Question
The owner of a bagel shop recorded the number of bagels sold each day for a week. Her results were 699, 751, 810, 842, 923, 1,061, 1,214. Describe the spread of number of bagels sold each day using the mean absolute deviation.
Answers
GPT-4o mini
To calculate the Mean Absolute Deviation (MAD) of the number of bagels sold each day, we will follow these steps:
1. **Calculate the Mean**:
First, we find the mean (average) of the bagels sold.
\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
\]
The total number of bagels sold over the week:
\[
699 + 751 + 810 + 842 + 923 + 1061 + 1214 = 5,310
\]
The number of days is 7. Therefore, the mean is:
\[
\text{Mean} = \frac{5310}{7} \approx 759.143
\]
2. **Calculate the Absolute Deviations**:
Next, we find the absolute deviation of each day from the mean:
\[
|699 - 759.143| \approx 60.143
\]
\[
|751 - 759.143| \approx 8.143
\]
\[
|810 - 759.143| \approx 50.857
\]
\[
|842 - 759.143| \approx 82.857
\]
\[
|923 - 759.143| \approx 163.857
\]
\[
|1061 - 759.143| \approx 301.857
\]
\[
|1214 - 759.143| \approx 454.857
\]
3. **Sum the Absolute Deviations**:
Now, we sum these absolute deviations.
\[
60.143 + 8.143 + 50.857 + 82.857 + 163.857 + 301.857 + 454.857 \approx 822.714
\]
4. **Calculate the Mean Absolute Deviation**:
Finally, we calculate the MAD by dividing the total absolute deviation by the number of values (days).
\[
\text{MAD} = \frac{822.714}{7} \approx 117.5306
\]
Therefore, the Mean Absolute Deviation (MAD) of the number of bagels sold each day is approximately **117.53**. This indicates the average deviation from the mean sales across the week, providing insight into the variability in daily bagel sales.
1. **Calculate the Mean**:
First, we find the mean (average) of the bagels sold.
\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
\]
The total number of bagels sold over the week:
\[
699 + 751 + 810 + 842 + 923 + 1061 + 1214 = 5,310
\]
The number of days is 7. Therefore, the mean is:
\[
\text{Mean} = \frac{5310}{7} \approx 759.143
\]
2. **Calculate the Absolute Deviations**:
Next, we find the absolute deviation of each day from the mean:
\[
|699 - 759.143| \approx 60.143
\]
\[
|751 - 759.143| \approx 8.143
\]
\[
|810 - 759.143| \approx 50.857
\]
\[
|842 - 759.143| \approx 82.857
\]
\[
|923 - 759.143| \approx 163.857
\]
\[
|1061 - 759.143| \approx 301.857
\]
\[
|1214 - 759.143| \approx 454.857
\]
3. **Sum the Absolute Deviations**:
Now, we sum these absolute deviations.
\[
60.143 + 8.143 + 50.857 + 82.857 + 163.857 + 301.857 + 454.857 \approx 822.714
\]
4. **Calculate the Mean Absolute Deviation**:
Finally, we calculate the MAD by dividing the total absolute deviation by the number of values (days).
\[
\text{MAD} = \frac{822.714}{7} \approx 117.5306
\]
Therefore, the Mean Absolute Deviation (MAD) of the number of bagels sold each day is approximately **117.53**. This indicates the average deviation from the mean sales across the week, providing insight into the variability in daily bagel sales.