Question
find the MAD (mean absolute deviation) of the following instruments.
woodwind: 30,30,40,45,45,60,60,60,80,90
Brass: 20,20,30,30,40,40,45,60,75,90
percussion: 20,30,30,30,40,45,45,60,60,60
woodwind: 30,30,40,45,45,60,60,60,80,90
Brass: 20,20,30,30,40,40,45,60,75,90
percussion: 20,30,30,30,40,45,45,60,60,60
Answers
GPT-4o mini
To find the Mean Absolute Deviation (MAD) for each set of instruments, we'll follow these steps:
1. Calculate the mean of the data set.
2. Subtract the mean from each data point to find the absolute deviations.
3. Calculate the mean of those absolute deviations.
Let's compute the MAD for each instrument category one by one.
### 1. Woodwind Instruments
Data: 30, 30, 40, 45, 45, 60, 60, 60, 80, 90
**Step 1: Calculate the mean**
\[
\text{Mean} = \frac{30 + 30 + 40 + 45 + 45 + 60 + 60 + 60 + 80 + 90}{10} = \frac{ 450}{10} = 45
\]
**Step 2: Calculate absolute deviations**
\[
|30 - 45| = 15
\]
\[
|30 - 45| = 15
\]
\[
|40 - 45| = 5
\]
\[
|45 - 45| = 0
\]
\[
|45 - 45| = 0
\]
\[
|60 - 45| = 15
\]
\[
|60 - 45| = 15
\]
\[
|60 - 45| = 15
\]
\[
|80 - 45| = 35
\]
\[
|90 - 45| = 45
\]
**Absolute Deviations: 15, 15, 5, 0, 0, 15, 15, 15, 35, 45**
**Step 3: Calculate the Mean Absolute Deviation**
\[
\text{MAD}_{\text{Woodwind}} = \frac{15 + 15 + 5 + 0 + 0 + 15 + 15 + 15 + 35 + 45}{10} = \frac{145}{10} = 14.5
\]
### 2. Brass Instruments
Data: 20, 20, 30, 30, 40, 40, 45, 60, 75, 90
**Step 1: Calculate the mean**
\[
\text{Mean} = \frac{20 + 20 + 30 + 30 + 40 + 40 + 45 + 60 + 75 + 90}{10} = \frac{ 450}{10} = 45
\]
**Step 2: Calculate absolute deviations**
\[
|20 - 45| = 25
\]
\[
|20 - 45| = 25
\]
\[
|30 - 45| = 15
\]
\[
|30 - 45| = 15
\]
\[
|40 - 45| = 5
\]
\[
|40 - 45| = 5
\]
\[
|45 - 45| = 0
\]
\[
|60 - 45| = 15
\]
\[
|75 - 45| = 30
\]
\[
|90 - 45| = 45
\]
**Absolute Deviations: 25, 25, 15, 15, 5, 5, 0, 15, 30, 45**
**Step 3: Calculate the Mean Absolute Deviation**
\[
\text{MAD}_{\text{Brass}} = \frac{25 + 25 + 15 + 15 + 5 + 5 + 0 + 15 + 30 + 45}{10} = \frac{ 165}{10} = 16.5
\]
### 3. Percussion Instruments
Data: 20, 30, 30, 30, 40, 45, 45, 60, 60, 60
**Step 1: Calculate the mean**
\[
\text{Mean} = \frac{20 + 30 + 30 + 30 + 40 + 45 + 45 + 60 + 60 + 60}{10} = \frac{ 410}{10} = 41
\]
**Step 2: Calculate absolute deviations**
\[
|20 - 41| = 21
\]
\[
|30 - 41| = 11
\]
\[
|30 - 41| = 11
\]
\[
|30 - 41| = 11
\]
\[
|40 - 41| = 1
\]
\[
|45 - 41| = 4
\]
\[
|45 - 41| = 4
\]
\[
|60 - 41| = 19
\]
\[
|60 - 41| = 19
\]
\[
|60 - 41| = 19
\]
**Absolute Deviations: 21, 11, 11, 11, 1, 4, 4, 19, 19, 19**
**Step 3: Calculate the Mean Absolute Deviation**
\[
\text{MAD}_{\text{Percussion}} = \frac{21 + 11 + 11 + 11 + 1 + 4 + 4 + 19 + 19 + 19}{10} = \frac{ 120}{10} = 12
\]
### Summary of Mean Absolute Deviations
- MAD (Woodwind): **14.5**
- MAD (Brass): **16.5**
- MAD (Percussion): **12**
1. Calculate the mean of the data set.
2. Subtract the mean from each data point to find the absolute deviations.
3. Calculate the mean of those absolute deviations.
Let's compute the MAD for each instrument category one by one.
