Mean deviation is a measure of central tendency. We can calculate it from Arithmetic Mean, Median or Mode. It shows us how far are all the observations from the middle, on average? Each deviation is an absolute deviation as it is an absolute value which implies that we ignore the negative signs. Also, the deviations on both the sides of the Mean shall be equal. Let us start learning the mean deviation formula in detail.

## Mean Deviation Formula

The mean deviation is the mean of the absolute deviations of the observations or values from a suitable average. This suitable average may be the mean, median or mode. We also know it as the mean absolute deviation. We shall now learn more about some important formulas, for example, the mean deviation formula for an individual series or a continuous series, etc.

### 1] Individual Series

M.D. = \(\frac{\sum \left | X -\overline{X} \right |}{N}\)

Where,

\(\sum\) | Summation |

X | Observations or values |

\(\overline{X}\) | Mean |

N | Number of observations |

### 2] Discrete Series

M.D. = \(\frac{\sum f \left | X -\overline{X} \right |}{\sum f}\)

Where,

\(\sum\) | Summation |

X | Observations or values |

\(\overline{X}\) | Mean |

f | frequency of observations |

### 3] Continuous Series

M.D.= \(\frac{\sum f \left | X -\overline{X} \right |}{\sum f}\)

Where,

\(\sum\) | Summation |

X | Mid-value of the class |

\(\overline{X}\) | Mean |

f | frequency of observations |

**Mean deviation from Median**

### 1] Individual Series

M.D. = \(\frac{\sum \left | X – M \right |}{N}\)

Where,

\(\sum\) | Summation |

X | Observations or values |

M | Median |

N | Number of observations |

### 2] Discrete Series

M.D. = \(\frac{\sum f \left | X – M \right |}{\sum f}\)

Where,

\(\sum\) | Summation |

X | Observations or values |

M | Median |

f | frequency of observations |

### 3] Continuous Series

M.D.= \(\frac{\sum f \left | X -\overline{X} \right |}{\sum f}\)

Where,

\(\sum\) | Summation |

X | Mid-value of the class |

M | Median |

f | frequency of observations |

## Mean deviation from Mode

### 1] Individual Series

M.D. = \(\frac{\sum \left | X – Mode \right |}{N}\)

Where,

\(\sum\) | Summation |

X | Observations or values |

M | Mode |

N | Number of observations |

### 2] Discrete Series

M.D. = \(\frac{\sum f \left | X – Mode \right |}{\sum f}\)

Where,

\(\sum\) | Summation |

X | Observations or values |

Mode | Mode |

f | frequency of observations |

### 3] Continuous Series

M.D.= \(\frac{\sum f \left | X – Mode \right |}{\sum f}\)

Where,

\(\sum\) | Summation |

X | Mid-value of the class |

Mode | Mode |

f | frequency of observations |

**Steps to Calculate the Mean Deviation:**

- Calculate the mean, median or mode of the series.
- Calculate the deviations from the Mean, median or mode and ignore the minus signs.
- Multiply the deviations with the frequency. This step is necessary only in the discrete and continuous series.
- Sum up all the deviations.
- Apply the formula.

**The formula for the Co-efficient of Mean Deviation**

- Co-efficient of Mean Deviation from Mean = \(\frac{M.D.}{ \overline{X}}\)
- Co-efficient of Mean Deviation from Median = \(\frac{M.D.}{M}\)
- The Co-efficient of Mean Deviation from Mode = \(\frac{M.D.}{Mode}\)

## Solved Examples

Q.1. Calculate the mean deviation from the median and the co-efficient of mean deviation from the following data:

Marks of the students: 86, 25, 87, 65, 58, 45, 12, 71, 35.

Solution: Arrange the data in ascending order: 12, 25, 35, 45, 58, 65, 71, 86, 87.

Median = Value of the \(\frac{N+1}{2}^{th} term\)

= Value of the \(\frac{9+1}{2}^{th} term = 58\)

Calculation of mean deviation:

X | \(\left | X – M \right |\) |

12 | 46 |

25 | 33 |

35 | 23 |

45 | 13 |

58 | 0 |

65 | 7 |

71 | 13 |

86 | 28 |

87 | 29 |

N = 9 | \(\sum \left | X – M \right | = 460\) |

M.D. = \(\frac{\sum \left | X – M \right |}{N}\)

= \(\frac{460}{9}\)

= 51.11

Co-efficient of Mean Deviation from Median = \(\frac{M.D.}{M}\)

= \(\frac{51.11}{58}\)

= 0.881

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