Find the mean absolute deviation for the set
of values.
11, 16, 70, 66, 14, 15
The Correct Answer and Explanation is :
To find the mean absolute deviation (MAD) for the set of values (11, 16, 70, 66, 14, 15), follow these steps:
Step 1: Calculate the Mean
First, we need to calculate the mean of the data set. The mean ((\mu)) is calculated by summing all the values and dividing by the number of values.
[
\text{Mean} = \frac{11 + 16 + 70 + 66 + 14 + 15}{6} = \frac{192}{6} = 32
]
Step 2: Calculate the Absolute Deviations
Next, we find the absolute deviations of each data point from the mean:
- ( |11 – 32| = 21 )
- ( |16 – 32| = 16 )
- ( |70 – 32| = 38 )
- ( |66 – 32| = 34 )
- ( |14 – 32| = 18 )
- ( |15 – 32| = 17 )
Step 3: Calculate the Mean of the Absolute Deviations
Now, we calculate the mean of these absolute deviations:
[
\text{Sum of Absolute Deviations} = 21 + 16 + 38 + 34 + 18 + 17 = 144
]
Now, divide by the number of values (6):
[
\text{Mean Absolute Deviation (MAD)} = \frac{144}{6} = 24
]
Summary
The mean absolute deviation (MAD) for the set of values (11, 16, 70, 66, 14, 15) is (24).
Explanation
The mean absolute deviation is a measure of the dispersion or variability within a data set. It quantifies how spread out the values are around the mean. By calculating the mean of the absolute deviations, we can understand how much the individual data points deviate from the average value.
MAD is particularly useful because it provides a simple, intuitive understanding of variability without being affected by extreme values or outliers, unlike standard deviation. In this case, the calculated MAD of (24) indicates that, on average, the values in the set differ from the mean by (24) units. This can help in assessing consistency and reliability of data points in various fields, including finance, quality control, and research, where understanding variability is crucial for making informed decisions.