Question

What is the mean absolute deviation of the following data set: {4, 6, 8, 10, 12}?

Answer 1

Find the outlier of the given data set: 5, 2, 6, 5, 6, 5, 7, 8, 1, 40, 3. Please show all of your work in order to receive full credit.

Answer 2

The first step to finding the mean absolute deviation is to find the mean (average) of the data set:

Mean = (4 + 6 + 8 + 10 + 12) ÷ 5 = 8

Next, find the absolute deviation of each number from the mean. To do this, subtract the mean from each number and take the absolute value (ignore any negative signs):

|4 - 8| = 4
|6 - 8| = 2
|8 - 8| = 0
|10 - 8| = 2
|12 - 8| = 4

Then, find the mean of these absolute deviations:

Mean absolute deviation = (4 + 2 + 0 + 2 + 4) ÷ 5 = 2.4

Therefore, the mean absolute deviation of the data set {4, 6, 8, 10, 12} is 2.4.

Answer 3

To identify the outlier in a data set, we can use the formula:

outlier = Q1 - 1.5 × IQR or outlier = Q3 + 1.5 × IQR

where Q1 is the first quartile, Q3 is the third quartile, and IQR is the interquartile range.

Step 1: Put the data set in order:

1, 2, 3, 5, 5, 5, 6, 6, 7, 8, 40

Step 2: Find the median (middle value) of the lower half of the data set (Q1) and the median of the upper half of the data set (Q3):

Q1 = median(1, 2, 3, 5, 5) = 3
Q3 = median(6, 6, 7, 8, 40) = 7

Step 3: Find the interquartile range (IQR), which is the difference between Q3 and Q1:

IQR = Q3 - Q1 = 7 - 3 = 4

Step 4: Calculate the lower and upper bounds for potential outliers using the formula:

Lower bound = Q1 - 1.5 × IQR = 3 - 1.5 × 4 = -3
Upper bound = Q3 + 1.5 × IQR = 7 + 1.5 × 4 = 13

Any value outside this range is considered a potential outlier. In this case, only 40 is outside the upper bound, so it is the outlier in the data set.

Therefore, the outlier in the data set {5, 2, 6, 5, 6, 5, 7, 8, 1, 40, 3} is 40.