Mean Absolute Deviation Another measure of variation is the mean absolute deviation. It is computed using the formula:


MAD = (Σ |xi − x̄|) / n


Compute the mean absolute deviation of the data in Problem 11 and compare the results with the sample standard deviation.

Verified step by step guidance

1. Identify the data set from Problem 11. You will need the individual data points \( x_i \) and the number of data points \( n \).
2. Calculate the mean \( \bar{x} \) of the data set using the formula: \[ \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \]
3. Find the absolute deviations of each data point from the mean by computing \( |x_i - \bar{x}| \) for each \( i \).
4. Sum all the absolute deviations: \[ \sum_{i=1}^n |x_i - \bar{x}| \]
5. Calculate the mean absolute deviation (MAD) by dividing the sum of absolute deviations by the number of data points \( n \): \[ \text{MAD} = \frac{\sum_{i=1}^n |x_i - \bar{x}|}{n} \] After finding the MAD, recall the formula for the sample standard deviation and compare the two measures to understand how they describe variation differently.

Key Concepts

Mean Absolute Deviation (MAD)

Mean Absolute Deviation measures the average distance between each data point and the mean of the dataset. It is calculated by taking the absolute differences between each value and the mean, summing them, and dividing by the number of observations. MAD provides a straightforward measure of variability that is less sensitive to extreme values than variance or standard deviation.

Sample Standard Deviation

Sample standard deviation quantifies the spread of data points around the sample mean by calculating the square root of the average squared deviations. It is sensitive to outliers due to squaring differences, making it useful for understanding variability in normally distributed data. Comparing it with MAD helps assess the impact of extreme values on data dispersion.

Comparison of Measures of Variation

Comparing MAD and sample standard deviation helps evaluate different aspects of data variability. While MAD uses absolute values and is more robust to outliers, standard deviation squares deviations, emphasizing larger differences. Understanding their differences aids in selecting the appropriate measure based on data characteristics and analysis goals.