-

The mean deviation or the average deviation is a measure of dispersion that can be used to measure the degree of variability in the data. It can be calculated by summing up the absolute values of the difference of each observation from the average value and then dividing by the number of observations.

### Mean Deviation Formula:

The value of the mean deviation can be calculated using the formula, \text{Mean Deviation }= \frac{\sum_{i=1}^{N}|x_i - A|}{N} Here, N is the total number of observations and A denotes the average value of the observations.

1. If we choose A = Mean, then the above formula gives us the Mean Deviation about the Mean.
2. If we choose A = Median, then the above formula gives us the Mean Deviation about the Median.
3. If we choose A = Mode, then the above formula gives us the Mean Deviation about the Mode.

We now list out some of the merits and demerits of mean deviation.

1. The advantage of mean deviation as a measure of dispersion is that it is based on all observations. This is in sharp contrast to the other measures of dispersion such as range and quartile deviation which are not calculated using all values.
2. It is simple to calculate. This is because the calculation involves easy and elementary operations such as taking absolute difference with average, adding them and then dividing by the number of observations.
3. The method of averaging the differences smoothes out all the irregularities and gives us a true picture regarding the degree of variability in the data.
4. It is a better measure of dispersion than the standard deviation. This is because the standard deviation does not measure the true value of the deviations but squares them instead. This also makes it more likely that standard deviation get affected by the presence of extreme values.
5. It can be used to calculate the coefficient of mean deviation which is a relative measure of dispersion. Such relative measures of dispersion are helpful to compare two sets of data values.

1. If the distribution is bimodal, then the mode is not well defined. In this case, the values of mean deviation about the mode becomes ill defined since there are two choices for the value of mode.
2. Since the mean deviation involves taking the absolute value, the formula cannot be mathematically manipulated any further. On the other hand when calculating variance we can always expand the square and manipulate the formula further.
3. It is not widely used in sciological studies.
4. If the data is highly skewed then the mean deviation does not give a true measure of the dispersion in the data.
5. It cannot be calculated for frequency distributions with open ended class intervals.
6. When studying a population by taking samples, the sample mean deviation is not a good estimator of the population mean deviation.

### Properties of Mean Deviation:

1. Mean deviation has the lowest value when calculated about the median. Therefore, the mean deviation about the median is always smaller than the mean deviation about the mean or the mean deviation about the mode.
2. If the distribution is symmetrical the mean deviation about the median covers roughly 57.5% of the data values.
3. Mean deviation is not affected by the change of scale. This means that if we add a single fixed value to all the observations the vakue of the mean deviation is unchanged.
4. On the other hand mean deviation is affected by the change of scale. If we multiply each value by the sasme positive number, then the mean deviation also gets multiplied by that same number.

### Uses of Mean Deviation:

1. It is widely used by economists beacuse of its accuracy and simplicity.
2. It is useful when calculating the distribution of wealth in a community. This is because it takes all values such as extremely poor ot extremely rich people into consideration.
3. It is used in forecasting business cycles since it is the most accurate measure of variability for this purpose.

Hey 👋

I'm currently pursuing a Ph.D. in Maths. Prior to this, I completed my master's in Maths & bachelors in Statistics.

I created this website for explaining maths and statistics concepts in the simplest possible manner.

If you've found value from reading my content, feel free to support me in even the smallest way you can.