Measures of Dispersion: Variance, Mean Deviation and Standard Deviation

Understanding Measures of Dispersion

Measures of dispersion help us understand how spread out the data is. They give us an idea about the variability within a dataset. The main measures of dispersion include:

• Range: The difference between the highest and lowest values.
• Variance: The average of the squared differences from the mean.
• Mean Deviation: The average of the absolute differences from the mean.
• Standard Deviation: The square root of the variance, representing the average distance from the mean.

 

Calculating Variance and Standard Deviation

To calculate variance and standard deviation, follow these steps:

1. Find the Mean: Calculate the average of the dataset.
2. Calculate Differences: Subtract the mean from each data point.
3. Square the Differences: Square each of the differences.
4. Find the Variance: Calculate the average of these squared differences.
5. Calculate the Standard Deviation: Take the square root of the variance.

 

Worked Example

 

Mean Deviation

The mean deviation is calculated by finding the average of the absolute differences between each data point and the mean.

 

Example

Find the mean deviation for the dataset: 4, 8, 6, 5, 3.

• Calculate the mean: $$\text{Mean} = 5.2$$
• Find absolute differences: $$|4-5.2|, |8-5.2|, |6-5.2|, |5-5.2|, |3-5.2|$$
• Mean deviation: $$\frac{|4-5.2| + |8-5.2| + |6-5.2| + |5-5.2| + |3-5.2|}{5} = 1.36$$