and $n$ is the total number or summation of the frequencies.

When the values are extremely large or are in fractions, the use of the direct method is generally very cumbersome. The short-cut method would be preferable where calculations are considerably reduced.

Formula: Grouped Data (Short-cut Method)

The short-cut method follows the concept of the arbitrary mean (also known as the assumed mean or presumed mean).

$$\bar{x}=A+\frac{\sum~fd}{n}$$

Where $A$ is the arbitrary or assumed mean, $f$ the frequency, and $d$ the deviation from the assumed mean.

Formula: Grouped Data (Step-deviation Method)

$$\bar{x}=A+\frac{\sum~fd^{'}}{n}\times~C$$

Of these three methods, the step-deviation is the most convenient and simplest to calculate.

Geometric Mean

Geometric mean is the $n^{th}$ root of the product of $n$ observations of a distribution. That means, if there are two values a square root is calculated; if there are three a cube root, and so on.

In most cases, the geometric mean is more useful than the harmonic mean.

Formula

$$GM=\sqrt[n]{x_1\times~x_1\times...\times~x_n}=Antilog~(\frac{\sum~log~x}{n})$$

Advantages

1. It is based on every observation in the data set.
2. It is rigidly defined.
3. It is more suitable while averaging ratios and percentages or calculating growth rates.
4. It gives more weight to smaller values (compared to arithmetic mean). This is why is less than or equal to the arithmetic mean.
5. It is capable of easy algebraic manipulation e.g. is the GM or more than one series is known wlong with their respective frequencies, a combined geometric mean can be calculated using logarithms.

Limitations

1. It is sowehat difficult to understand, compared to arithmetic mean.
2. It is to compute and to interprete.
3. It cannot be calculated in a series with negative or zero values.

Harmonic Mean

Harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of the individual observations.

Formula: Individual Series

$$HM=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}=Reciprocal~of~\frac{\sum\frac{1}{x}}{n}$$

Formula: Continuous Series

$$HM=\frac{n}{\sum_{i=1}^{n}(f_i\times\frac{1}{x_i})}=Reciprocal~of~\sum_{i=1}^{n}(f_i\times\frac{1}{x_i})$$

Where $n$ is the total number of observations.

Advantages

1. It is based on all observations in a distribution
2. It is amenable to further algebraic treatment.
3. It is suitable when we want to allocate greater weights to smaller observations and lesser weights to larger observations.

Limitations

1. It is difficult to understand and to calculate.
2. It cannot be calculated if any of the observations is zero or negative.
3. It is just a summary value that may not be an observation in the distribution.

Quadratic Mean

The quadratic mean is the square root of the arithmetic mean of the squares of items in a distribution.

Formula

$$QM=\sqrt\frac{x_1^2+x_2^2+...+x_n^2}{n}$$

Relative Positions of Different Means

The geometric mean of a distribution is always lower than the arithmetic mean of the same distribution.

The harmonic mean of a distribution is always lower than the geometric and arithmetic means of the same distribution. This is because the harmonic mean assigns lesser importance to higher values — reciprocals of higher values are lower than those of lower values.

Thus;

$$QM>\bar{x}>GM>HM$$

Worked Examples

Example 1

Consider the following series:

10, 15, 30, 7, 42, 79 and 83

1.1. Calculate the arithmetic, geometric, harmonic and quadratic means.

Solution 1.1

Arithmetic mean:

$$\mu=\frac{\sum~x}{n}$$

$$\mu=\frac{10+15+30+7+42+79+83}{7}=\frac{266}{7}$$

$$\mu=38$$

Geometric mean:

$$GM=\sqrt[n]{x_1\times~x_1\times...\times~x_n}$$

$$GM=\sqrt[7]{10\times15\times30\times7\times42\times79\times83}=\sqrt[7]{8674911000}$$

$$GM\approx~26.2877$$

Harmonic mean:

$$HM=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$$

$$HM=\frac{7}{\frac{1}{10}+\frac{1}{15}+\frac{1}{30}+\frac{1}{7}+\frac{1}{42}+\frac{1}{79}+\frac{1}{83}}$$

$$HM\approx~17.8857$$

Quadratic mean:

$$QM=\sqrt\frac{x_1^2+x_2^2+...+x_n^2}{n}$$

$$QM=\sqrt\frac{10^2+15^2+30^2+7^2+42^2+79^2+83^2}{7}=\sqrt\frac{16168}{7}$$

$$QM\approx~48.0595$$

Example 2

Consider the following data:

2.1 Calculate the weighted arithmetic mean.

Solution 2.1

Weighted arithmetic mean:

$$\bar{x}=\frac{\sum wx}{\sum w}=\frac{235}{10}$$

$$\bar{x}=23.5$$

Example 3

Consider the following data:

3.1 Calculate the arithmetic mean using the direct, short-cut and step deviation methods.

3.2 Calculate the geometric, harmonic means.

Solution 3.1

Arithmetic mean (direct method):

$$\bar{x}=\frac{\sum fm}{n}=\frac{1940}{58}$$

$$\bar{x}\approx~33.4483$$

Arithmetic mean (short-cut method):

$$\bar{x}=A+\frac{\sum~fd}{n}$$

$$\bar{x}=35+\frac{-90}{58}$$

$$\bar{x}\approx~33.4484$$

Arithmetic mean (step-deviation method):

$$\bar{x}=A+\frac{\sum~fd^{'}}{n}\times~C$$

$$\bar{x}=35+\frac{-9}{58}\times~10$$

$$\bar{x}\approx~33.4483$$

Solution 3.2

Geometric mean:

$$GM=Antilog~(\frac{\sum~(f\times~log~x)}{n})$$

$$GM=Antilog~(\frac{84.6495}{58})=Antilog~1.4595$$

$$GM\approx~28.8054$$

Harmonic mean:

$$HM=Reciprocal~of~\sum_{i=1}^{n}(f_i\times\frac{1}{x_i})$$

$$HM=Reciprocal~of~2.6085$$

$$HM\approx~0.3834$$

References