Question
The data set shows the admission prices at several museums. $20, $20, $16, $12, $15, $25, $11
Find and interpret the range, interquartile range, and mean absolute deviation of the data
Find and interpret the range, interquartile range, and mean absolute deviation of the data
Answers
GPT-4o mini
Let's first summarize the steps for calculating the range, interquartile range (IQR), and mean absolute deviation (MAD) for the given set of admission prices: $20, $20, $16, $12, $15, $25, and $11.
### Step 1: Calculate the Range
The range is calculated as the difference between the maximum and minimum values in the data set.
- **Maximum value**: $25
- **Minimum value**: $11
\[
\text{Range} = \text{Maximum} - \text{Minimum} = 25 - 11 = 14
\]
### Step 2: Calculate the Interquartile Range (IQR)
The interquartile range is defined as the difference between the first quartile (Q1) and the third quartile (Q3) of the data set.
1. **Sort the Data**:
$11, 12, 15, 16, 20, 20, 25$
2. **Find Q1 and Q3**:
- **Q1** (the median of the first half of the data): The first half is $11, 12, 15$. The median (Q1) is $12$.
- **Q3** (the median of the second half of the data): The second half is $16, 20, 20, 25$. The median (Q3) is the average of $20$ and $20$, which is $20$.
3. **Calculate IQR**:
\[
\text{IQR} = Q3 - Q1 = 20 - 12 = 8
\]
### Step 3: Calculate the Mean Absolute Deviation (MAD)
The mean absolute deviation is calculated as the average of the absolute deviations from the mean.
1. **Calculate the Mean**:
\[
\text{Mean} = \frac{\sum \text{data}}{n} = \frac{11 + 12 + 15 + 16 + 20 + 20 + 25}{7} = \frac{119}{7} \approx 17
\]
2. **Calculate the Absolute Deviations**:
- For $11$: |11 - 17| = 6
- For $12$: |12 - 17| = 5
- For $15$: |15 - 17| = 2
- For $16$: |16 - 17| = 1
- For $20$: |20 - 17| = 3
- For $20$: |20 - 17| = 3
- For $25$: |25 - 17| = 8
Absolute deviations: $6, 5, 2, 1, 3, 3, 8$
3. **Calculate the Mean Absolute Deviation (MAD)**:
\[
\text{MAD} = \frac{\sum \text{absolute deviations}}{n} = \frac{6 + 5 + 2 + 1 + 3 + 3 + 8}{7} = \frac{28}{7} = 4
\]
### Summary of Results
- **Range**: 14
- **Interquartile Range (IQR)**: 8
- **Mean Absolute Deviation (MAD)**: 4
### Interpretation
- The range of $14$ indicates the difference between the highest and lowest admission prices. This shows a general spread in the prices.
- The interquartile range of $8$ signifies the spread of the middle 50% of the admission prices, suggesting that the central tendency of the prices is more consistent, as the majority of prices fall within this range.
- The mean absolute deviation of $4$ implies that, on average, individual museum admission prices deviate from the mean by $4. This reflects how much variation exists in the admission prices around the average price.
### Step 1: Calculate the Range
The range is calculated as the difference between the maximum and minimum values in the data set.
- **Maximum value**: $25
- **Minimum value**: $11
\[
\text{Range} = \text{Maximum} - \text{Minimum} = 25 - 11 = 14
\]
### Step 2: Calculate the Interquartile Range (IQR)
The interquartile range is defined as the difference between the first quartile (Q1) and the third quartile (Q3) of the data set.
1. **Sort the Data**:
$11, 12, 15, 16, 20, 20, 25$
2. **Find Q1 and Q3**:
- **Q1** (the median of the first half of the data): The first half is $11, 12, 15$. The median (Q1) is $12$.
- **Q3** (the median of the second half of the data): The second half is $16, 20, 20, 25$. The median (Q3) is the average of $20$ and $20$, which is $20$.
3. **Calculate IQR**:
\[
\text{IQR} = Q3 - Q1 = 20 - 12 = 8
\]
### Step 3: Calculate the Mean Absolute Deviation (MAD)
The mean absolute deviation is calculated as the average of the absolute deviations from the mean.
1. **Calculate the Mean**:
\[
\text{Mean} = \frac{\sum \text{data}}{n} = \frac{11 + 12 + 15 + 16 + 20 + 20 + 25}{7} = \frac{119}{7} \approx 17
\]
2. **Calculate the Absolute Deviations**:
- For $11$: |11 - 17| = 6
- For $12$: |12 - 17| = 5
- For $15$: |15 - 17| = 2
- For $16$: |16 - 17| = 1
- For $20$: |20 - 17| = 3
- For $20$: |20 - 17| = 3
- For $25$: |25 - 17| = 8
Absolute deviations: $6, 5, 2, 1, 3, 3, 8$
3. **Calculate the Mean Absolute Deviation (MAD)**:
\[
\text{MAD} = \frac{\sum \text{absolute deviations}}{n} = \frac{6 + 5 + 2 + 1 + 3 + 3 + 8}{7} = \frac{28}{7} = 4
\]
### Summary of Results
- **Range**: 14
- **Interquartile Range (IQR)**: 8
- **Mean Absolute Deviation (MAD)**: 4
### Interpretation
- The range of $14$ indicates the difference between the highest and lowest admission prices. This shows a general spread in the prices.
- The interquartile range of $8$ signifies the spread of the middle 50% of the admission prices, suggesting that the central tendency of the prices is more consistent, as the majority of prices fall within this range.
- The mean absolute deviation of $4$ implies that, on average, individual museum admission prices deviate from the mean by $4. This reflects how much variation exists in the admission prices around the average price.