Mean deviation, also known as the mean absolute deviation (MAD), is a measure of dispersion that quantifies the average absolute difference between each data point and the mean of the dataset. Unlike the variance or standard deviation, which consider the squared differences, the mean deviation uses the absolute differences, making it a measure of the average distance of data points from the mean.

Formula for Mean Deviation:

For a sample:

For a population:

Mean Deviation=N∑∣xi​−μ∣​

Where:

• xi​ represents each individual data
  $$

• $\mu$ represents the population mean

• $n$ represents the sample size

• $N$ represents the population size

• 
  $\mathrm{\mid }{\mathrm{<br>}}_{\mathrm{<br>}}-\bar{x}\mathrm{\mid }$

  ∣xi​−xˉ∣ or

  $\mathrm{\mid }{\mathrm{<br>}}_{}-\mathrm{<br>}\mathrm{\mid <span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord"></span></span></span></span></span>}$
• $∣xi​−μ∣ represents the absolute difference between each data point and the mean$

Characteristics and Considerations:

1. Absolute Differences: Mean deviation considers the absolute differences between each data point and the mean, giving equal weight to deviations above and below the mean.
2. Interpretability: Mean deviation provides a measure of dispersion in the original units of measurement, making it interpretable and relevant in various contexts.
3. Simplicity: Mean deviation is relatively simple to calculate and understand, making it accessible for assessing the variability of a dataset.
4. Comparison with Variance: Compared to the variance or standard deviation, mean deviation is less commonly used due to its simplicity and the computational convenience of squared differences in variance and standard deviation calculations.
5. Sensitivity to Outliers: Like other measures of dispersion, mean deviation is sensitive to extreme values or outliers in the dataset, potentially affecting its value and interpretation.

Example:

Consider the following dataset representing the ages (in years) of a sample of individuals:

25

25,30,35,40,45

To calculate the mean deviation:

1. Calculate the mean age:
   
   $\bar{x}=\frac{25+30+35+40+45}{5}=\frac{175}{5}=35\text{ years}$

    

    xˉ=525+30+35+40+45​=5175​=35 years

2. Calculate the absolute differences between each data point and the mean:
   
   $\mathrm{\mid }25-35\mathrm{\mid }=10$

   ∣25−35∣=10

   $\mathrm{\mid }30-35\mathrm{\mid }=5$

    ∣30−35∣=5

   $\mathrm{\mid }35-35\mathrm{\mid }=0$

    

    

   ∣35−35∣=0

   |40−35∣=5

    ∣40−35∣=5

   $\mathrm{\mid }45-35\mathrm{\mid }=10$

    ∣45−35∣=10

3. Calculate the mean deviation:
   
   $\text{Mean Deviation}=\frac{10+5+0+5+10}{5}=\frac{30}{5}=6\text{ years}$

   Mean Deviation=510+5+0+5+10​=530​=6 years

The mean deviation of the ages in the sample is 6 years, indicating the average absolute difference between each individual’s age and the mean age of 35 years.

Mean deviation is a measure of dispersion that quantifies the average absolute difference between each data point and the mean of the dataset. While it provides a straightforward and interpretable measure of variability, mean deviation is less commonly used compared to the variance or standard deviation, particularly in statistical analyses and applications where the properties of squared differences are advantageous. Nonetheless, mean deviation offers a valuable perspective on data variability and can complement other measures of dispersion in descriptive analyses and exploratory data analysis.