Calculate the mean absolute deviation of 15, five, 17, seven, 14, five, 15, and 20. Round your answer to the nearest tenth if necessary.
First, let’s remember the steps to find the mean absolute deviation. First, we find the mean of all values. Next, we find the distance of each value from that mean. Finally, we find the mean of the distances in step two. Our data set consists of eight values, and our first step is to find the mean of all values. To do that, we add all the values together,15 plus five plus 17 plus seven plus 14 plus five plus 15 plus 20, and divide by the number of data points we have, which is eight. All the values in the numerator add up to 98. And we need to divide 98 by eight, which gives us 12.25.
Now, we need to move on to step two, where we find the distance of each value from our mean of 12.25. To find the distance, we’ll subtract the mean and take the absolute value. We begin subtracting 12.25 from each point. 15 minus 12.25 equals 2.75. Five minus 12.25 equals negative 7.25. 17 minus 12.25 equals 4.75. Seven minus 12.25 equals negative 5.25. 14 minus 12.25 equals 1.75. Five minus 12.25 equals negative 7.25. 15 minus 12.25 equals 2.75. And finally, 20 minus 12.25 equals 7.75.
But remember, in step two, we’re interested in the distance of a value from the mean. And distances cannot be negative. And so, we need to take the absolute value of the differences we found. The positive differences stay the same. And negative 7.25 becomes positive 7.25. Negative 5.25 becomes positive 5.25.
Now that we have those distances, we’re ready for step three. Find the mean of the distances in step two. The mean of the distances will be equal to the sum of the distances divided by eight, as that is the number of points we have. When we add all the values in the numerator, we get 39.5. And the denominator doesn’t change, eight. 39.5 divided by eight equals 4.9375.
Our instructions tell us we’re interested in this value to the nearest tenth. To round to the nearest tenth, we’ll need to look at the digit to the right of the tenths place, the hundredths place. Since there’s a three in the hundredths place, we will round down to 4.9. The mean absolute deviation of these eight values rounded to the nearest tenth is 4.9.