Miniature golf scores for a group of friends follows:
30, 32, 36, 40, 44, 45, 46, 48, 55, 58, 59, 60, 76, 82, 90
Which measure of variability—IQR or MAD—best describes the spread of the dataset? Enter 1 for IQR or 2 for MAD.
To determine whether the Interquartile Range (IQR) or Mean Absolute Deviation (MAD) best describes the spread of the dataset, we need to calculate both measures.
The IQR is calculated by finding the difference between the upper quartile (Q3) and the lower quartile (Q1).
The MAD is calculated by finding the average of the absolute differences between each data point and the mean.
To calculate the IQR, we first need to find the median. The median is the middle value when the data is arranged in ascending order. Since we have 15 data points, the median will be the average of the 7th and 8th values.
(30, 32, 36, 40, 44, 45, 46, 48, 55, 58, 59, 60, 76, 82, 90)
Arranging the data in ascending order:
30, 32, 36, 40, 44, 45, 46, 48, 55, 58, 59, 60, 76, 82, 90
The median is the average of the 46th and 48th values:
Median = (45 + 46) / 2
Median = 45.5
Next, we need to find Q1 and Q3. Q1 represents the lower quartile, which is the median of the lower half of the data points. It will be the average of the 3rd and 4th values.
Q1 = (36 + 40) / 2
Q1 = 38
Q3 represents the upper quartile, which is the median of the upper half of the data points. It will be the average of the 11th and 12th values.
Q3 = (59 + 60) / 2
Q3 = 59.5
Now we can calculate the IQR by finding the difference between Q3 and Q1.
IQR = Q3 - Q1
IQR = 59.5 - 38
IQR = 21.5
To calculate the MAD, we first need to find the mean. The mean is calculated by summing all the values and dividing by the total number of values.
Mean = (30 + 32 + 36 + 40 + 44 + 45 + 46 + 48 + 55 + 58 + 59 + 60 + 76 + 82 + 90) / 15
Mean = 674 / 15
Mean = 44.93 (approximately)
Next, we find the absolute differences between each data point and the mean:
|30 - 44.93| = 14.93
|32 - 44.93| = 12.93
|36 - 44.93| = 8.93
|40 - 44.93| = 4.93
|44 - 44.93| = 0.93
|45 - 44.93| = 0.07
|46 - 44.93| = 1.07
|48 - 44.93| = 3.07
|55 - 44.93| = 10.07
|58 - 44.93| = 13.07
|59 - 44.93| = 14.07
|60 - 44.93| = 15.07
|76 - 44.93| = 31.07
|82 - 44.93| = 37.07
|90 - 44.93| = 45.07
Now, we calculate the average of these differences:
MAD = (14.93 + 12.93 + 8.93 + 4.93 + 0.93 + 0.07 + 1.07 + 3.07 + 10.07 + 13.07 + 14.07 + 15.07 + 31.07 + 37.07 + 45.07) / 15
MAD = 264.39 / 15
MAD = 17.63 (approximately)
Comparing the IQR and MAD, if we interpret "spread" as referring to the range between the middle 50% of data, the IQR of 21.5 may be more appropriate. However, if we interpret "spread" as referring to the average deviation of each data point from the mean, the MAD of 17.63 would be more suitable.
Therefore, the best measure of variability to describe the spread of this dataset would be the IQR (1).