Mean Absolute Deviation Another useful measure of variation for a data set is the mean absolute deviation (MAD). It is calculated by the formula
MAD = Σ |x − x̄| / n.
b. Find the mean absolute deviation of the data set in Exercise 16. Compare your result with the sample standard deviation obtained in Exercise 16.

Verified step by step guidance

1. Step 1: Recall the formula for the Mean Absolute Deviation (MAD): $MAD\; =\; (\Sigma \; |x\; -\; x̄|)\; /\; n$, where $x̄$ is the mean of the data set, $x$ represents each data point, and $n$ is the number of data points.
2. Step 2: Calculate the mean ($x̄$) of the data set from Exercise 16 by summing all the data points and dividing by the total number of data points ($n$).
3. Step 3: For each data point in the set, compute the absolute deviation from the mean, which is $|x\; -\; x̄|$. This involves subtracting the mean from each data point and taking the absolute value of the result.
4. Step 4: Sum all the absolute deviations calculated in Step 3 to get $\Sigma \; |x\; -\; x̄|$.
5. Step 5: Divide the sum of absolute deviations from Step 4 by the total number of data points ($n$) to find the Mean Absolute Deviation (MAD). Compare this value to the sample standard deviation obtained in Exercise 16 to analyze the differences in the measures of variation.

Key Concepts

Mean Absolute Deviation (MAD)

Mean Absolute Deviation (MAD) is a measure of the dispersion of a data set. It quantifies the average distance between each data point and the mean of the data set, providing insight into the variability of the data. The formula MAD = Σ |x − x̄| / n involves summing the absolute differences between each data point (x) and the mean (x̄), then dividing by the number of observations (n). This measure is particularly useful because it treats all deviations equally, without squaring them as in variance.

Sample Standard Deviation

The sample standard deviation is a statistic that measures the amount of variation or dispersion in a sample data set. It is calculated using the formula s = √(Σ (x - x̄)² / (n - 1)), where x is each data point, x̄ is the sample mean, and n is the sample size. Unlike MAD, the standard deviation squares the deviations, which emphasizes larger differences and can be more sensitive to outliers. It is widely used in statistics to understand the spread of data points around the mean.

Comparison of MAD and Standard Deviation

Comparing Mean Absolute Deviation (MAD) and sample standard deviation provides insights into the nature of data variability. While both measures indicate how spread out the data points are, MAD offers a straightforward interpretation as it uses absolute values, making it less sensitive to extreme values. In contrast, the standard deviation, by squaring the deviations, can be influenced more by outliers. Understanding these differences helps in selecting the appropriate measure of variability based on the data characteristics.