# 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

- 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. - 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. - 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. - 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—

- Direct method
- Short-cut method
- 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

- It is based on every observation in the data set.
- It is rigidly defined.
- It is more suitable while averaging ratios and percentages or calculating growth rates.
- It gives more weight to smaller values (compared to arithmetic mean). This is why is less than or equal to the arithmetic mean.
- 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

- It is sowehat difficult to understand, compared to arithmetic mean.
- It is to compute and to interprete.
- 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

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

### Limitations

- It is difficult to understand and to calculate.
- It cannot be calculated if any of the observations is zero or negative.
- 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:

Type of Test | Relative Weight | Marks |
---|---|---|

Mid-term | 2 | 30 |

Laboratory | 3 | 25 |

Final | 5 | 20 |

2.1 Calculate the weighted arithmetic mean.

#### Solution 2.1

Type of Test | Relative Weight $(w)$ | Marks $(x)$ | $wx$ |
---|---|---|---|

Mid-term | 2 | 30 | 60 |

Laboratory | 3 | 25 | 75 |

Final | 5 | 20 | 100 |

Total | 10 | 235 |

Weighted arithmetic mean:

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

$$\bar{x}=23.5$$

### Example 3

Consider the following data:

Marks | No. of Students |
---|---|

0-10 | 4 |

10-20 | 8 |

20-30 | 11 |

30-40 | 15 |

40-50 | 12 |

50-60 | 6 |

60-70 | 2 |

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

Marks | No. of Students $(f)$ | Midpoint $(m)$ | $fm$ | $d=x-A$, (A=35) | $fd$ | $d'=d/C$, (C=10) | $fd'$ |
---|---|---|---|---|---|---|---|

0 - 10 | 4 | 5 | 20 | -30 | -120 | -3 | -12 |

10 - 20 | 8 | 15 | 120 | -20 | -160 | -2 | -16 |

20 - 30 | 11 | 25 | 275 | -10 | -110 | -1 | -11 |

30 - 40 | 15 | 35 | 525 | 0 | 0 | 0 | 0 |

40 - 50 | 12 | 45 | 540 | 10 | 120 | 1 | 12 |

50 - 60 | 6 | 55 | 330 | 20 | 120 | 2 | 12 |

60 - 70 | 2 | 65 | 130 | 30 | 60 | 3 | 6 |

Total | 58 | 1940 | -90 | -9 |

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

Marks | No. of Students $(f)$ | Midpoint $(x)$ | $f(log x)$ | $f(\frac{1}{x})$ |
---|---|---|---|---|

0 - 10 | 4 | 5 | 2.7959 | 0.8 |

10 - 20 | 8 | 15 | 9.4087 | 0.5333 |

20 - 30 | 11 | 25 | 15.3773 | 0.44 |

30 - 40 | 15 | 35 | 23.1610 | 0.4286 |

40 - 50 | 12 | 45 | 19.8386 | 0.2667 |

50 - 60 | 6 | 55 | 10.4422 | 0.1091 |

60 - 70 | 2 | 65 | 3.6258 | 0.0308 |

Total | 58 | 84.6495 | 2.6085 |

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:

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

$$HM=Reciprocal~of~2.6085$$

$$HM\approx~0.3834$$