Averaging the differences from the mean.
Number of Newspapers Delivered: 19, 14, 19, 21, 17
[tex]
\begin{array}{l}
|19-18|=1 \\
|14-18|=4 \\
|19-18|=1 \\
|21-18|=3 \\
|17-18|=1
\end{array}
[/tex]
MAD = [?]
Answer (1)
Calculate the absolute differences from the mean: ∣19 − 18∣ = 1 , ∣14 − 18∣ = 4 , ∣19 − 18∣ = 1 , ∣21 − 18∣ = 3 , ∣17 − 18∣ = 1 .
Sum the absolute differences: 1 + 4 + 1 + 3 + 1 = 10 .
Divide the sum by the number of data points (5) to find the MAD: 5 10 = 2 .
The Mean Absolute Deviation (MAD) is 2 .
Explanation
Understanding the Problem We are given a set of numbers representing the number of newspapers delivered: 19, 14, 19, 21, 17. We are also given the absolute differences of each number from the mean, which is 18. Our goal is to find the Mean Absolute Deviation (MAD).
Formula for MAD The Mean Absolute Deviation (MAD) is the average of the absolute differences between each data point and the mean of the data set. The formula for MAD is: M A D = n ∑ i = 1 n ∣ x i − μ ∣ where x i represents each data point, μ is the mean of the data, and n is the number of data points.
Calculating the Sum of Absolute Differences We are given the absolute differences from the mean: ∣19 − 18∣ = 1 ∣14 − 18∣ = 4 ∣19 − 18∣ = 1 ∣21 − 18∣ = 3 ∣17 − 18∣ = 1 We need to sum these absolute differences: 1 + 4 + 1 + 3 + 1 = 10 .
Calculating the MAD Now, we divide the sum of the absolute differences by the number of data points, which is 5: M A D = 5 10 = 2 Therefore, the Mean Absolute Deviation (MAD) is 2.
Final Answer The Mean Absolute Deviation (MAD) for the given data set is 2.
Examples
Understanding MAD is useful in many real-world scenarios. For example, in quality control, MAD can help assess the consistency of product measurements. If a company claims that each produced item should weight 100 grams, MAD can be used to measure how much the weight of the items deviates from 100 grams on average. A smaller MAD indicates higher consistency and better quality control. Similarly, in finance, MAD can be used to measure the volatility of stock prices or the risk associated with an investment portfolio. A lower MAD suggests a more stable and predictable investment.