### 1. Woodwind Instruments
Data: 30, 30, 40, 45, 45, 60, 60, 60, 80, 90
**Step 1: Calculate the mean**
\[
\text{Mean} = \frac{30 + 30 + 40 + 45 + 45 + 60 + 60 + 60 + 80 + 90}{10} = \frac{ 450}{10} = 45
\]
**Step 2: Calculate absolute deviations**
\[
|30 - 45| = 15
\]
\[
|30 - 45| = 15
\]
\[
|40 - 45| = 5
\]
\[
|45 - 45| = 0
\]
\[
|45 - 45| = 0
\]
\[
|60 - 45| = 15
\]
\[
|60 - 45| = 15
\]
\[
|60 - 45| = 15
\]
\[
|80 - 45| = 35
\]
\[
|90 - 45| = 45
\]
**Absolute Deviations: 15, 15, 5, 0, 0, 15, 15, 15, 35, 45**
**Step 3: Calculate the Mean Absolute Deviation**
\[
\text{MAD}_{\text{Woodwind}} = \frac{15 + 15 + 5 + 0 + 0 + 15 + 15 + 15 + 35 + 45}{10} = \frac{145}{10} = 14.5
\]
### 2. Brass Instruments
Data: 20, 20, 30, 30, 40, 40, 45, 60, 75, 90
**Step 1: Calculate the mean**
\[
\text{Mean} = \frac{20 + 20 + 30 + 30 + 40 + 40 + 45 + 60 + 75 + 90}{10} = \frac{ 450}{10} = 45
\]
**Step 2: Calculate absolute deviations**
\[
|20 - 45| = 25
\]
\[
|20 - 45| = 25
\]
\[
|30 - 45| = 15
\]
\[
|30 - 45| = 15
\]
\[
|40 - 45| = 5
\]
\[
|40 - 45| = 5
\]
\[
|45 - 45| = 0
\]
\[
|60 - 45| = 15
\]
\[
|75 - 45| = 30
\]
\[
|90 - 45| = 45
\]
**Absolute Deviations: 25, 25, 15, 15, 5, 5, 0, 15, 30, 45**
**Step 3: Calculate the Mean Absolute Deviation**
\[
\text{MAD}_{\text{Brass}} = \frac{25 + 25 + 15 + 15 + 5 + 5 + 0 + 15 + 30 + 45}{10} = \frac{ 165}{10} = 16.5
\]
### 3. Percussion Instruments
Data: 20, 30, 30, 30, 40, 45, 45, 60, 60, 60
**Step 1: Calculate the mean**
\[
\text{Mean} = \frac{20 + 30 + 30 + 30 + 40 + 45 + 45 + 60 + 60 + 60}{10} = \frac{ 410}{10} = 41
\]
**Step 2: Calculate absolute deviations**
\[
|20 - 41| = 21
\]
\[
|30 - 41| = 11
\]
\[
|30 - 41| = 11
\]
\[
|30 - 41| = 11
\]
\[
|40 - 41| = 1
\]
\[
|45 - 41| = 4
\]
\[
|45 - 41| = 4
\]
\[
|60 - 41| = 19
\]
\[
|60 - 41| = 19
\]
\[
|60 - 41| = 19
\]
**Absolute Deviations: 21, 11, 11, 11, 1, 4, 4, 19, 19, 19**
**Step 3: Calculate the Mean Absolute Deviation**
\[
\text{MAD}_{\text{Percussion}} = \frac{21 + 11 + 11 + 11 + 1 + 4 + 4 + 19 + 19 + 19}{10} = \frac{ 120}{10} = 12
\]
### Summary of Mean Absolute Deviations
- MAD (Woodwind): **14.5**
- MAD (Brass): **16.5**
- MAD (Percussion): **12**