There are many different ways of characterizing distribution curves in statistics. The easiest of them almost always include some or the other measure of dispersion (i.e. the *spread)* of the distribution. Therefore, depending on the need of the hour, different measures of dispersion like range, mean deviation(precisely- mean deviation formula), standard deviation, the coefficient of variation etc. might be employed.

Some of them are absolute in nature (have units, and are distribution specific), while the others are relative (unit-less and can be used to compare different distributions). Whatever they might be, each of them finds various applications in different sectors of Economics, Mathematics, Data Analysis etc. and is important to study. Therefore, begins our discussion to this topic from the basic measures of dispersion like the range and the mean deviation, and move on to advanced topics at a later stage. So let’s begin!

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## Range

The Range of a distribution gives a measure of the width (or the spread) of the data values of the corresponding random variable. For example, if there are two random variables X and Y such that X corresponds to the age of human beings and Y corresponds to the age of turtles, we know from our general knowledge that the variable corresponding to the age of turtles should be larger.

Since the average age of humans is 50-60 years, while that of turtles is about 150-200 years; the values taken by the random variable Y are indeed spread out from 0 to at least 250 and above; while those of X will have a smaller range. Thus, qualitatively you’ve already understood what the Range of distribution means. The mathematical formula for the same is given as – $$ Range = L – S $$

where L – the largest/maximum value attained by the random variable under consideration and S – the smallest/minimum value.

**Browse more Topics under Measures Of Central Tendency And Dispersion**

- Arithmetic Mean
- Median and Mode
- Partition Values or Fractiles
- Harmonic Mean and Geometric Mean
- Measure of Dispersion
- Quartiles, Quartile Deviation and Coefficient of Quartile Deviation
- Standard deviation and Coefficient of Variation

### Properties

- The Range of a given distribution has the same units as the data points.
- If a random variable is transformed into a new random variable by a change of scale and a shift of origin as –

Y = aX + b

where Y – the new random variable, X – the original random variable and a,b – constants. Then the ranges of X and Y can be related as –

R_{Y} = |a|R_{X}

Clearly, the shift in origin doesn’t affect the shape of the distribution, and therefore its spread (or the width) remains unchanged. Only the scaling factor is important.

- For a grouped class distribution, the Range is defined as the difference between the two extreme class boundaries.
- A better measure of the spread of a distribution is the Coefficient of Range, given by – $$ \text{Coefficient of Range (expressed as a percentage)} = \frac{L – S}{L + S} \times 100 $$ Clearly, we need to take the ratio between the Range and the total (combined) extent of the distribution. Besides, since it is a ratio, it is dimensionless, and can, therefore, one can use it to compare the spreads of two or more different distributions as well.
- The range is an
*absolute measure*of Dispersion of a distribution while the Coefficient of Range is a*relative measure*of dispersion.

Due to the consideration of only the end-points of a distribution, the Range never gives us any information about the shape of the distribution curve between the extreme points. Thus, we must move on to better measures of dispersion. One such quantity is Mean Deviation which is we are going to discuss now.

## Mean Deviation and Mean Deviation Formula

The mean deviation accommodates the spread imparted to the distribution by each individual element separately. This absolute measure of dispersion usually considers the arithmetic mean as the reference point, following which, we calculate the modulus of the difference between a data point and the mean value. This difference for all of the elements is then averaged over the entire dataset to finally give the Mean Deviation.

Thus, for a variable distribution X, with an arithmetic mean A and a total of n data points, the mean deviation formula is – $$ M.D. = \frac{\Sigma |X_i – A|}{n} $$

If the variable assumes a grouped frequency distribution, we modify the mean deviation formula as – $$ M.D. = \frac{\Sigma |X_i – A|f_i}{n} $$

Do note that in this case, the X_{i} and f_{i} are the central points and the frequencies of the i’th class; and \(n – \Sigma f_i\).

We may take mean deviation about the median of the data as well, in which case we’ll just need to use A = the median, in the formula above. Then we call the resulting absolute measure of dispersion as the mean deviation about the median of the distribution.

## Variation with Respect to a Linear Transformation of Coordinates

If a random variable is transformed into a new random variable by a change of scale and a shift of origin as –

Y = aX + b

where Y – the new random variable, X – the original random variable and a,b – constants. Then the ranges of X and Y can be related as –

MD_{Y} = |a|MD_{X}

Clearly, the shift in origin doesn’t affect the shape of the distribution, and therefore its spread (or the width) remains unchanged. Only the scaling factor is important. Thus, the mean deviation changes correspondingly.

### The Coefficient of Mean Deviation

The construction of a relative measure of dispersion using the mean deviation is possible, and we can construct it as follows –

Coefficient of Mean Deviation (expressed as a percentage) = $$ \frac{\text{The Mean Deviation about A}}{A} \times 100 $$

Usually, we take A as the arithmetic mean or the median.

## Solved Examples on Mean Deviation Formula

**Question – **Calculate the range, mean deviation, and the coefficient of mean deviation of the given dataset –

Marks | 0 – 10 | 10 – 20 | 20 – 30 | 30 – 40 |

Number of students | 2 | 5 | 1 | 3 |

**Solution:** We can start by finding the range of the dataset first, which we can easily calculate as –

Range = Difference of the extreme class boundaries

= 40 – 0 (marks)

= 40 marks

Now, to find the coefficient of mean deviation, we first need to know the mean of the distribution. We can calculate it as –

$$ Mean = \frac{\Sigma f_i x_i}{N}$$

Let us find the required values now –

Marks | x_{i }(mid-point of the class) |
Number of students (f_{i}) |
f_{i}x_{i} |

0 – 10 | 5 | 2 | 10 |

10 – 20 | 15 | 5 | 75 |

20 – 30 | 25 | 1 | 25 |

30 – 40 | 35 | 3 | 105 |

Σ f_{i} = N = 11 |
Σ f_{i}x_{i} = 215 |

Then,

A (Arithmetic Mean) = 215/11

= 19.54 marks

Now, let us analyze the data to find the mean deviation from A.

x_{i} |
|x_{i} – A| |
f_{i} |
f_{i}|x_{i} – A| |

5 | 14.54 | 2 | 29.08 |

15 | 4.54 | 5 | 22.7 |

25 | 5.46 | 1 | 5.46 |

35 | 15.46 | 3 | 46.38 |

Σ f_{i}|x_{i }– A| = 103.62 |

Now, we find, the mean deviation using the mean deviation formula – $$M.D. = \frac{f_i|x_i – A|}{N}$$ $$ = \frac{103.62}{11} $$ $$ = 9.42 \text{ marks} $$

Also, we can find the coefficient of mean deviation (about the mean) as – $$ \text{Coefficient of Mean Deviation} = \frac{M.D.}{A} \times 100 $$ $$ = \frac{9.42}{19.54} \times 100 $$ $$ = 48.2 \text{ percent} $$

Thus, this concludes our discussion on range and mean deviation. Also, we learnt the application of mean deviation formula.

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