Alex is 16 years old and his brother is 5 years old. In many years will Alex be twice the age of his brother?
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Answer:
The interquartile range of the data set is 5
The mean absolute deviation of the data set is 3.6
Step-by-step explanation:
* Lets explain how to find the interquartile range and the mean absolute
deviation (MAD)
- The steps to find the interquartile range is:
1- Arrange the values from the smallest to the largest
∴ The values are 5 , 6 , 10 , 11 , 12 , 13 , 14 , 15 , 18 , 20
2- Find the median
- The median is the middle value after arrange them
* If there are two values in the middle take their average
∵ The values are 10 then the 5th and the 6th are the values
∵ The 5th is 12 and the 6th is 13
∴ The median =
∴ The median is 12.5
3- Calculate the median of the lower quartile
- The lower quartile is the median of the first half data values
∵ There are 10 values
∴ The first half is the first five values
∴ The first half values are 5 , 6 , 10 , 11 , 12
∵ The middle value is 10
∴ The median of lower quartile = 10
- Similar find the median of the upper quartile
- The upper quartile is the median of the second half data values
∵ There are 10 numbers
∴ The second half is the last five values
∴ The second half values are 13 , 14 , 15 , 18 , 20
∵ The middle value is 15
∴ The median of upper quartile = 15
4- The interquartile range (IQR) is the difference between the upper
and the lower medians
∴ The interquartile range = 15 - 10 = 5
* The interquartile range of the data set is 5
* Lets talk about the mean absolute deviation
- Mean absolute deviation (MAD) of a data set is the average distance
between each data value and the mean
- To find the mean absolute deviation of the data, start by finding
the mean of the data set.
1- Find the sum of the data values, and divide the sum by the
number of data values.
∵ The data set is 5 , 6 , 10 , 11 , 12 , 13 , 14 , 15 , 18 , 20
∵ Its sum = 5 + 6 + 10 + 11 + 12 + 13 + 14 + 15 + 18 + 20 = 124
∵ The mean = the sum of the data values/the number of the data
∵ The set has 10 numbers
∴ The mean = 124/10 = 12.4
2- Find the absolute value of the difference between each data value
and the mean ⇒ |data value – mean|
# I5 - 12.4I = 7.4
# I6 - 12.4I = 6.4
# I10 - 12.4I = 2.4
# I11 - 12.4I = 1.4
# I12 - 12.4I = 0.4
# I13 - 12.4I = 0.6
# I14 - 12.4I = 1.6
# I15 - 12.4I = 2.6
# I18 - 12.4I = 5.6
# I20 - 12.4I = 7.6
3- Find the sum of the absolute values of the differences.
∵ Their sum = 7.4 + 6.4 + 2.4 + 1.4 + 0.4 + 0.6 + 1.6 + 2.6 + 5.6 + 7.6
∴ Their sum = 36
4- Divide the sum of the absolute values of the differences by the
number of data values to find MAD
∴ MAD = The sum of the absolute values/number of the values
∵ The sum = 36
∵ The data set has 10 values
∴ MAD = 36/10 = 3.6
* The mean absolute deviation of the data set is 3.6