What is the MAD and IQR for the given dataset?
1, 4, 6, 7, 9, 10, 10, 11, 12, 12, 14, 15, 17, 18, 19
To find the MAD (mean absolute deviation) and IQR (interquartile range) for the given dataset, first arrange the numbers in ascending order:
1, 4, 6, 7, 9, 10, 10, 11, 12, 12, 14, 15, 17, 18, 19
Next, calculate the mean of the dataset:
(1 + 4 + 6 + 7 + 9 + 10 + 10 + 11 + 12 + 12 + 14 + 15 + 17 + 18 + 19) / 15 = 10.4
Calculate the absolute deviations from the mean for each data point:
|1 - 10.4| = 9.4
|4 - 10.4| = 6.4
|6 - 10.4| = 4.4
|7 - 10.4| = 3.4
|9 - 10.4| = 1.4
|10 - 10.4| = 0.4
|10 - 10.4| = 0.4
|11 - 10.4| = 0.6
|12 - 10.4| = 1.6
|12 - 10.4| = 1.6
|14 - 10.4| = 3.6
|15 - 10.4| = 4.6
|17 - 10.4| = 6.6
|18 - 10.4| = 7.6
|19 - 10.4| = 8.6
Calculate the MAD by finding the mean of these absolute deviations:
(MAD) = (9.4 + 6.4 + 4.4 + 3.4 + 1.4 + 0.4 + 0.4 + 0.6 + 1.6 + 1.6 + 3.6 + 4.6 + 6.6 + 7.6 + 8.6) / 15 = 4.56
The MAD for the dataset is 4.56.
To find the IQR, first find the median of the dataset, which is the middle number:
Median = 10
Next, divide the dataset into two halves:
Lower half: 1, 4, 6, 7, 9
Upper half: 12, 12, 14, 15, 17, 18, 19
Find the median of each half:
Lower median = 6
Upper median = 15
Calculate the IQR by finding the difference between the upper and lower median:
IQR = Upper median - Lower median
IQR = 15 - 6
IQR = 9
The IQR for the dataset is 9.