Measures of Dispersion

Measures of dispersion help us understand how spread out data is in a dataset. They include range, variance, standard deviation, and more, giving insights into data variability and helping us interpret statistical results effectively.

1. Range

   • The range is the simplest measure of dispersion, calculated as the difference between the maximum and minimum values in a dataset.
   • It provides a quick sense of the spread of data but can be heavily influenced by outliers.
   • The formula for range is: Range = Maximum value - Minimum value.

2. Variance

   • Variance measures the average squared deviation of each data point from the mean, indicating how spread out the values are.
   • It is calculated by taking the sum of squared differences from the mean and dividing by the number of observations (for population variance) or by one less than the number of observations (for sample variance).
   • A higher variance indicates greater dispersion in the dataset.

3. Standard Deviation

   • Standard deviation is the square root of variance, providing a measure of dispersion in the same units as the original data.
   • It helps to understand how much individual data points typically deviate from the mean.
   • A smaller standard deviation indicates that data points are closer to the mean, while a larger standard deviation indicates more spread.

4. Interquartile Range (IQR)

   • The IQR measures the range of the middle 50% of the data, calculated as the difference between the first quartile (Q1) and the third quartile (Q3).
   • It is a robust measure of dispersion that is less affected by outliers compared to the range.
   • The formula for IQR is: IQR = Q3 - Q1.

5. Coefficient of Variation

   • The coefficient of variation (CV) is a standardized measure of dispersion, calculated as the ratio of the standard deviation to the mean, expressed as a percentage.
   • It allows for comparison of variability between datasets with different units or means.
   • A higher CV indicates greater relative variability in relation to the mean.

6. Mean Absolute Deviation

   • Mean absolute deviation (MAD) measures the average absolute deviations from the mean, providing a straightforward interpretation of dispersion.
   • It is calculated by taking the average of the absolute differences between each data point and the mean.
   • MAD is less sensitive to outliers compared to variance and standard deviation.

7. Percentiles and Quartiles

   • Percentiles divide a dataset into 100 equal parts, while quartiles divide it into four equal parts, providing insights into the distribution of data.
   • The first quartile (Q1) represents the 25th percentile, the second quartile (Q2) is the median (50th percentile), and the third quartile (Q3) is the 75th percentile.
   • These measures help identify the position of data points within the overall distribution, aiding in understanding data spread and central tendency.